Quantum Field Theory II, David Skinner CAMBRIDGE 2018

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Quantum Field Theory II
University of Cambridge Part III Mathematical Tripos
David Skinner
Department of Applied Mathematics and Theoretical Physics,
Centre for Mathematical Sciences,
Wilberforce Road,
Cambridge CB3 0WA
United Kingdom
d.b.skinner@damtp.cam.ac.uk
http://www.damtp.cam.ac.uk/people/dbs26/
Abstract: These are the lecture notes for the Advanced Quantum Field Theory course
given to students taking Part III Maths in Cambridge during Lent Term of 2018. The main
aims are to discuss Path Integrals, the Renormalization Group, Wilsonian E↵ective Field
Theory and non–Abelian Gauge Theories.
mailto:d.b.skinner@damtp.cam.ac.uk
http://www.damtp.cam.ac.uk/people/dbs26/
Contents
1 Introduction 1
1.1 Choosing a QFT 1
1.2 What do we want to compute? 4
1.2.1 The partition function 5
1.2.2 Correlation functions 6
1.2.3 Boundaries and Hilbert space 9
1.2.4 Scattering amplitudes 11
2 QFT in zero dimensions 13
2.1 Partition functions and correlation functions in d = 0 13
2.2 Free field theory 14
2.3 Perturbation theory 17
2.3.1 Feynman diagrams 20
2.4 Fermions and Grassmann variables 23
2.4.1 Supersymmetry and localization 23
2.5 E↵ective theories: a toy model 27
3 QFT in one dimension (= QM) 31
3.1 Quantum Mechanics 32
3.1.1 The partition function 34
3.1.2 Operators and correlation functions 35
3.2 The continuum limit 37
3.2.1 The path integral measure 38
3.2.2 Discretization and non–commutativity 39
3.2.3 Non–trivial measures? 41
3.3 Locality and E↵ective Quantum Mechanics 43
3.4 The worldline approach to perturbative QFT 46
4 Symmetries in Quantum Field Theory 50
4.1 Symmetries of the quantum theory 50
4.1.1 Ward identities 52
4.1.2 Currents and charges 54
4.1.3 The Ward–Takahashi identity in QED 57
5 The Renormalization Group 61
5.1 The regularized path integral 61
5.2 Integrating out degrees of freedom 62
5.2.1 Running couplings and their �-functions 63
5.2.2 Anomalous dimensions 65
5.3 Renormalization group flow 67
– i –
5.4 Calculating RG evolution 71
5.4.1 Polchinski’s equation 71
5.4.2 The local potential approximation 74
5.4.3 The Gaussian critical point 77
5.4.4 The Wilson–Fisher critical point 79
5.4.5 Zamolodchikov’s C–theorem 81
6 Quantum Field Theory in the Continuum 84
6.1 Counterterms 85
6.2 One–loop renormalization of ��4 theory 87
6.2.1 The on–shell renormalization scheme 89
6.2.2 Dimensional regularization 91
6.2.3 Renormalization of the quartic coupling 93
6.3 One–loop renormalization of QED 95
6.3.1 Vacuum polarization: loop calculation 96
6.3.2 Counterterms in QED 100
6.3.3 The �-function of QED 102
6.3.4 Physical interpretation of vacuum polarization 103
7 E↵ective Field Theory 107
7.1 Propagators in Background Fields 107
7.2 The Euler–Heisenberg Action 108
7.3 Symmetries of the E↵ective Action 108
7.3.1 Why is the sky blue? 109
7.3.2 Why does light bend in glass? 111
7.3.3 Quantum Gravity as an EFT 113
7.4 Emergent symmetries 114
8 Non–Abelian Gauge Theory: Classical Aspects 117
8.1 Principal bundles and vector bundles 117
8.1.1 Vector bundles from representations 119
8.1.2 Connections and curvature 121
8.1.3 Holonomy 124
8.2 Classical Yang–Mills theory 125
8.2.1 The Yang–Mills action 126
8.2.2 Minimal coupling 129
8.3 Charges and gauge transformations 130
9 Non-Abelian Gauge Theory: Perturbative Quantization 131
9.1 A ghost story 132
9.2 BRST cohomology 140
9.3 Perturbative renormalization of Yang–Mills theory 142
9.3.1 Feynman rules in R⇠ gauges 143
9.3.2 Yang–Mills is perturbatively renormalizable! 145
– ii –
9.3.3 The �-function and asymptotic freedom 145
9.3.4 Topological terms and the vacuum angle 147
– iii –
Acknowledgments
Nothing in these lecture notes is original. In particular, my treatment is heavily influenced
by several of the textbooks listed below, especially Vafa et al. and the excellent lecture notes
of Neitzke in the early stages, then Schwartz and Weinberg’s textbooks and Hollowood’s
lecture notes later in the course.
I am supported by the European Union under an FP7 Marie Curie Career Integration
Grant.
– iv –
Preliminaries
This course is the second course on Quantum Field Theory o↵ered in Part III of the Maths
Tripos, so I’ll feel free to assume you’ve already taken the first course in Michaelmas Term
(or else an equivalent course elsewhere). You’ll also find it helpful to know about groups
and representation theory, say at the level of the Symmetries, Fields and Particles course
last term. Last term’s General Relativity and Statistical Field Theory courses may also be
helpful, but I won’t assume you attended these.
There may be some overlap between this course and certain other Part III courses
this term. In particular, I’d expect the material here to complement the courses on The
Standard Model and on Applications of Di↵erential Geometry to Physics very well. In
turn, I’d also hope this course is useful preparation for courses on Supersymmetry and
String Theory.
Books & Other Resources
There are many (too many!) textbooks and reference books available on Quantum Field
Theory. Di↵erent ones emphasize di↵erent aspects of the theory, or applications to di↵erent
branches of physics or mathematics – indeed, QFT is such a huge subject nowadays that
it is probably impossible for a single textbook to give an encyclopedic treatment (and
absolutely impossible for a course of 24 lectures to do so). Here are some of the ones I’ve
found useful while preparing these notes; you might prefer di↵erent ones to me.
• Nair, V.P., Quantum Field Theory: A Modern Perspective, Springer (2005).
Although it isn’t so well known, this is perhaps my favourite QFT book. It begins
with a clear, concise discussion of all the standard perturbative material you’ll find
in any QFT course. However, unlike many books, it also makes clear that there’s
far more to QFT than just perturbation theory. Contains excellent discussions of
the configuration space of field theories, ambiguities in quantization, approaches to
strong coupling limits in QCD, and QFT at finite temperature.
This next list contains the stalwart QFT textbooks. You will certainly want to consult (at
least) one of these repeatedly during the course. They’ll also be very helpful for people
taking the Standard Model course.
• Peskin, M. and Schroeder, D., An Introduction to Quantum Field Theory,
Addison–Wesley (1996).
An excellent QFT textbook, containing extensive discussions of both gauge theories
and renormalization. Many examples worked through in detail, with a particular
emphasis on applications to particle physics.
• Schwartz, M., Quantum Field Theory and the Standard Model, CUP (2014).
The new kid on the block, honed during the author’s lecture courses at Harvard. I re-
ally like this book – it strikes an excellent balance between formalism and applications
(mostly to high energy physics), with fresh and clear explanations throughout.
– v –
• Srednicki, M., Quantum Field Theory, CUP (2007).
This is also an excellent, very clearly written and very pedagogical textbook, with
clearly compartmentalised chapters breaking the material up into digestible chunks.
However, our route through QFT in this course will follow a slightly di↵erent path.
• Zee, A., Quantum Field Theory in a Nutshell, 2nd edition, PUP (2010).
QFT is notorious for containing many technical details, and its easy to get lost. This
is a great book if you want to keep the big picture of what QFT is all about firmly
in sight. It will put you joyfully back on track and remind you why you wanted to
learn the subject in the first place. It’s not the best place to work through detailed
calculations, but that’s not the point.
There are also a large number of books that are more specialized. Many of these are rather
advanced, so I do not recommend you use them as a primary text. However, you may well
wish to dip into them occasionally to get a deeper perspective on topics you particularly
enjoy. This list is particularlybiased towards my (often geometric) interests:
• Banks, T. Modern Quantum Field Theory: A Concise Introduction, CUP (2008).
I particularly enjoyed its discussion of the renormalization group and e↵ective field
theories. As it says, this book is probably too concise to be a main text.
• Cardy, J., Scaling and Renormalization in Statistical Physics, CUP (1996).
A wonderful treatment of the Renormalization Group in the context in which it
was first developed: calculating critical exponents for phase transitions in statistical
systems. The presentation is extremely clear, and this book should help to balance
the ‘high energy’ perspective of many of the other textbooks.
• Coleman, S., Aspects of Symmetry, CUP (1988).
Legendary lectures from one of the most insightful masters of QFT. Contains much
material that is beyond the scope of this course, but so engagingly written that I
couldn’t resist including it here!
• Costello, K., Renormalization and E↵ective Field Theory, AMS (2011).
A pure mathematician’s view of QFT. The main aim of this book is to give a rigorous
definition of (perturbative) QFT via path integrals and Wilsonian e↵ective field the-
ory. Another major achievement is to implement this for gauge theories by combining
BV quantization with the ERG. Repays the hard work you’ll need to read it – for
serious mathematicians only.
• Deligne, P., et al., Quantum Fields and Strings: A Course for Mathematicians,
vols. 1 & 2, AMS (1999).
Aimed at professional mathematicians wanting an introduction to QFT. They thus
require considerable mathematical maturity to read, but most certainly repay the
e↵ort. Almost everything here is beyond the level of this course, but I can promise
you’re appreciation of QFT will be deepened immeasurably by reading the lectures
– vi –
of Deligne & Freed on Classical Field Theory (vol. 1), Gross on the Renormalization
Group (vol. 1), Gadwezki on CFTs (vol. 2), and especially Witten on Dynamics of
QFT (vol. 2). (I recommend you read Witten on anything.)
• Polyakov, A., Gauge Fields and Strings, Harwood Academic (1987).
A very original and very deep perspective on QFT, building a form of synthesis
of Polyakov’s approach to strongly coupled QCD. Several of the most important
developments in theoretical physics over the past couple of decades have been (directly
or indirectly) inspired by ideas in this book.
• Schweber, S., QED and the Men Who Made It: Dyson, Feynman, Schwinger and
Tomonaga, Princeton (1994).
Not a textbook, but a tale of the times in which QFT was born, and the people who
made it happen. It doesn’t aim to dazzle you with how very great these heroes were1,
but rather shows you how puzzled they were, how human their misunderstandings,
and how tenaciously they had to fight to make progress. Inspirational stu↵.
• Vafa, C., and Zaslow, E., (eds.), Mirror Symmetry, AMS (2003).
A huge book comprising chapters written by di↵erent mathematicians and physicists
with the aim of understanding Mirror Symmetry in the context of string theory.
Chapters 8 – 11 give an introduction to QFT in low dimensions from a perspective
close to the one we will start with in this course. The following chapters could well
be useful if you’re taking the String Theory Part III course.
• Weinberg, S., The Quantum Theory of Fields, vols. 1 & 2, CUP (1996).
Penetrating insight into everything it covers and packed with many detailed examples.
The perspective is always deep, but it requires strong concentration to follow a story
that sometimes plays out over several chapters. Weinberg’s thesis is that QFT is the
inevitable consequence of marrying Quantum Mechanics, Relativity and the Cluster
Decomposition Principle (that distant experiments yield uncorrelated results). In
this telling, particles play a primary role, with fields coming later; for me, that’s
backwards.
• Zinn–Justin, J. Quantum Field Theory and Critical Phenomena,
4th edition, OUP (2002).
Contains a very insightful discussion of the Renormalization Group and also a lot
of information on Gauge Theories. Most of its examples are drawn from either
Statistical or Condensed Matter Physics.
Textbooks are expensive. Fortunately, there are lots of excellent resources available freely
online. I like these:
• Dijkgraaf, R., Les Houches Lectures on Fields, Strings and Duality,
http://arXiv.org/pdf/hep-th/9703136.pdf
1I should say ‘are’; even now in 2017, Freeman Dyson still works at the IAS almost every day.
– vii –
http://arXiv.org/pdf/hep-th/9703136.pdf
An modern perspective on what QFT is all about, and its relation to string theory.
For the most part, the emphasis is on more mathematical topics (e.g. TFT, dualities)
than we will cover in the lectures, but the first few sections are good for orientation.
• Hollowood, T., Six Lectures on QFT, RG and SUSY,
http://arxiv.org/pdf/0909.0859v1.pdf
An excellent mini–series of lectures on QFT, given at a summer school aimed at end–
of–first–year graduate students from around the UK. They put renormalization and
Wilsonian E↵ective Theories centre stage. While the final two lectures on SUSY go
beyond this course, I found the first three very helpful when preparing the current
notes. We’ll follow parts of these notes closely.
• Neitzke, A., Applications of Quantum Field Theory to Geometry,
https://www.ma.utexas.edu/users/neitzke/teaching/392C-applied-qft/
Lectures aimed at introducing mathematicians to Quantum Field Theory techniques
that are used in computing Seiberg–Witten invariants. I very much like the per-
spective of these lectures, and we’ll Neitzke’s notes closely for the first part of the
course.
• Osborn, H., Advanced Quantum Field Theory,
http://www.damtp.cam.ac.uk/user/ho/Notes.pdf
The lecture notes for a previous incarnation of this course, delivered by Prof. Hugh
Osborn. They cover similar material to the current ones, but from a rather di↵erent
perspective. If you don’t like the way I’m doing things, or for extra practice, take a
look here!
• Polchinski, J., Renormalization and E↵ective Lagrangians,
http://www.sciencedirect.com/science/article/pii/0550321384902876
• Polchinksi, J., Dualities of Fields and Strings,
http://arxiv.org/abs/1412.5704
The first paper gives a very clear description of the ‘exact renormalization group’
and its application to scalar field theory. The second is a recent survey of the idea of
‘duality’ in QFT and beyond. We’ll explore this if we get time.
• Segal, G., Quantum Field Theory lectures,
YouTube lectures
Recorded lectures aiming at an axiomatization of QFT by one of the deepest thinkers
around. I particularly recommend the lectures “What is Quantum Field Theory?”
from Austin, TX, and “Three Roles of Quantum Field Theory” from Bonn (though
the blackboards are atrocious!).
• Tong, D., Quantum Field Theory,
http://www.damtp.cam.ac.uk/user/tong/qft.html
The lecture notes from the Michaelmas QFT course in Part III. If you feel you’re
– viii –
http://arxiv.org/pdf/0909.0859v1.pdf
https://www.ma.utexas.edu/users/neitzke/teaching/392C-applied-qft/
http://www.damtp.cam.ac.uk/user/ho/Notes.pdf
http://www.sciencedirect.com/science/article/pii/0550321384902876
http://arxiv.org/abs/1412.5704
https://www.youtube.com/results?search_query=%22graeme+segal%22+%22quantum+field+theory%22
http://www.damtp.cam.ac.uk/user/tong/qft.html
missing some background from last term, this is an excellent place to look. There
are also some video lectures from when the course was given at Perimeter Institute.
• Weinberg, S., What Is Quantum Field Theory, and What Did We Think It Is?,
http://arXiv.org/pdf/hep-th/9702027.pdf
• Weinberg, S., E↵ective Field Theory, Past and Future,
http://arXiv.org/pdf/0908.1964.pdf
These two papers provide a fascinating account of the origins of e↵ective field the-
ories in current algebras for soft pion physics, and how the Wilsonian picture of
Renormalization gradually changed our whole perspective of what QFT is about.
• Wilson, K., and Kogut, J. TheRenormalization Group and the ✏-Expansion,
Phys. Rep. 12 2 (1974),
http://www.sciencedirect.com/science/article/pii/0370157374900234
One of the first, and still one of the best, introductions to the renormalization group
as it is understood today. Written by somone who changed the way we think about
QFT. Contains lots of examples from both statistical physics and field theory.
That’s a huge list, and only a real expert in QFT would have mastered everything on it.
I provide it here so you can pick and choose to go into more depth on the topics you find
most interesting, and in the hope that you can fill in any background you find you are
missing.
– ix –
http://pirsa.org/index.php?p=speaker&name=David_Tong
http://arXiv.org/pdf/hep-th/9702027.pdf
http://arXiv.org/pdf/0908.1964.pdf
http://www.sciencedirect.com/science/article/pii/0370157374900234
1 Introduction
Quantum Field Theory is, to begin with, exactly what it says it is: the quantum version of
a field theory. But this simple statement hardly does justice to what is the most profound
description of Nature we currently possess. As well as being the basic theoretical framework
for describing elementary particles and their interactions (excluding gravity), QFT also
plays a major role in areas of physics and mathematics as diverse as string theory, condensed
matter physics, topology, geometry, combinatorics, astrophysics and cosmology. It’s also
extremely closely related to statistical field theory, probability and from there even to
(quasi–)stochastic systems such as finance.
1.1 Choosing a QFT
To build a QFT, we start by picking the space on which it lives. Usually, this will be some
smooth, Riemannian (or pseudo–Riemannian) manifold (M, g) of dimension dim(M) = d.
For example, for most applications to particle physics, we’d choose (M, g) = (R4, ⌘) where
⌘ is the Minkowski metric. However, this is far from being the only interesting choice.
For many applications in condensed matter, one sets either (M, g) = (R3, �) with � the
flat Euclidean metric, or perhaps M = U ⇢ R3 to study field theory living in a sample of
material that occupies some region U . As a further example, the worldsheet description of
string theory involves a QFT living on a Riemann surface (⌃, [g]) where only the conformal
class
[g] = {g 2 Met(⌃) with g ⇠ e2�g for � : ⌃ ! R}
of the metric needs to be specified, while applications of QFT to topological problems such
as knot invariants make use of a certain gauge theory (known as Chern–Simons theory)
living on an arbitrary orientable three–manifold M with no metric at all. Whatever choice
we make, in QFT the metric g is regarded as fixed – studying what happens when the
metric itself has quantum fluctuations requires quantum gravity.
Having decided which universe we live in, our next choice is to pick which objects we
wish to study. That is, we must choose the fields. The simplest choice is a scalar field,
which is just a function on M . It’ll often be useful to think of this a map
� : M ! R, C, . . . ,
according to whether the scalar is real– or complex–valued. More generally, � could describe
a map
� : M ! N
from our space to some other (Riemannian) manifold (N, G), known as the target space.
For example, we’ll see that we can think of ordinary non–relativistic Quantum Mechanics
in terms of a d = 1 QFT living on an interval I = [0, 1] known as the worldline, where the
fields describe a map � : I ! R3. In particle physics, the pion field ⇡(x) describes a map
M ! G/H where M is our Universe and G and H are Lie groups. (In the specific case of
pions in the Standard Model, it turns out that G/H = (SU(2)⇥SU(2))/SU(2).) In string
– 1 –
1
2 3
4 1
2 3
41
2 3
4
�
⌃
Figure 1: String Theory involves a QFT describing maps from a Riemann surface ⌃ to a
Calabi–Yau manifold.
theory, some of the worldsheet fields are scalars describing a map � : ⌃ ! N embedding
the worldsheet in a certain special type of Riemannian manifold N called a Calabi–Yau
manifold.
There are many further options. In a gauge theory, as we’ll see in chapter 8, the basic
field is a connection r on a principal G-bundle P ! M . We could also choose to include
charged matter, described mathematically in terms of sections of vector bundles E ! M
associated to P ! M by a choice of representation. For example, scalar QED involves a
photon Aµ and a scalar �, defined up to the gauge transformations
Aµ ⇠ Aµ + @µ� � ⇠ ei��
This is just the local description of a connection on a principal U(1) bundle, together with
a section of a rank one complex vector bundle E ! M whose fibres are equipped with a
Hermitian metric. As you learned if you took the General Relativity course, Riemannian
manifolds naturally come along with various bundles, such as the tangent and cotangent
bundles TM and T ⇤M . Under mild topological conditions, we might also be able to define
spin bundles over M . In physics, we’d think of sections of these bundles as being fields that
transform non-trivially under Lorentz transformations; i.e. they carry non–zero ‘spin’ and
are described (at least locally) by functions such as V µ, ↵̇, B[µ⌫] and �
↵̇
µ with various types
of vector and/or spinor indices. I don’t want to get into any details in the introduction
— we’ll explore these objects and what the mathematical words mean in detail as we go
along. My only point here is there’s lots of choice in what type fields we might like to
include in our QFT, and that all the most common choices (certainly all the ones I expect
you to have heard of so far, and all the ones we’ll meet in this course) are very natural
geometrical objects.
Whatever fields we pick, I’ll let C denote the space of field configurations on M .
That is, every point � 2 C corresponds to a configuration of the field – a picture of what
(every component of) the field looks like across the whole universe M . Since we allow
our fields to have arbitrarily small bumps and ripples, C is typically an infinite dimensional
function space. Trying to understand the geometry and topology of this infinite dimensional
– 2 –
space of fields, and then trying to do something useful with it is fundamentally what makes
QFT di�cult, but it’s also what makes it interesting and powerful.
The next ingredient we need is to specify the action for our theory. This is a function
S : C ! R (1.1)
on the space of fields. In other words, given a field configuration, the action produces
a real number. We often write S[�] for this number, as opposed to S(�), and say the
action is a functional. The word is just to remind us that the domain C of S is itself and
infinite–dimensional function space. The critical set2
Crit C(S) = {� 2 C | �S[�] = 0} (1.2)
correspond to fields that solve the classical field equations, the Euler–Lagrange equations.
In the simplest circumstances, these critical points are isolated.
When setting up our QFT, we often assume that S[�] is local, meaning that it can be
written as
S[�] =
Z
M
ddx
p
g L(�(x), @�(x), . . .) (1.3)
where the Lagrangian3 L depends on the value of � and finitely many derivatives at just a
single point in M . As a consequence, the classical field equations become nonlinear pdes
of an order determined by the number of derivatives of � appearing in L.
You’ve doubtless been writing down local actions for so long – motivated by either
classical mechanics or classical field theories such as electromagnetism – that you now do
it without thinking. However, it’s worth pointing out that, purely from the point of view
of functions on C, locality on M is actually a very strong restriction. Even a monomial
function on C generically looks like
Z
M⌦n
ddx1 d
dx2 · · · ddxn ⇤(x1, x2, . . . , xn)�(x1)�(x2) · · ·�(xn)
involving the integral of the field at many di↵erent points, with some choice of function
⇤ : M⌦n ! R. (You can think of this as an infinite dimensional analogue of a monomial
X
ijk···l
⇤ijk...lz
izjzk · · · zl
2Here, � is properly viewedas the exterior derivative on C and obeys �2 = 0. Thus � =
R
M
��(x) �/��(x)
where ��(x) is a one-form on C and the derivative �/��(x) on C acts e.g. as
�
��(x)
�(y) = �d(x� y) , �
��(x)
Z
M
ddy �(y)2 = 2�(x) ,
�
��(x)
Z
M
ddy @µ�@µ� = 2
Z
M
ddy @µ�d(x� y) @µ� = �22�(x)
(the last example holding in the case that there is no boundary term).
3What I’m calling the Lagrangian here is really the Lagrangian density, with the Lagrangian itself being
the integral L over a Cauchy surface in M . The abuse of terminology is standard.
– 3 –
in finitely many variables zi, with the function ⇤(x1, x2, . . . , xn) playing the role of the
‘coe�cients’.) Locality means that we restrict to ‘functions’ ⇤ of the form
⇤(x1, x2, . . . , xn) = �(x) @
(p1)�d(x1 � x2) @(p2)�d(x2 � x3) · · · @(pn�1)�d(xn�1 � xn) (1.4)
that are supported on the main diagonal M ⇢ M⌦n, with finitely many derivatives allowed
to act on the �–functions. Integrating by parts if necessary, these derivatives can be made
to act on the fields, leaving us with an expression of the general form
Z
M
ddx �(x) @(q1)�(x) @(q2)�(x) · · · @(qn)�(x)
of a monomial of degree n in the fields, acted on again by some derivatives, with all
fields and derivatives evaluated at the same x 2 M . The only remnant of the original
⇤ : M⌦n ! R is the function � : M ! R. If this monomial is higher than quadratic
in the fields, it leads to a non–linear term in the classical field equations, meaning we no
longer have superposition of solutions (i.e. the space of solutions to the Euler–Lagrange
equations will no longer be expected to be a vector space). Physically, we interpret this
as an interaction, either between several di↵erent fields or between a field and itself. In
many cases, we restrict further and choose the functions � to be constant, �(x) = � with
the constant � known as a coupling constant.
If we were to allow multi-local terms in our action, the resulting classical field equations
would be integro–di↵erential equations, so the behaviour of our field at one point x 2 M
would depend what the field configuration looks like across all of M . This ‘action at a
distance’ is usually thought to be unphysical, at least in classical physics. However, we’ll
see later that QFT forces us to consider certain non-local terms even if we try to rule them
out when setting up the theory.
1.2 What do we want to compute?
In this course, the main tools we’ll use to study QFT are path integrals. Heuristically,
these are integrals such as Z
C
[D�] exp
✓
�1~S[�]
◆
(1.5)
that are taken over the infinite dimensional space of fields C, with some sort of measure
[D�] e�S[�]/~ that weights the contribution of each field configuration � 2 C by e�S/~. The
vague idea of this measure is for the exponential to suppress field configurations that are
‘wild’, so that for example we might optimistically hope that configurations in which �
jumps around rapidly between very di↵erent values (perhaps even being discontinuous, or
worse) play a ‘negligible’ role.
However, it’s very far from clear that this hope will be realised. Even if you have only
an anecdotal knowledge of functional analysis, you likely know that there are vastly more
discontinuous functions than continuous ones, vastly more continuous than continuously
di↵erentiable, vastly more functions that are Ck than there are functions that are Ck+1,
– 4 –
and vastly more smooth functions than analytic ones4. Going in the other direction, there
are vastly more distributions than even discontinuous functions. Despite the best e↵orts
of the suppression factor e�S/~, the contribution of the enormous variety of ‘wild’ field
configurations can easily overwhelm the much smaller set of ‘nice’ (e.g. smooth) fields.
This idea is familiar in statistical mechanics, where the contribution of any given
configuration (e.g. the spin state of electrons located at each site of a lattice) is weighted
by e��E(�) with � the inverse temperature of the system. However, there are typically
many more ‘disordered’ states (e.g. with random alignments of neighbouring spins) than
ordered ones (e.g. all spins aligned) and, depending on the details of the model (e.g.
types of interactions allowed between nearby / distant spins, and the dimensionality and
connectivity of the lattice on which they sit) there can be a complicated structure of phase
transitions as parameters such as the temperature ��1 is varied. In the poetic words of
Bryce de Witt, the balance between the tendency of the exponential factor to suppress
rapidly varying configurations and the fact that there are simply many more of them is
“the eternal struggle between energy and entropy”.
In a field theory, the fact that we’re dealing with infinite–dimensional spaces makes
path integrals such as (1.5) even more delicate to study. Since we’re integrating over an
infinite dimensional space, it’s far from obvious that we can get any sort of finite answer out
of a path integral at all. Indeed, much of the hard work we need to do in this course is about
understanding how to achieve this (even at low orders in perturbation theory). However,
living on the edge is exciting, and when a path integral exists it has a rich character that
is often goes far beyond what one can see in the classical action.
Incidentally, I’ve chosen the argument of the exponential in (1.5) to be �S[�]/~, as ap-
propriate when (M, g) is Riemannian (such as Euclidean space). For a pseudo–Riemannian
manifold (such as Minkowski space) we’d instead use iS[�]/~. It should be clear already
that any di�culties we have in making sense of the doubtfully convergent integral (1.5) are
only going to be worse for a doubtfully–conditionally convergent integral. For this reason,
we’ll mostly stick to Riemannian signature spaces in this course5.
1.2.1 The partition function
Let’s now take a look at the path integral in a little more detail, though still completely
heuristically. If M is closed and compact (such as a sphere Sd or torus T d = S1 ⇥ S1 ⇥
· · · ⇥ S1), the most important object to compute in any QFT is the partition function
Z(M,g)(�, · · · ) =
Z
C
[D�] exp
✓
�S[�]~
◆
. (1.6)
4Recall that a function on M is in Ck(M) if, at every p 2 M , the function and its first k derivatives
exist and are each continuous. A continuous function is said to be in C0(M). A function is in C1(M)
(smooth) if it is in Ck(M) for all k. It is C! (analytic) if its Taylor expansion around any point converges
to the function itself.
5You’ll find that most QFT textbooks do too: even though they may claim to start out writing things
in Minkowksi space, before any real calculation is done they will ‘Wick rotate’ to Euclidean space. Working
in Euclidean space is also essentially the same thing as studying Statistical Field Theory, except here we’ll
take d = dim(M) to be the total space–time dimension.
– 5 –
As we’ve indicated here, the partition function depends on all the choices we made in
setting up our theory, such as the space (M, g) on which the theory lives and the values
of the couplings �, as well as of course on ~. Note however that Z does not depend on
the fields! These are just dummy variables that we’ve integrated out in computing the
partition function.
1.2.2 Correlation functions
After the partition function, the most important objects we wish to compute in any QFT
are correlation functions. These are path integrals with further insertions, of the general
form Z
C
[D�] exp
✓
�S[�]~
◆ nY
i=1
Oi[�] (1.7)
where the insertions Oi are again functions on C. We sometimes normalise the correlation
functions by the partition function, writing
*
nY
i=1
Oi[�]
+
=
1
Z
Z
C
[D�] exp
✓
�S[�]~
◆ nY
i=1
Oi[�] . (1.8)
The idea of this normalisation (as we’ll see in detail later) is both to ensure that h1i = 1 and
to separate out the e↵ect of inserting the operator O into the path integral from e↵ects that
are there in the basic partition function already. Mathematically,normalised correlation
functions compute various moments of the probability distribution [D�] e�S/~/Z. From
the point of view of physics, we choose the functions we insert to correspond to some
quantity of physical interest that we wish to measure; perhaps the energy of the quantum
field in some region, or the total angular momentum carried by some electrons, or perhaps
temperature fluctuations in the CMB at di↵erent angles on the night sky.
In the QFT context, we often call these extra functions operator insertions in the
path integral, for reasons that will become apparent. The most common examples of
operator insertions are local operators that depend on the value of the field (and perhaps
finitely many derivatives) at a single point in M . Examples include
Oi(xi) = �4(xi) , O(xj) = �3 @µ�@µ�(x) , O(xk) = e�(xk)
and very many more. It’s also perfectly possible to have operator insertions such as
Z
M
ddx (@µ�@µ�)
2
that depend on the value of the field over all of M . For example, if you take the Part III
String Theory course, you’ll often compute correlation functions of a mixture of operator
insertions, some of which are inserted at points x 2 M = ⌃, and others that are integrated
over all of the worldsheet ⌃. As you’ll learn, these correlation functions correspond to
dynamical processes in the target space of the string. We can also have operators that
depend on the value of the field along a curve � ⇢ M , or some other subspace K ⇢ M .
Indeed, the operator
W�[A] = tr P exp
✓
�
I
A
◆
,
– 6 –
that depends on the value of gauge field A = Aµ dxµ along a curve � is one of the most
fundamental operators present in any gauge theory, known to physicists as a Wilson loop
and to mathematicians as the trace of the holonomy of the connection. Note again that
although the operator insertions depend on the values of the fields, the correlation functions
themselves do not. Rather, our correlators are functions
F(M,g)(x1, . . . , xn;�, · · · ) =
*
nY
i=1
Oi(xi)
+
that depend on all the same data as the partition function Z, together with any extra
choices (such as the choice of points xi 2 M or subspaces �i, or Ki and general form of
the operator) that were made in choosing which correlator to compute.6
Because local operators also appear in the action S[�], correlation functions are closely
related to the partition function. Indeed, if the action includes a term
O = �
Z
M
�4(x) ddx
with coupling constant �, then di↵erentiating formally7 we get
� ~Z
@
@�
Z = � ~Z
@
@�
✓Z
C
[D�] e�S[�]/~
◆
=
1
Z
Z
C
✓
[D�] e�S[�]/~
Z
M
�4 ddx
◆
= hOi ,
(1.9)
so that the normalised correlator is (�~ times) the derivative of ln Z with respect to the
coupling. Thus, knowing Z as a function of the all couplings in the action is equivalent to
knowing all the correlators of operators appearing in the action.
The operators that appear in the action are integrated over all of M . It’s often con-
venient to extend the idea above so as to obtain correlators of local operators O(x) that
depend on the value of � (and perhaps finitely many derivatives) just at one point x 2 M .
To do this, we include source terms such as
S[�] ! S[�] +
Z
M
ddx Ji(x)Oi(x)
in the action. The source Ji(x) is, like the field �, a function on M . Really, this is just
another case of the choices we made in picking our (local) action, allowing the coupling
6Skipping very far ahead of our story, if our real interest is in objects such as these functions, which
are independent of any fields, we might hope to avoid our troubles in actually defining the path integral
by instead trying to give some other, perhaps axiomatic, way to compute and manipulate such functions.
This turns out to be successful in various special cases such as topological QFT and minimal models (a
special class of conformal field theory in two dimensions), but despite much e↵ort no one has yet come up
with a set of axioms that are rich and flexible enough to allow the great variety of phenomena we see in
QFTs, whilst still being useful enough that we can actually calculate with them. The path integral is the
best description we have.
7That is, without worrying about whether the di↵erentiation and integration commute. Since we haven’t
properly defined our path integral and don’t even know yet whether it actually exists, there’s no point in
worrying about this seriously at this stage anyway.
– 7 –
‘constant’ � ! �(x) to still vary over M , but the name ‘source’ and use of the letter J(x)
is conventional.
We do not integrate over J in performing the path integral, so the partition function
itself becomes a functional
Z ! Z(M,g)[Ji]
depending on the choice of functions Ji in addition to the other data. Varying this partition
function wrt the value of the source at some point xi 2 M , we obtain formally
�~ �
�Ji(xi)
Z[Ji] =
Z
C
✓
[D�] e�(S[�]+
R
JjOj)/~ �
�Ji(xi)
Z
M
ddy Ji(y)Oi(y)
◆
=
Z
C
⇣
[D�] e�(S[�]+
R
JjOj)/~ Oi(xi)
⌘
,
(1.10)
and thus
hO1(x1) O2(x2) · · · On(xn)i =
(�~)n
Z
�nZ[J ]
�J1(x1) �J2(x2) · · · �Jn(xn)
����
J=0
. (1.11)
Probably the most common use of this formula is when the sources couple to single
powers of the field, such as
S[�] ! S[�] +
Z
M
ddx J(x)�(x) (1.12)
for just one field. Computing Z[J ] here is equivalent to knowing all the correlation functions
h�(x1)�(x2) · · · �(xn)i =
(�~)n
Z
�nZ[J ]
�J(x1) �J(x2) · · · �J(xn)
����
J=0
. (1.13)
of the field itself. However, there’s no reason we can’t choose the source to couple to a
composite operator – i.e. a non–linear function of the fields. For example, since we
chose a Riemannian metric g to define our QFT, we could vary the partition function with
respect to the value of this metric. You should recall from last term’s QFT lectures (or
any course on classical field theory) that the stress tensor Tµ⌫(x) is defined by
�gS[�] =
1
2
Z
M
Tµ⌫(x) �g
µ⌫(x)
p
g ddx (1.14)
in terms of the variation of the (matter part of the) action wrt the metric. The stress
tensor is indeed a non-linear function of the fields and their derivatives. Varying inside the
path integral we obtain
� ~ �g(ln Z(M,g)) =
1
2
Z
M
hTµ⌫(x)i �gµ⌫(x)
p
g ddx (1.15)
and hence
� 2~p
g(x)
� ln Z(M,g)
�gµ⌫(x)
= hTµ⌫(x)i(M,g) (1.16)
where on the rhs we emphasize that the correlation function is computed using the original
metric g.
Relations such as these show the close connection between correlation functions and
the partition function. We see that correlators probe the response of the partition function
to a change in the background structures we chose in setting up the theory.
– 8 –
1.2.3 Boundaries and Hilbert space
If M has boundaries, say @M = [iBi, then to specify the path integral we must choose
some boundary conditions for the fields on each component of @M . We’ll see below that
on each boundary component Bi, the possible configurations of the field naturally form a
Hilbert space Hi. Thus, on a manifold with boundary the path integral really defines a
map
⌦i Hi ! C . (1.17)
To compute this object, the idea is that once we decide what our fields look like on each Bi
(in other words once we pick a state in each Hi) we obtain a complex number by performing
the path integral Z
�|Bi= 'i
D� e�S[�]/~ (1.18)
over those fields on M that agree with our chosen profiles 'i on each boundary component.
As a very important special case, suppose M = N ⇥ I, where N is some d � 1 dimensional
manifold and I is just an interval of length T with respect to the metric g on M . In this
case the path integral gives us a map8
U(T ) : H ! H (1.19)
from the Hilbert space associated to the incoming boundary of M to that associated to the
outgoing boundary, where
h'1|U(T )|'0i =
�|N⇥{T}='1Z
�|N⇥{0}= '0
D� e�S[�]/~ . (1.20)
evaluates the map acting on |'0i 2 H and ending on |'1i 2 H.
The fact that Hilbert spaces are associated to boundaries of M is completely natural
– we’ll see that it’s exactly what happens in the path integralapproach to Quantum
Mechanics as well as QFT9. An important hint of this can already be seen in classical
mechanics. In the case that @M = [iBi, varying the classical action leads to10
�S[�] = (bulk eom) ��+
X
i
Z
Bi
nµi
�L
�(@µ�)
��
p
g dd�1x (1.21)
where (unlike usual) I haven’t assumed the variation obeys ��|@M = 0. We define the call
the field momentum ⇡ conjugate to � along Bi as the variation
⇡ =
p
g nµi �L/�(@
µ�) (1.22)
8In fact, in Minkowskian signature, this map is unitary. Unitarity is di�cult to see from the path
integral perspective and is why you spent time studying canonical quantization last term.
9It’s also the starting point for Atiyah’s axiomatic approach to topological QFTs, and to Segal’s approach
to d = 2 CFTs.
10In the boundary term, nµ
i
is the outward–pointing normal to the boundary component Bi; ‘normal’
means wrt g.
– 9 –
of the Lagrangian. The standard example you’re probably most used to is when @M is
a constant time slice of flat Minkowski space, and
p
g nµ�L/�(@µ�) = �L/��̇, but the
statement holds more generally.
Suppose M = N ⇥I so that @M has just two components. Like any exterior derivative,
the exterior derivative � on the space of fields obeys �2 = 0. Thus, if the classical equations
of motion are satisfied,
0 = �2S[�]
��
eom
=
Z
N⇥{T}
�⇡ ^ �� dd�1x �
Z
N⇥{0}
�⇡ ^ �� dd�1x (1.23)
showing that the quantity
⌦ =
Z
N
�⇡ ^ �� dd�1x (1.24)
is conserved under evolution via the classical equations. (The minus sign arises because the
natural orientation of nµ points into M at one end, and out at the other.) ⌦ is a 2-form on
the space of boundary field configurations C[N ] and is obviously closed. It is a symplectic
form on the space of fields, and the fact that it is conserved is the usual fact that
in classical field theory (as in classical mechanics) time evolution is symplectic. It is this
symplectic structure on the space of fields that we wish to quantise in QFT, exactly as you
quantised the symplectic structure (R2n,!) with ! = dpi ^ dxi in quantum mechanics.
Note that (for eoms that are 2nd order), specifying boundary values of � and ⇡ deter-
mines a unique solution to the classical field equations, so the space of boundary values
(�|N ,⇡) is isomorphic to the space of classical solutions (at least in a small neighbour-
hood of N). This is why, when studying canonical quantisation on (R4, ⌘) last term, you
expanded fields in terms of modes
�(x) =
Z
d3p
(2⇡)3/2
1p
2E
h
eipµx
µ
a(p) + e�ipµx
µ
a†(p)
i
(1.25)
that satisfied the equations of motion. In fact, since you were working perturbatively, fields
satisfied the free equations of motion, meaning in the relativistic context that their energy
was fixed in terms of their mass and momentum by E =
p
p2 + m2. It’s also why, when
you introduced commutation relations
[�(x),�(y)] = 0 = [⇡(x),⇡(y)]
[�(x),⇡(y)] = i �3(x � y)
(1.26)
for the fields and their momenta, these were defined only when the fields were evaluated
at equal time; the symplectic structure ⌦ whose form they reflect is only defined on a
co-dimension 1 slice of M (such as a boundary).
States in which the fields take definite values on the boundary are the analogue of
position eigenstates in Quantum Mechanics. Just as |xi represents a quantum mechanical
state in which the particle is definitely located at x, so |'ii represents a state in quantum
field theory in which the field on the boundary component Bi definitely takes some profile
'i. In quantum mechanics we can have more general states, written in Dirac notation as
| i =
Z
dnx |xihx| i , (1.27)
– 10 –
where (x) = hx| i 2 L2(Rn, dnx) is a wavefunction. So too in QFT we can have more
general states
| i =
Z
C[B]
[d'] |'ih'| i (1.28)
where the integral is taken over the space C[B] of all possible boundary field configurations,
and ['] = h'| i is, heuristically, a wavefunction on this space of fields.
Again, you saw this already in last term’s course when studying canonical quantisation
of a field theory. There, rather than general functions [�] you studied polynomials on C[N ],
with a monomial Z
N⌦n
 (x1, . . . ,xn)�(x1) · · · �(xn)
of degree n in the fields being interpreted as an n-particle state. Via (1.25), in canoni-
cal quantisation the fields themselves were written in terms of creation and annihilation
operators, whilst the ‘co-e�cients’ of (x1, . . . ,xn) this monomial were interpreted as the
wavefunction of an n-particle quantum mechanical state. Indeed, if V is the Hilbert space
associated to a single particle, then you decomposed the Hilbert space H of QFT as11
H = C � V � Sym2V � Sym3V � · · ·
=
1M
n=0
SymnV .
(1.29)
where C represents the vacuum state, V the one–particle state, Sym2V the two–particle
state, and so on. This is known as the Fock basis of H.
Restricting to polynomials is somewhat like expanding a general state of the quantum
harmonic oscillator in Hermite polynomials, which are square–integrable wrt to a Gaussian
measure (i.e. they form a basis of the Hilbert space L2(R, e�x2/2 dx)). The di�culties
of defining what we mean by the infinite–dimensional path integral are reflected in the
canonical quantisation approach to QFT in the di�culties of defining what is meant by the
‘Hilbert space’ L2(C[B], dµ) for functions on the infinite–dimensional space of boundary
field configurations C[B].
1.2.4 Scattering amplitudes
If M is non–compact then there may be a region that is asymptotically far away in the
metric g, such as the region kxk ! 1 in Rd, or the asymptotic past and future in Minkowski
space. On a non–compact manifold case, to define the path integral we have to specify
asymptotic values of the fields, and the result of the path integral then depends on our
choice of these asymptotic values. The simplest choice is just to ask that � ! 0 in this
asymptotic region, and we use this to define the partition function Z on a non–compact
space. Another standard example is the case of Minkowski space, where we choose initial
11Sym means symmetric power, e.g. a⌦b+b⌦a 2 Sy2V ⇢ V ⌦V . I’m assuming the field is bosonic so that
the wavefunction of identical particles is symmetrized; multi-particle fermion states involve antisymmetric
powers of V .
– 11 –
and field profiles
� ! �i as t ! �1
� ! �f as t ! +1
in the asymptotic past and future, and obtain a path integral written as
h�f |�ii =
1
Z
Z
C(�i,�f)
[D�] eiS[�]/~ (1.30)
taken over the space of fields that approach these asymptotic profiles, normalised by the
partition function computed again using the ‘trivial’ profiles. This is known as the scat-
tering amplitude, and represents the quantum amplitude for a state that initially looks
like the field is �i to evolve throughout space–time and emerge looking like �f .
Of course, even on a non–compact manifold we can still define correlation functions,
where we assume � ! 0 in the asymptotic region. More generally still form factors,
which are simply correlation functions taken in the presence of non–trivial asymptotic con-
ditions on the fields (and thus are a sort of mixture of scattering amplitudes and correlation
functions). Remarkably, as we’ll learn later in the course, scattering amplitudes are them-
selves related to correlation functions of the form h�(x1)�(x2) · · · �(xn)i through the LSZ
theorem. This is a very useful theorem, as it’s typically easier to understand correlation
functions than to work with boundary conditions on the path integral.
Naively, you might think it’s easier to work with a QFT on Rd than on a compact
space M , particularly if we just require � ! 0 as kxk ! 1. Whilst there’s some truth in
this (and we’ll certainly mostly just consider (M, g) = (Rd, �) in this course), you should
be aware that putting QFT on a non–compact space introduces new di�culties in defining
the path integral, in part associated with infra–red divergences. The issue is somewhat like
an infinite–dimensional analogue of the fact that convergence of Fourier integralsis even
more subtle than convergence of Fourier series.
– 12 –
2 QFT in zero dimensions
We’ll embark on our journey from the simplest possible starting point: we’ll study QFT
in a space–time with zero dimensions. That’s a very drastic simplification, and much
of the richness of QFT will be absent here. Indeed, I expect many of the ideas in this
chapter will be things you’ve met (long) before, although perhaps in a di↵erent context.
Still, you shouldn’t sneer. We’ll see that even this simple case contains baby versions of
ideas we’ll study more generally later in the course, and it will provide us with a safe
playground in which to check we understand what’s going on. Furthermore, it has been
seriously conjectured that full, non-perturbative string theory is itself a zero–dimensional
QFT (though admittedly with infinitely many fields).
2.1 Partition functions and correlation functions in d = 0
If our space–time M is zero–dimensional and connected, then it must be just a single point:
M = {pt} . (2.1)
In zero dimensions, there are no lengths, so there is no notion of a metric. Similarly, the
Lorentz group is trivial, hence all its representations are trivial. In other words, all fields
must be scalars: there is no notion of the ‘spin’ of a field, simply because there is no notion
of a Lorentz transformation.
In the simplest case, a ‘field’ on M is a map � : {pt} ! R, or in other words just a
real variable. The space C of all field configurations is also easy to describe: it’s again just
R, because our entire universe M is just one point, so we completely specify what the field
looks like by giving its value at this one point.
Now let’s choose our action. In zero dimensions, there are no space–time directions
along which we could di↵erentiate our ‘field’, so there can be no kinetic terms. Thus, the
action is just a function S(�) of this one real variable. All that really matters is that S is
chosen so that the partition function (2.2) converges, but we’ll typically take S(�) to be a
polynomial (with highest term of even degree), such as
S(�) =
m2
2
�2 or perhaps S(�) =
m2
2
�2 +
�
4!
�4 .
The coupling constants are just the coe�cients of the various powers of � in the action.
The coe�cients of �p with p = 0, 1, 2 have a slightly special status, and are known as the
vacuum energy, the tadpole and the mass of the field, respectively, although they are
also just coupling constants from our general point of view.
Because C ⇠= R, the path integral measure D� becomes just the standard (Lebesgue)
measure d� on R and the partition function
Z =
Z
R
d� e�S(�)/~ , (2.2)
is just a standard integral over the real line. Similarly, correlation functions are
hfi := 1Z
Z
R
d� f(�) e�S(�)/~ (2.3)
– 13 –
http://arXiv.org/abs/hep-th/9610043
http://arXiv.org/abs/hep-th/9610043
where we’ve inserted some other function f(�) into the basic integral. We’ll assume that
f is su�ciently well-behaved that the integral (2.3) still exists. In particular, f should
not grow so rapidly as |�| ! 1 as to overcome the decay of e�S(�)/~. In practice, we’ll
restrict ourselves to the case that f is just a polynomial. In this dim(M) = 0 case, so long
as the action S(�) is real, e�S/~ � 0 so we really can think of e�S/~/Z as a probability
density on the space of fields, with the factor of 1/Z ensuring that the probability measure
is normalized. The correlation function (2.3) is just the expectation value hfi of f(�)
averaged over the space of fields with this measure.
As before, what we get for Z depends on which action we picked, so the partition
function depends on the values of the coupling constants
Z = Z(m2,�, · · · ) . (2.4)
Correlation functions depend on the coe�cients of the polynomial f(�) as well as the
couplings in the action. Again, we can think of the correlator as probing the response
of the partition function to an infinitesimal change in the couplings in the action. For
example, in the simplest case that f(�) = �p is monomial, we have formally
1
p!
h�pi = � ~Z
@
@�p
Z(m2,�i)
����
⇤
(2.5)
where �p is the coupling to �p/p! in the general action, and ⇤ is the point in theory space
where the couplings are set to their values in the specific action that appears in (2.3).
Finally, as a piece of notation, I’ll often write Z0 for the partition function in the free
theory, where the couplings of all but the term quadratic in the field(s) are set to zero.
2.2 Free field theory
The simplest QFTs are free, meaning that the action is (at most) quadratic in the fields.
As an example, suppose we have n fields �a with a = 1, . . . , n, thought of as a map
� : {pt} ! Rn. We choose the action to be the quadratic function
S(�) =
1
2
M(�,�) =
1
2
Mab �
a�b , (2.6)
where M : Rn ⇥Rn ! R is represented by a real, positive–definite, symmetric matrix. The
partition function of this free, zero–dimensional QFT is the basic Gaussian integral
Z0 =
Z
Rn
dn� e�M(�,�)/2~ (2.7)
with the standard (Lebesgue) measure dn� on the space of fields, which is now just Rn. To
evaluate this, note that since M is a real symmetric matrix, its eigenvectors are orthogonal
so M can be diagonalized by some orthogonal transformation O : Rn ! Rn. The path
integral measure is the standard measure dn� on Rn, which is invariant under such an
orthogonal transformation. In the basis of eigenvectors, the integral is just a product of n
independent Gaussian integrals
Z
R
d� e�m�
2/2~ =
r
2⇡~
m
, (2.8)
– 14 –
where m is the eigenvalue of M . Multiplying all the contributions, we obtain
Z0 =
Z
Rn
dn� e�M(�,�)/2~ =
(2⇡~)n/2p
det M
, (2.9)
where we have written the product of eigenvalues more invariantly as the determinant.
Note that M being positive–definite ensures that det M > 0 and the integral exists.
We also want to compute the partition function in the presence of a source for �. Thus,
we include a linear source term Ja�a in the action:
S(�) =
1
2
M(�,�) + J · � . (2.10)
Completing the square, we have
1
2
M(�,�) + J · � = 1
2
M(�̃, �̃) � 1
2
M�1(J, J) (2.11)
where �̃ := � + M�1(J, · ) are some translated coordinates on Rn. (Our assumption that
M was positive–definite also guarantees that M�1 exists.) Since �̃ di↵ers from � by a
translation, the measure dn�̃ = dn�. Therefore, in the presence of the source J the
partition function is
Z(J) =
Z
Rn
dn� exp
✓
�1~
✓
1
2
M(�,�) + J · �
◆◆
= exp
✓
1
2~M
�1(J, J)
◆ Z
Rn
dn�̃ e�M(�̃,�̃)/2~ = exp
✓
1
2~M
�1(J, J)
◆
Z0
(2.12)
where Z0 is the original partition function (2.9).
To see how this generalization allows us to compute correlation functions, suppose
hP (�)i is a polynomial P : Rn ! R. By linearity of the integral, evaluation of the correla-
tion function hP (�)i reduces to the case that P is a product of linear factors `(�) = ` · �,
so we just need to compute
h`1(�) · · · `p(�)i =
1
Z0
Z
Rn
dn� e�M(�,�)/2~
pY
i=1
`i(�) . (2.13)
If p is odd, then the integrand is an odd function of (at least one direction of) �, so vanishes
when integrated over Rn. Let’s evaluate the remaining case p = 2k. We have that
h`1(�) · · · `2k(�)i =
1
Z0
Z
Rn
dn�
2kY
i=1
`i(�) e
�M(�,�)/2~�J(�)/~
�����
J=0
=
(�~)2k
Z0
Z
Rn
dn�
2kY
i=1
`i ·
@
@J
h
e�M(�,�)/2�J(�)/~
i�����
J=0
=
2kY
i=1
`i ·
@
@J

~2k
Z0
Z
Rn
dn� e�M(�,�)/2�J(�)/~
������
J=0
= ~2k
2kY
i=1
`i ·
@
@J
h
eM
�1(J,J)/2~
i�����
J=0
(2.14)
– 15 –
The first line is a triviality: we want to know the correlation function in the original theory
where J = 0. In going to the second line here we di↵erentiated the action wrt J to bring
down each factor of �, in going to the third line we note that the integrand is absolutely
convergent so the order of integration and di↵erentiation may safely be exchanged, and the
final line uses the result (2.12).
Let’s first think about this formula in the important special case k = 1, where it
reduces to
h`1(�) `2(�)i = ~ M�1(`1, `2) or equivalently h�a�bi = ~ (M�1)ab . (2.15)
This says that (inthis free theory) the two–point function is just the inverse of (minus)
the quadratic term in the exponential. In dimensions d > 0, we’ll consider correlation
functions where the fields are inserted at di↵erent points in our space–time. In this case, the
coe�cient M of the term quadratic in the fields is a di↵erential operator, whose inverse M�1
is the Green’s function or propagator of the theory. We can think of the propagator as
representing the response of one field insertion to the presence of another. It’ll be useful
to represent this result by the picture with the solid line keeping track of the fact that the
= +
=
+
+
· · ·
· · ·
~
m2e�
�4~2
2m6e�
�
1
2
h�2i
�a �b ~ (M�1)ab==h�a�bi
field insertions are joined by a copy of M�1. This picture is a (rather trivial) example of
Feynman diagram.
Now let’s return to the general case of (2.14). For every derivative ~ ` · @/@J that acts
on the exponential we get a factor of M�1(`, J). Because we’ll set J = 0 at the end of the
calculation, we can get a non–vanishing contribution to (2.14) only when exactly half the
derivatives bring down such factors, while the other half then removes the J dependence in
front of the exponential. We’ll then be left with a product of k factors of M�1, contracted
into `’s (or having free indices) in a way that depends on how we paired up the way the
derivatives act. Let � be a way of joining the elements of the set {1, 2, . . . , 2k} into pairs,
and let ⇧2k denote the set of all possible (complete) pairings. Then the correlation function
is
h`1(�) · · · `2k(�)i = ~k
X
�2⇧2k
Y
i2{1,...,2k}/�
M�1(`i, `�(i)) , (2.16)
in other words, a sum over products of all inequivalent ways of connecting pairs of `i using
M�1.
We can use our Feynman diagrams to help keep track of the possible pairings. For
example, the 4-point function
h�a�b�c�di = ~2
⇣
(M�1)ab(M�1)cd + (M�1)ac(M�1)db + (M�1)ad(M�1)bc
⌘
(2.17)
can be represented by the Feynman diagrams
– 16 –
1
2 3
4 1
2 3
41
2 3
4
In general, there are |⇧2k| = (2k)!/(2kk!) ways of joining 2k elements into pairs, so the
2k-point function receives (2k)!/(2kk!) contributions. In particular, we find
h(`·�)2ki = (2k)!
2kk!
�
~M�1(`, `)
�k
(2.18)
when all of the `i’s are the same.
Our result (2.16) for the correlation function is known as Wick’s theorem in QFT,
though in the d = 0 context of Gaussian distributions it’s called Isserlis’ theorem by
probabalists. You met Wick’s theorem last term from the point of view of canonical quan-
tization, where it arose from decomposing the field operator � into creation and annihilation
operators, and commuting these operators past one another. Of course, in d = 0, there’s no
sense in which the fields ‘propagate’ anywhere, so the Feynman diagrams are just a nifty
way to keep track of the combinatorics. Also, since we’re currently thinking just about free
theory, our diagrams have no (internal) vertices at present.
2.3 Perturbation theory
Interesting theories involve interactions, so that the action S(�) is not merely quadratic.
In this case, integrals such as Z
Rn
dn� f(�) e�S(�)/~ (2.19)
become transcendental, even for simple actions S(�) – including most of physical interest
– and simple choices of f(�). Typically, we do not know how to evaluate such integrals
analytically. We may hope to approximate such integrals perturbatively by expanding
around the classical limit ~ ! 0. However, our integral cannot have a Taylor expansion
around ~ = 0, since any such Taylor expansion would have to converge for all ~ in a disc
D ⇢ C centered on the origin. But if the action is chosen so that the integral converges
whenever ~ > 0, then (2.19) surely diverges if we formally attempt continue into the region
Re(~) < 0. Barring numerical methods, the best we can do is to obtain an asymptotic
expansion for such path integrals. (Recall that a series
P
n an~n is asymptotic to a
function I(~) if, for all N 2 N,
lim
~!0+
1
~N
�����I(~) �
NX
n=0
an~n
����� = 0 . (2.20)
In other words, with fixed N , for su�ciently small ~ 2 R�0 the first N terms of the series
di↵er from the exact answer by less than ✏�N for any ✏ > 0. (The di↵erence is o(N)). We
– 17 –
write12
I(~) ⇠
1X
n=0
an~n as ~ ! 0 (2.21)
to mean that the series on the right is an asymptotic expansion of I(~) as ~ ! 0. It’s
important to remember that the true function may di↵er from its asymptotic series by
transcendental terms; for example, the function e�1/~
2 ⇠ 0 as ~ ! 0, but clearly e�1/~2 6= 0.
Thus, if we instead fix a value of ~, however small, and include more and more terms in the
sum, we will eventually get worse and worse approximations to the answer. Perturbation
theory thus tells us important, but not complete, information about our QFT.
Now suppose S(�) is a smooth function that has a unique global minimum at a unique
point � = �0 2 Rn, so that the Hessian matrix @a@bS|�0 is positive–definite. Then (2.19)
has an asymptotic expansion
Z
Rn
dn� f(�) e�S(�)/~ ⇠ (2⇡~)n/2 f(�0) e
�S(�0)/~
p
det(@a@bS|�0)
�
1 + ~A1 + ~2A2 + · · ·
�
(2.22)
as ~ ! 0+. The proof of this is known as steepest descent and should be familiar if
you’ve taken a course such as Part II Asymptotic Methods13. The leading term in this
12In this course, all the asymptotic expansions we consider will be valid as ~ ! 0, so we’ll usually take
this limit as understood.
13In case you didn’t take such a course, here’s an outline of a proof in the case of a single field: Let
A(~) = e
+S(�0)/~
p
~
Z
b
a
e�S(�)/~ f(�) d�
and let ✏ 2 (0, 12 ). Define B(~) in the same way as A(~), but where the integral is taken over the range
[�0 � ~
1
2�✏,�0 + ~
1
2�✏]. As ~ ! 0, we have that A(~) � B(~) is smaller than ~N for any N 2 N. (We say
the di↵erence is rapidly decaying in ~.) Now let � = (�� �0)/
p
~, so
B(~) =
Z ~✏
�~✏
e(S(�0)�S(�0+�
p
~))/~
f(�0 + �
p
~) d� .
Provided the action S(�) and insertion f(�) were smooth, the integrand of this expression is a smooth
function of
p
~ when ~ � 0. Let C(~) be the same integral as for B(~), but with the integrand replaced by
its Taylor expansion around 0 in
p
~, modulo terms of order ~N . Then
|B(~)� C(~)|  K ~N�✏
for some constant K � 0. Finally, let D(~) be the same as C(~), but where the limits of the integral are
�1 and 1. Then D(~) is a polynomial in
p
~, while C(~)�D(~) is rapidly decaying in ~. Since D(~) is a
polynomial in
p
~, it admits a Taylor expansion in
p
~ modulo ~N�✏. Also, the coe�cients of odd powers of
p
~ in D(~) are given by integrals of an odd function of � over all of R, and hence vanish. Finally, we have
D(0) =
Z
R
e�@
2
S|�0 �
2
/2
f(�0) d� =
p
2⇡ f(�0)p
@2S|�0
.
Putting all these facts together shows that
Z
R
e�S(�)/~ f(�) d� = e�S(�0)/~
p
~A(~) ⇠
p
2⇡~e
�S(�0)/~ f(�0)p
@2S|�0
1X
n=0
An~n ,
where A0 = 1. This proves (2.22) in the case of a single field. The generalization to finitely many fields �
a
is straightforward. But don’t worry, neither of these proofs are examinable for this course.
– 18 –
expansion is known as the semi-classical term. In particular, expanding � around the
classical solution �0 as �a = �a0 + ��
a, we have
S(�) = S(�0) +
1
2
@a@bS|�0��a ��b + · · · (2.23)
so that the leading term
Z0 = (2⇡~)n/2
e�S(�0)/~p
det(@a@bS|�0)
(2.24)
in the asymptotic series of the partition function is just what we’d obtain as the partition
function of a theory a purely quadratic theory. We’ll see that it arises in perturbation
theory from the 1-loop approximation, while terms at higher order in ~ in the series (2.22)
arise from multi-loop diagrams.
Let’s understand how this works in an example. Consider the d = 0 QFT with a
single scalar field � and action S(�) = m2�2/2 + ��4/4!. We need to take � > 0 for the
partition function to converge, and we’ll also assume m2 > 0 so that the action has a
unique minimum at �0 = 0. Then the leading term in our asymptotic expansion is
(2⇡~)1/2 e
�S(�0)/~
p
@2S|�0
=
p
2⇡~
m
, (2.25)
sinceS(�0) = 0 and @2S|�0 = m2. As claimed, this is just the partition function Z(m, 0)
of the free theory. Going further, since
Z(m2,�) =
Z
R
d� e
� 1~
⇣
m
2
2 �
2+ �4!�
4
⌘
=
Z
R
d�
"
e�m
2�2/2~
1X
n=0
1
n!
✓
��
4!~
◆n
�4n
#
, (2.26)
we obtain an asymptotic series for Z(m2,�) by truncating to the first N + 1 terms of this
expansion, whereupon
Z(m2,�) ⇠
Z
R
d�
"
e�m
2�2/2~
NX
n=0
1
n!
✓
��
4!~
◆n
�4n
#
=
p
2~
m
NX
n=0
1
n!
✓
�~�
3! m4
◆n Z 1
0
dx e�x x2n+
1
2�1
=
p
2~
m
NX
n=0
1
n!
✓
�~�
3! m4
◆n
�
✓
2n +
1
2
◆
.
(2.27)
In going to the second line we substituted x = m2�2/2~ and exchanged the order of the
finite summation and integration. Note that it would not be legitimate to exchange the
order of the integral and the infinite sum in (2.26), because the original integral does
not converge if ~ < 0. The final line recognizes the integral as a representation of the
gamma function. (Somewhat more laboriously, this integral can be computed by repeated
– 19 –
integration by parts.) Using the value of �(z) at positive half-integers we have finally
Z(m2,�) ⇠
p
2⇡~
m
NX
n=0
(�)n ~
n�n
m4n
1
(4!)n n!
(4n)!
4n(2n)!
= Z0

1 � ~�
8m4
+
35
384
~2�2
m8
+ · · ·
� (2.28)
as our asymptotic series for the partition function, where Z0 = Z(m2, 0) =
p
2⇡~/m.
Let me make a couple of remarks. Firstly, the fact that each term in the expansion
of Z(m2,�)/Z0 is proportional to (�~�/m4)2 is essentially fixed by dimensional analysis.
The coe�cient
1
(4!)n n!
(4n)!
4n(2n)!
can be understood as a product of the factor 1(4!)nn! that comes straightforwardly from
expanding the �4 term in the exponential, and the remaining factor (4n!)/4n(2n)! is the
number of ways of joining 4n elements (the � insertions) into distinct pairs; indeed, we
saw in the discussion of Wick’s theorem that the integral
R
e��
2/2 �4 d� had a combinatoric
interpretation in terms of pairings. Note that we can see the divergence of the perturbation
series directly from these coe�cients: From Stirling’s approximation n! ⇡ en ln n, we see
that
1
(4!)n n!
(4n)!
4n(2n)!
⇡ en ln n
for large n. Thus these coe�cients asymptotically grow faster than exponentially with n,
so the series (2.28) has zero radius of convergence. It’s interesting to ask whether it is
possible to recover the exact value of Z(m2,�) from its asymptotic series. Remarkably, a
technique known as Borel resummation allows one to achieve this, at least in certain
circumstances. You’re invited to explore it for this example in the problem sheets.
As a second remark, observe that Z(m2,�) itself should exist even if m2 < 0, pro-
vided ~ and � are strictly positive, because the exponential enhancement from the factor
e+|m
2|�2/2~ at small � is eventually suppressed by the quartic term in the action. However,
the asymptotic series (2.28) is not valid in this case, as we can see from the fact that
the (Gaussian) integrals in the second line of (2.27) require m2 > 0 to converge. More
fundamentally, the problem is that when m2 < 0, the point � = 0 which we took to give
the dominant contribution to the integral is now a (local) maximum of the action, the
global minima being at �0 = ±
p
6m2/�. In physics terminology we are expanding around
the wrong vacuum. Particles with m2 < 0 are called tachyons, and they always signal
an instability. Whether or not this instability is just due to a poor choice of perturbative
expansion (as here), or whether the whole theory is unstable (meaning Z(m2,�) does not
exist for m2 < 0) is not always clear. The situation where the minimum of the action
involves a non–zero value for some field is often associated with spontaneous symmetry
breaking. You can learn more about this e.g. in the Part III course on the Standard
Model.
– 20 –
2.3.1 Feynman diagrams
Above, we gave a combinatoric interpretation of the numerical coe�cients of the asymptotic
series
Z(m2,�)/Z0 ⇠
NX
n=0
(�)n ~
n�n
m4n
1
(4!)n n!
(4n)!
4n(2n)!
(2.29)
in terms of ways of pairing up the � insertions in the integral. Let’s now reconsider this
from the point of view of Feynman diagrams. With the action S(�) = m2�2/2 + ��4/4!
the ingredients of our Feynman diagrams are
)
�1
�2
�3
�4
+1
+1
�1
�
h(�)
�1
T1 T2
T3
x y
z
~
m2
��~
for the propagator and vertex. Note again that the propagator is just a constant since we
are in zero dimensions, while the minus sign in the vertex comes from the fact that we are
expanding e�S/~.
To compute perturbation series in this theory, Feynman tells us to start by constructing
all possible graphs (not necessarily connected) using this propagator and vertex. In the
case of the partition function, we want vacuum graphs, i.e., those with no external14 edges.
Let Dn be the set of all labelled vacuum graphs containing n vertices, and let there be |Dn|
elements in this set. By a labelled graph, I mean that individual vertices carry their own
unique ‘label’, so that we can tell them apart. Likewise, each of the four legs emanating
from a given vertex carries its own label.
Since each end of every edge in a vacuum graph is attached to a vertex, and the vertex
is 4-valent in this theory, every graph in Dn must contain precisely 2n edges. Thus, using
the propagator and vertex given above, every graph in Dn contributes a term proportional
to (�~�/m4)n, as indeed we saw in (2.29). For example, in this theory the set D1 consists
of the three graphs
)
�1
�2
�3
�4
corresponding to the three possible ways to join up the four � fields into pairs. Thus, the
term proportional to � receives contributions from these three individual graphs.
Joining up our labelled vertices in every possible way means that the set Dn may
contain several elements that are identical as unlabelled topological graphs, but di↵er just
in the labelling of their vertices or edges. For example, all three graphs displayed above
are equivalent as topological graphs. Identical topological graphs correspond to identical
14An internal edge of a graph is one in which both ends of the edge are attached to vertices, which may
be distinct vertices or the same. An external edge is an edge that is not internal, and so has at least one
end not attached to a vertex.
– 21 –
physical processes15, and the original integral knew nothing of our choice of labels, so in
working out the perturbation series we need to remove this overcounting. To do so, observe
that Dn is naturally acted on by the group Gn = (S4)n o Sn that permutes each of the
four fields present at a given vertex (n copies of the permutation group S4 on 4 elements)
and also permutes the labels of each of the n vertices (the permutation group Sn)16. This
group has order |Gn| = (4!)nn!, which is the same factor we saw before from expanding
e�S/~ in powers of �. Thus the asymptotic series (2.29) may be rewritten as
Z
Z0
⇠
NX
n=0
✓
��
m4
◆n |Dn|
|Gn|
. (2.30)
In detail, the power (��)n is the contribution of the coupling constants in each graph,
the power of (1/m2)2n comes from the fact that any vacuum diagram with exactly n 4–
valent vertices must have precisely 2n edges, each of which contributes a factor of 1/m2.
The factor |Dn|/|Gn| is the number of diagrams that contribute at this order, counting as
equivalent those diagrams that merely permute the labels of the fields at a given vertex,
or the labelling of the vertices.
There’s another way to think of |Dn|/|Gn| that is sometimes convenient17. An orbit
� of Gn in Dn is a set of labeled graphs in Dn that are identical except for a relabelling
of their fields and vertices, so that we can move from one labelled graph to another in
the orbit using an element of Gn (i.e. by permuting these labels). Thus an orbit � is a
topologically distinct graph in Dn. Let On be the set of such orbits �; that is On is the
set of topologically distinct vacuum graphs on n vertices. The orbit stabilizer theorem
says that18
|Dn|
|Gn|
=
X
�2On
1
|Aut �| , (2.31)
whereAut � is the stabilizer of any element in � in Gn, i.e., the elements of the per-
mutation group Gn that don’t alter the labelled graph. For example, if a graph in Dn
contains an edge both of whose edges are attached to the same vertex, then exchanging the
labelling of those fields doesn’t change the labelled graph. Similarly, if a pair of vertices
are connected by two (or more) propagators, then exchanging the labels of the two (or
more) legs on each vertex that are joined to these propagators does not change the labelled
15For this statement to hold true, we need to be careful to account for all the quantum numbers, and to
give precise meaning to ‘topological equivalence’. For example, we often draw Feynman graphs representing
matrix–valued fields as graphs whose edges are thickened into ribbons. Graphs that would be equivalent
as line graphs but that di↵er by twisting of the ribbons should be counted separately (they correspond to
di↵erent ways to tie up the matrix indices of the fields), and the relevant notion of topological equivalence
is called ambient isotopy.
16Exercise: why is the full group the semi-direct product of these two subgroups?
17In practice, at least for the simple graphs we’ll meet in this course, it’s often just as quick to think
through the possible ways a given topological graph � may be obtained by expanding out the vertices in
e�S/~ and joining pairs of fields by propagators, as to work out the symmetry factor |Aut�|. I’ll leave it to
your taste.
18If you don’t know this already, you can find a nicely explained proof on Gowers’s Weblog.
– 22 –
https://gowers.wordpress.com/2011/11/09/group-actions-ii-the-orbit-stabilizer-theorem/
graph. Finally then, we can rewrite our asymptotic series (2.29) as
Z
Z0
⇠
1X
n=0
"✓
�~�
m4
◆n X
�2On
1
|Aut �|
#
=
X
�
~|e(�)|�|v(�)|
|Aut �|
(��)|v(�)|
(m2)|e(�)|
,
(2.32)
in terms of a sum over Feynman graphs �, where |v(�)| and |e(�)| are respectively the
number of vertices and edges of the graph �. The factor |Aut �| is often known as the
symmetry factor of the graph.
We’ve rederived the Feynman rule that we should weight each topologically distinct
graph by |v(�)| powers of (minus) the coupling constant ��/~ and |e(�)| powers of the
propagator ~/m2, then divide by the symmetry factor |Aut �| of the graph. Thus, the
asymptotic expansion of the partition function is given by the Feynman diagrams
(N, g)
0 T
x(t)
��
y
1/(k2 + M2)
x
1/(k2 + m2)
; + ++ + + · · ·
= 1 ++ + + · · ·
=Z/Z0
~�
8m4
� ~
2�2
48m8
~2�2
16m8
~2�2
128m8
where we include both connected and disconnected graphs, with the contribution of a
disconnected graph being the product of the contributions of the two connected graphs.
Notice that this requires that we assign a factor 1 to the trivial graph ; (no vertices or
edges), which is also included as the zeroth–order term in the sum.
More generally, our theory may involve a di↵erent types of field, each associated with a
propagator 1/Pa. These fields could interact via various di↵erent vertices v↵, either joining
di↵erent types of field or di↵erent powers of the same field. Let’s suppose that a vertex
of type ↵ (where ↵ labels the types and multiplicities of the fields at this vertex) has a
coupling constant �↵ in the action. Then a graph � containing |ea(�)| edges representing
propagators of the type a field and |v↵(�)| vertices of type ↵ is associated with a weight
factor
F (�) =
Y
a,↵
(��↵)|v↵(�)|
(Pa)|ea(�)|
(2.33)
by the Feynman rules. Let |e(�)| =
P
a |ea(�)| and |v(�)| =
P
↵ |v↵(�)| be the total
number of edges and vertices in the graph, and let
b(�) = |e(�)| � |v(�)| (2.34)
be the di↵erence. Since each propagator contributes a factor of ~ and each vertex a factor
of 1/~, a graph with |e(�)| edges and |v(�)| vertices, � comes with a power ~|b(�)|. Thus
the partition function has the perturbative expansion
Z
Z0
⇠
X
�
1
|Aut �|~
b(�)F (�) (2.35)
– 23 –
as ~ ! 0, where we sum this expression over both connected and disconnnected vacuum
graphs, including the trivial graph with no vertices. In particular, if our action includes
sources J , then the Feynman diagrams may involve a vertex which joins the fields to these
external sources. If J couples to a single power of the field, then the vertex is 1-valent and
is associated with a Feynman rule
= +
=
+
+
· · ·
· · ·
~
m2e�
�4~2
2m6e�
�
1
2
h�2i
�a �b ~ (M�1)ab==h�a�bi
J �J/~
whereas if J sources a composite operator involving pa powers of the field of type a, then
this vertex will be (
P
a pa)-valent, absorbing pa factors of �
a. We include such vertices in
what we mean by a ‘vacuum’ graph, so edges that terminate on a (green) source are not
considered external.
2.4 E↵ective actions
In this section I want to introduce the very important notion of an e↵ective action, which
will help us to develop a better feel for the partition function. We’ll see that there are (at
least) two distinct definitions that are related to eachother by a Legendre transform, very
much analogous to the relation between the Helmholz (F ) and Gibbs (G) free energies in
statistical physics.
These e↵ective actions will turn out to be central to our understanding of QFT in
higher dimensions. The ‘Helmholz’ version plays a key role in Wilson’s approach to renor-
malization, first developed in condensed matter systems, whereas the ‘Gibbs’ version is
more closely related to the approach of Goldstone, Salam, Weinberg and Jona-Lasinio that
was developed in parallel with high energy physics in mind.19
2.4.1 Connected graphs and a loop expansion
We start with a pragmatic observation: In computing the asymptotic expansion of Z, we
needed to take both connected and disconnected graphs into account. For example, in
computing the partition function of the theory S(�) = m2�2/2 + ��4/4! above, both
(N, g)
0 T
x(t)
��
y
1/(k2 + M2)
x
1/(k2 + m2)
; + ++ + + · · ·
= 1 ++ + + · · ·
=Z/Z0
~�
8m4
� ~
2�2
48m8
~2�2
16m8
~2�2
128m8
and
appeared (and higher powers of this and all other diagrams would occur further down the
perturbative expansion). This is a duplication of e↵ort – a disconnected graph is made
up of several connected graphs, each of whose contributions we’ve already included. We’ll
now show that
W = �~ ln Z , (2.36)
is given asymptotically by a sum of connected Feynman graphs, avoiding this extra e↵ort.
In QFT, W is known as the Wilsonian e↵ective action, and as I mentioned it’s closely
19I’d love you to already have profound physical insight and a strong mathematical grasp of the uses and
definitions of F vs G, but I’m a realist, so we’ll try to develop these as we go along.
– 24 –
analogous to the Helmholz free energy in statistical physics. Knowing W is equivalent to
knowing Z, so of course W also depends on all the choices we made in setting up our QFT
and in particular may depend on the sources.
To understand how W involves only connected graphs, suppose {�j} is the set of all
possible connected vacuum graphs we can build using our propagators and vertices, where
the label j tells us the topology of the graph. We define the product �1�2 of any two
graphs �1, �2 to be their disjoint union, and similarly we interpret (�j)n as the disjoint
union of n copies of the same connected graph �j . Any disconnected graph � is specified
by a set of numbers {nj} (with each nj 2 N0) telling us how many copies of the connected
graph �j it contains.
Now, the symmetry factor of a disconnected graph consisting of n1 copies of �1, n2
copies of �2 etc. is
|Aut(�n11 �
n2
2 · · · �
nk
k )| =
kY
j=1
(nj !) |Aut(�j)|nj , (2.37)
because this is just a product of all the symmetry factors for the individual graph compo-
nents, times a factor of nj ! arising because we get an identical disconnected graph if we
exchange any of the nj copies of graph �j . Also,
F
0
@
Y
j
�
nj
j
1
A =
Y
j
F (�j)
nj and b
0
@
Y
j
�
nj
j
1
A =
X
j
njb(�j) , (2.38)
since the vertices and propagatorscontribute multiplicatively to an individual graph.
Putting these facts together, we can write the partition function as
Z
Z0
⇠
X
� 2 disconn
~b(�)
|Aut �|F (�) =
X
{nj}
~b
⇣Q
j
�
nj
j
⌘
���Aut
⇣Q
j �
nj
j
⌘���
F
0
@
Y
j
�
nj
j
1
A
=
X
{nj}
Y
j
1
nj !
~njb(�j)
|Aut(�j)|nj
F (�j)
nj =
Y
j
0
@
1X
nj=0
1
nj !
 
~b(�j)
|Aut(�j)|
F (�j)
!nj1
A
=
Y
j
exp
 
~b(�j)
|Aut(�j)|
F (�j)
!
= exp
 
X
� 2 conn
~b(�)
|Aut(�)|F (�)
!
.
(2.39)
Comparing with the definition (2.36) we have shown that the Wilsonian e↵ective action is
given by
W ⇠ W0 � ~
X
� 2 conn
~b(�)
|Aut(�)|F (�) , (2.40)
where W0 = �~ ln Z0. As promised, the e↵ective action has an asymptotic expansion in
terms of connected graphs built from the same propagators and vertices as the partition
function itself.
– 25 –
Euler’s theorem tells us that, for a connected graph,
b(�) = |e(�)| � |v(�)| = `(�) � 1 (2.41)
where `(�) is the number of loops20 in the graph. Comparing to (2.40) shows that an
`-loop connected Feynman graph contributes a term of order ~` to the expansion of the
Wilsonian e↵ective action. For this reason, the asymptotic expansion of the partition
function is often known as the loop expansion of the QFT. We can say more by doing a
little more elementary graph theory: If a vertex v↵(�) involves n↵a fields of type a, then
for vacuum graphs
2|ea(�)| =
X
↵
n↵a|v↵(�)| , (2.42)
because each end of every edge must be attached to some vertex. Let’s also suppose that
these vertices all represent genuine interactions, so
P
a n↵a > 2 as at least three fields
(possibly of di↵erent types) meet at each vertex. Then
`(�) = 1 +
X
a
|ea(�)| �
X
↵
|v↵(�)| = 1 +
X
↵
 
�1 +
X
a
n↵a
2
!
|v↵(�)| > 1 . (2.43)
In other words, if all our vertices are at least 3-valent, then every non–trivial vacuum graph
contains at least 2 loops.
Using the definition (2.24) of Z0, this shows that
W ⇠ S(�0) +
~
2
ln det(@a@bS|�0) �
X
� 2 conn
~`(�)
|Aut(�)|F (�) , (2.44)
where `(�) � 2 in each term in the final sum. We see that the leading term in W, of order ~0,
is just the original classical action evaluated at its minimum �0. In (2.24) we saw that the
term of order ~ came from expanding the action to quadratic order around the minimum,
and integrating over the fluctuations. We’ll see shortly that this can indeed be interpreted
as a (sum of) 1-loop diagrams; as a quick plausibility check note that @a@bS|�0��a ��b
can be interpreted as action consisting of purely 2-valent vertices of the form @a@bS|�0 , and
that (2.43) says that if all n↵a = 2, we can only construct 1-loop graphs. As we said before,
the higher order terms in the asymptotic series correspond to multi-loop diagrams21.
I stress that the counting given above is valid for vacuum graphs in which all the
vertices are at least trivalent; Feynman diagrams associated to scattering amplitudes or
correlation functions, or those involving external sources corresponding to a 1-valent vertex,
may come with di↵erent powers of ~ depending on the number of external states, number
of field insertions in the correlator, or number of vertices involving the external source.
20A ‘loop’ is an independent 1-cycle in the sense of homology of the graph.
21I’m sorry to break the bad news, but in studying tree diagrams throughout last term’s course, you
weren’t really doing any quantum field theory at all. Rather, the tree diagrams you drew were just a
perturbative way to evaluate the classical action on a solution to the equations of motion. (Feynman tree
diagrams are very closely related to Picard iteration, a standard perturbative technique to solve non-
linear di↵erential equations.) Unlike our example above, you found S(�0) 6= 0 because you were working
on a non-compact space R3,1 and demanded the fields were non–trivial in the distant past and future: i.e.,
you computed a scattering amplitude. Nonetheless, the tree amplitude you obtained was purely classical —
indeed, QFT should agree with classical field theory as ~ ! 0.
– 26 –
2.4.2 Integrating out fields
Having seen that it’s computed using just connected graphs, let’s now try to get a feel for
the physical meaning of W. To begin, suppose we have two real–valued fields � and �, so
that the space of fields is R2, and let the action be
S(�,�) =
m2
2
�2 +
M2
2
�2 +
�
4
�2�2 (2.45)
so that � provides a coupling between the two fields. The Feynman rules are
= + + + +h�2i
1
m2
� �
2m4M2
+
�2
4m6M4
+
�2
2m6M4
+
�2
4m6M4
=
+ ++=
= + +
ln

Z
Z0
�
� ~�
4m2M2
~2�2
16m4M4
~2�2
16m4M4
~2�2
8m4M4
+
� �
~/m2 ~/M2 ��/~
and we may use these to compute perturbative expressions for correlation functions such
as
hfi = 1Z
Z
R2
d� d� e�S(�,�)/~ f(�,�)
in the usual way. For example, we have
+ ++
= + +� ~�
4m2M2
~2�2
16m4M4
~2�2
16m4M4
~2�2
8m4M4
+
�~�1W ⇠
� �
~/m2 ~/M2 ��/~
= + + + +h�2i
=
~
m2
�~2
2m4M2
�2~3
4m6M4
�2~3
2m6M4
�2~3
4m6M4
+ + +�
as the sum of connected vacuum diagrams, and also
+ ++=
= + +
ln

Z
Z0
�
� ~�
4m2M2
~2�2
16m4M4
~2�2
16m4M4
~2�2
8m4M4
+
� �
~/m2 ~/M2 ��/~
= + + + +h�2i
=
~
m2
�~2
2m4M2
�2~3
4m6M4
�2~3
2m6M4
�2~3
4m6M4
+ + +�
where the insertion of each power of � is represented by a blue dot.
I want to arrive at this result in a di↵erent way. Suppose we first perform the integral
over � whilst holding � fixed. In higher dimensions this step might be appropriate if,
for example, M � m so that our experiment isn’t powerful enough to observe real �
production so can only measure � directly. If we have no idea what � is doing, we perform
its path integral first, i.e., we average over the behaviour of � at each fixed �. From this
point of view, whilst performing the � integral, the coupling �2�2 acts as a background
source J = �2 for the composite operator �2. The � path integral then yields a W(�) that
depends on this source:
e�W(�)/~ =
Z
R
d� e�S(�,�)/~ . (2.46)
Once we’ve found this W(�), we can use it in the remaining � integral to compute hfi
for any observable f that depends only on � – i.e. the only quantities our low–energy
– 27 –
experiment is able to probe. Of course there’s nothing mysterious here, we’re simply
choosing in which order to do our integrals, writing
hf(�)i = 1Z
Z
d� d� e�S(�,�)/~ f(�) =
1
Z
Z
d� e�W(�)/~f(�) . (2.47)
Note that indeed W(�) plays the role of an e↵ective action for the � field – one in which
all the quantum e↵ects of � are taken into account.
In general, computing W(�) has to be done perturbatively in terms of a sum of con-
nected Feynman diagrams in the presence of the source J = �2. However, in our toy
example it’s straightforward to find W(�) exactly:
Z
R
d� e�S(�,�)/~ = e�m
2�2/2~
s
2⇡~
M2 + ��2/2
(2.48)
and therefore
W(�) = 1
2
m2�2 +
~
2
ln

1 +
�
2M2
�2
�
+
~
2
ln
M2
2⇡~ . (2.49)
This is exactly what we expect from above. At constant �, the original action has a
unique minimum at �0 = 0, where S(�,�0) = m2�2/2, the leading term in W(�). The
logarithms comes from the � integral, which is our example is purely Gaussian. The final
term in (2.49) is independent of the field �; such field–independent terms are irrelevant in
QFT, for example, they will cancel when we compute any correlation function normalized
by the partition function of the free (� = 0) theory. We will drop this term henceforth,
but note that the fact the constant term in the action changes as we integrate out fields is
actually the origin of the notorious cosmological constant problem.
Expanding the remaining logarithm, we write W(�) as an infinite series
W(�) =
✓
m2
2
+
~�
4M2
◆
�2 � ~�
2
16M4
�4 +
~�3
48M6
�6 + · · ·
=
m2e↵
2
�2 +
�4
4!
�4 +
�6
6!
�6 + · · · .
(2.50)
Thus the e↵ect of integrating out the ‘high energy’ field � is to change the structure of the
action seen by �. In particular, the mass term of the � field has been shifted
m2 ! m2e↵ = m2 +
~�
2M2
. (2.51)
Even more strikingly, we’ve generated an infinite seriesof new coupling terms
�4 = �
3~
2
�2
M4
, �6 = 15~
�3
M6
, �2k = (�1)k+1~
(2k)!
2k+1k
�k
M2k
(2.52)
describing self–interactions of �. It’s important to observe that the � mass shift and new
� self–interactions all vanish as ~ ! 0; they are quantum e↵ects. Notice also that they’re
each suppressed by powers of the (high) mass M .
– 28 –
Following our general story above, it’s useful to think in a little more detail about how
these new couplings arise. We can perform the � path integral using Feynman graphs,
using the ingredients
~
M2
���
2
2~
+ + + +
+
=
=
· · ·
· · ·�1
2
��2
2M2
�1
4
�2�4
4M4
� 1
3!
�3�6
8M6
�S(�)~
�W(�)~
which involve the same � propagator as before, but now account for the fact that we are
treating the interaction as a source, which takes the value ���2/2 from the point of view of
the � integral. These ingredients lead to the following perturbative construction of W(�)22:
~
M2
���
2
2~
+ + + + · · ·W(�) ⇠ �~
 
+= · · ·S(�) +
1
2
~�
2M2
�2
1
4
~�2
4M4
�4
1
3!
~�3
8M6
�6+�
where in the first term S(�) = S(�, 0) is the part of the original action that came straight
out of the � integral. Again since �~�1W(�) is the logarithm of the � integral, only
connected diagrams appear.
Just as we expected, the diagrammatic expansion reveals that the new interactions in
W are generated by the � field running around a loop, interacting with the ‘source’ as it
goes. In our e↵ective description that knows only about the behaviour of the � field, we can
no longer ‘see’ the � field ‘circulating’ around the loop. Instead, we perceive this just as a
new interaction vertex for �. As promised, the fact that � appears only quadratically in the
original action (2.45) means that in this example we can only construct 1-loop diagrams
from our propagator and 2-valent vertex. All these 1-loop diagrams sum up to give the
logarithm we obtained by direct integration. Starting from a more generic initial action
with higher valent vertices, we’d obtain contributions from higher loop graphs, each coming
with a factor of ~`(�).
Using this e↵ective action, we find
�a �b ~ (M�1)ab==h�a�bi
J �J/~
+
=
+
+
⇠ · · ·
· · ·
~
m2e�
�4~2
2m6e�
�
h�2i = 1Z
Z
d� e�W(�)/~ �2
where the propagator and vertices here are the ones appropriate for the e↵ective action
W(�). Using the definition (2.51)-(2.52) of the new couplings in terms of the original � and
22In evaluating these Feynman diagrams, I’ve kept the symmetry factors separate from the vertices and
propagators – check you understand them.
– 29 –
M , this unsurprisingly agrees with our answer before, correct to order �2. However, once
we had the e↵ective action, we arrived at this answer using just two diagrams, whereas
previously it required five. If we only care about a single correlation function then the
work involved in first computing W(�) and then using the new set of Feynman rules to
compute the low–energy correlator is roughly the same as just using the original action
to compute this correlator directly. On the other hand, if we wish to compute many
low–energy correlators then we’re clearly better o↵ investing a little time to work out the
e↵ective action first.
However, the real point I wish to make is this: the way we experience the world is
always through an e↵ective action. Naively at least, we have no idea what new physics may
be lurking just out of reach of our most powerful accelerators; there may be any number
of new, hitherto undiscovered species of particle, or new dimensions of space–time, or even
wilder new phenomena. However, when describing low–energy physics, we should only seek
to describe the behaviour of the degrees of freedom (fields) that are relevant and accessible
at the energy scale at which we’re conducting our experiments, even if we happen to know
what the more fundamental description is. For example, a glass of water certainly consists
of very many H2O molecules, these molecules are bound states of atoms, each of which
consist of many electrons orbiting around a central nucleus. In turn, this nucleus comprises
of protons and neutrons stuck together by a strong force mediated by pions, and all these
hadrons are themselves seething masses of quarks and gluons. But it would be very foolish
to imagine we should describe the properties of water that are relevant in everyday life by
starting from the Lagrangian for QCD.
Let me make one final comment. In the example above, we started from a very simple
action in equation (2.45) and obtained a more complicated e↵ective action (2.50) after
integrating out the unobserved degree of freedom �. A more generic case would start from
a general action (invariant under � ! �� and � ! �� for simplicity)
S0(�,�) =
X
i,j
�i,j
(2i)! (2j)!
�2i�2j (2.53)
in which all possible even monomials in � and � are allowed. For example, we may have
arrived at this action by integrating out some other field that was unknown in our above
considerations. In this generic case, the e↵ect of integrating out � will not generate new
interactions for � — all possible even self–interactions are included anyway — but rather
the values of the coupling constants �i,0 will get shifted, just as for the mass shift we saw
above. In addition, because the � path integral would now be very complicated, we can
only reasonably expect to describe the shifted couplings as an asymptotic series in ~, rather
than the single power of ~ we obtained above. Nonetheless, the main lesson to remember
is that integrating out degrees of freedom changes the values of the coupling constants in
the e↵ective action for the remaining fields.
2.4.3 The 1PI e↵ective action
Wilson’s e↵ective action is motivated by the idea of averaging over quantum fluctuations of
high energy fields that are beyond the reach of our experimental observations, and provides
– 30 –
us with a new action for the remaining, low energy degrees of freedom. The quantum e↵ects
of the remaining fields still need to be computed. We’d now like to construct a new type
of e↵ective action that takes account of the quantum fluctuations of the whole system.
You might think that this should just be W(J) itself: we couple our fields to sources,
integrate out all the quantum fields to obtain W(J) and then di↵erentiate wrt J to obtain
correlation functions. This point of view is indeed useful if our quantum system is immersed
in some background (the choice of sources) that we are able to vary. However, for an isolated
quantum system (such as the whole Universe, or a scattering experiment performed in
CERN) there is no obvious background.
We include a source term J� in the original action and let
� =
@W
@J
= � ~Z(J)
@
@J
Z
d� e�(S+J�)/~
�
=
1
Z(J)
Z
d� e�(S+J�)/~ � = h�iJ ,
(2.54)
so that � is the average value of the field �, including all quantum e↵ects. I emphasise
that this average is computed in the presence of a source J for � itself – i.e., we do not set
J = 0 after taking the derivative. Clearly, this means that � depends on what we choose
for J and, conversely, if we specify a value of � we want to obtain then the source is fixed
(at least if the relation �(J) is invertible).
We define the quantum e↵ective action �(�) as the Legendre transformation
�(�) = W(J) � �J (2.55)
of the Wilsonian e↵ective action. Note that
@�
@�
=
@W
@�
� J � �@J
@�
=
@W
@J
@J
@�
� J � � @J
@�
= �J . (2.56)
The relations
� =
@W
@J
and J = � @�
@�
(2.57)
allow us to transform between W(J) and �(�): If we are given W(J) as a function of
J , we define � by @W/@J as above and inverting23 this gives us J(�). Then �(�) =
W(J(�)) � � J(�) is a function of �. On the other hand, if we are presented with a
function �(�), we define J to be �@�/@�. Inverting gives �(J) and hence we reconstruct
W(J) = �(�(J)) + �(J) J as a function of J .
To understand the role of �(�), first note that
@�
@�
����
J=0
= 0 (2.58)
so that the possible quantum averaged values for � inthe absence of a source are just the
extrema of �(�). This is one sense in which �(�) is an e↵ective action – the extrema of
23The Legendre transform requires that the functions W(J) and �(�) are convex, which ensures the
derivatives are monotonically non-decreasing, so that these relations are invertible. This is known to be the
case in statistical mechanics, but is much less clear in the infinite dimensional context of QFT.
– 31 –
�(�) correspond to equations of motion (which, in our current d = 0 context will just be
algebraic) with all the quantum corrections taken into account.
Let’s go even further and consider a quantum theory defined via a (path) integral over
� where we let �(�) play the role of the classical action. We define a quantity W�(J, g) by
e�W�(J)/g =
Z
d� e�(�(�)+J�)/g , (2.59)
where the parameter g plays the role of ~ – I wish to keep g separate from the original
parameter ~ that is still present in the vertices of �(�). It follows from our previous results
that W�(J) can be computed in terms of a series of connected Feynman graphs, now built
using the propagators and vertices that follow from �(�), rather than the original classical
action S(�). As before, an `-loop diagram will contribute a term to W� that is proportional
to g`, so we can expand
W�(J) =
1X
`=0
g` W(`)� (J) (2.60)
where W(`)� (J) is the sum of all `-loop connected Feynman graphs present in (2.59). In
particular, the tree graphs we can construct using the propagators and vertices of the
quantum e↵ective action all appear in W(0)� (J). To extract these tree graphs, we take
the limit g ! 0. In this limit, by the method of steepest descent we know that the
integral (2.59) will be dominated by the minimum of the argument of the exponential, i.e.
the value of � for which
@�
@�
= �J (2.61a)
and that, to leading order,
W�(J) = W(0)� (J) = �(�) + J� (2.61b)
evaluated at this extremum. These are exactly the same equations as (2.55) & (2.57), so
we see that the tree level term W(0)� (J) is nothing other than W(J). In other words, the
sum of connected diagrams W(J) built from the classical action S(�) + J� can also be
obtained as a sum of tree diagrams using the e↵ective action �(�) + J�.
To understand how this can be possible, note that any connected graph can be viewed
as a tree whose ‘vertices’ are all possible one particle irreducible graphs. (An edge e
Figure 2: Any connected graph can be viewed as a tree whose vertices are 1PI graphs.
– 32 –
in a connected graph � is a bridge if �\e is disconnected. A connected graph is said to
be one particle irreducible, or 1PI, if it does not contain any bridges.) This is simple to
see: start from any connected graph and remove all bridges. The result is a product of
1PI graphs, which may be taken as vertices of a tree – see figure ?? for an example. This
tells us how to compute �(�) perturbatively from the original action: �(�) consists of all
possible 1PI Feynman graphs that may be constructed using the propagators and vertices
in S(�). These graphs may have arbitrarily many external lines, with each external line
associated with a factor of �. The number of external lines in a given 1PI graph thus tells
us the valency of a vertex in �(�).
As a check on this formalism, suppose we have several fields �a, each with sources Ja
and let e�W(Ja)/~ =
R
dn� e�(S(�
a)+Ja�a)/~. Then
�~ @
2W
@Ja @Jb
= �~ @
@Ja

1
Z(J)
Z
dn� e�(S(�
c)+Jc�c)/~ �b
�
=
1
Z(J)
Z
dn� e�(S(�)+J�)/~ �a�b
� 1Z(J)2
Z
dn� e�(S(�
c)+Jc�c)/~ �a
� Z
dn� e�(S(�
c)+Jc�c)/~ �b
�
,
(2.62)
where the terms in the final line come from letting the second derivative operator act on
1/Z(J). In keeping with the fact that W(J) involves only connected graphs, we see that
expression is the connected two–point function of the fields
h�a�biconnJ = h�a�biJ � h�aiJ h�biJ , (2.63)
since any contribution to h�a�biJ coming from a Feynman graph that does not somehow join
together the two � insertions will cancel against identical Feynman graphs in h�aiJh�biJ :
�a
�b
�a
�b
+h�a�bi = = h�a�biconn + h�ai h�bi
We can thus view h�a�biconnJ=0 as an expression for the exact propagator in the interacting
theory, including not just the inverse of the kinetic term in S(�), but also corrections due
to interactions.
Using the connecting relations (2.57) we can also express this as
h�a�biconnJ = �~
@2W
@Ja @Jb
= �~@�
b
@Ja
= �~
✓
@Ja
@�b
◆�1
= ~
✓
@2�
@�b @�a
◆�1
. (2.64)
This shows that the exact propagator in our interacting theory is indeed given by ~ times
the inverse of the quadratic term in the quantum e↵ective action. Di↵erentiating further
allows us to see that the connected n–point functions h�a�b · · ·�diconnJ of the fields are
exactly the tree graphs we obtain by connecting the 1PI vertices in �(�) with these exact
propagators. (We’re usually interested in the case that there is no background source, so
we set Ja = 0 at the end of the day.)
– 33 –
2.5 Fermions and Grassmann variables
Realistic theories contain fermions. In higher dimensions, the spin–statistics theorem says
that for a unitary theory, fermions must have half–integral spin. However, in d = 0 there
is no notion of spin, much less a spin–statistics theorem, and fermionic ‘fields’ are simply
Grassmann numbers. These are a set of n elements {✓a} obeying the algebra
✓a✓b = �✓b✓a and ✓a�b = �b✓a for all �b 2 C . (2.65)
Thus, Grassmann variables anticommute with eachother and commute with any bosonic
variable. In particular, this implies ✓a✓a = �✓a✓a = 0 for each a (no sum). This property
means that any function of a finite number of Grassmann variables has a finite expansion
F (✓) = f + ⇢a ✓
a +
1
2!
ga1a2 ✓
a1✓a2 + · · · + 1
n!
ha1a2···an ✓
a1✓a2 · · · ✓an , (2.66)
where we can take the coe�cients to be totally antisymmetric, e.g. ga1a2 = �ga2a1 .
We can also define di↵erentiation and integration for Grassmann variables. For di↵er-
entiation we have
@
@✓a
✓b + ✓b
@
@✓a
= �ba (2.67)
so that the derivative operator itself anticommutes with the variables. Since any function
of a single Grassmann variable ✓ is of the form f + ⇢ ✓, we only have to define
R
d✓ andR
d✓ ✓. We ask that our definition be translationally invariant, so that
Z
d✓ (✓ + ⌘) =
Z
d✓ ✓ (2.68)
and this implies Z
d✓ 1 = 0 . (2.69a)
We then choose to normalise our integration measure such that
Z
d✓ ✓ = 1 . (2.69b)
These rules are often known as Berezin integration. Note that these definitions imply
Z
d✓
@
@✓
F (✓) = 0 (2.70)
since the derivative removes the single power of ✓ that can appear in F (✓). This allows us
to integrate by parts, provided due care is taken of signs.
If we have n Grassmann variables ✓a, repeated application of the above rules shows
that the only non–vanishing integral is one whose integrand involves exactly one power of
every ✓a. Specifically, we have
Z
dn✓ ✓1✓2 · · · ✓n�1✓n =
Z
d✓n d✓n�1 · · · d✓1 ✓1✓2 · · · ✓n = 1 (2.71)
– 34 –
and, in general Z
dn✓ ✓a1✓a2 · · · ✓an = ✏a1a2···an (2.72)
with the sign coming from ordering the ✓s. Suppose we write ✓0a = Nab✓
b for some N 2
GL(n; C). Then, by linearity
Z
dn✓ ✓0a1✓0a2 · · · ✓0an = Na1b1N
a2
b2
· · · Nanbn
Z
dn✓ ✓b1✓b2 · · · ✓bn
= Na1b1N
a2
b2
· · · Nanbn✏
b1b2···bn
= det(N) ✏a1a2···an = det N
Z
dn✓0 ✓0a1✓0a2 · · · ✓0an .
(2.73)
Thus we see that for Berezin integration
✓0a = Nab✓
b ) dn✓ = det(N) dn✓0 (2.74)
where the Jacobian of the change of variables appears upside down (and without a modulus
sign) compared to the standard, bosonic rule dn� = dn�0/| det N | if �0a = Nab�b.
2.5.1 Fermionic free field theory
Let’s suppose our d = 0 QFT involves two fermionic fields, {✓1, ✓2}. The action is a bosonic
quantity, so each term has to involve and even number of fermions. Consequently, the only
non–constant action we can write down is
S(✓) =
1
2
A✓1✓2 , (2.75)
because since (✓1)2 = 0 = (✓2)2 there is no way to introduce a non–trivial interaction. The
partition function is then
Z0 =
Z
d2✓ e�S(✓)/~ =
Z
d2✓✓
1 � A
2~✓
1✓2
◆
= � A
2~ (2.76)
using the fact that the expansion of e�S(✓)/~ truncates at the first non–trivial term, and
the rule (2.71) of Berezin integration. More generally, if we have 2m fermionic fields ✓a
described by the quadratic action
S(✓) =
1
2
Aab✓
a✓b (2.77)
where A is an antisymmetric matrix, the partition function is given by the Berezin integral
Z0 =
Z
d2m✓ e�A(✓,✓)/2~ =
Z
d2m✓
mX
n=0
(�)n
(2~)nn! (Aab✓
a✓b)n
=
(�)m
(2~)mm!
Z
d2m✓ Aa1a2Aa3b4 · · · Aa2m�1a2m ✓a1✓a2 · · · ✓a2m�1✓a2m
=
(�)m
(2~)mm!✏
a1a2···a2m�1a2mAa1a2 · · · Aa2m�1a2m ,
(2.78)
– 35 –
where we note that only the mth term of the expansion can contribute. The Pfa�an of a
2m ⇥ 2m antisymmetric matrix A is given by24
Pfa↵(A) =
1
2mm!
✏a1a2···a2m�1a2mAa1a2 · · · Aa2m�1a2m (2.79)
and in the first problem set, I ask you to use Grassmann variables to show that (Pfa↵ A)2 =
det A.
In summary, the partition function of n = 2m free fermions can be written as
Z0 = ±
r
det(A)
~n , (2.80)
whereas for n free bosons we had Z0 =
p
(2⇡~)n/ det(M) with M a symmetric matrix.
Except for a numerical factor (which we could in any case include in the normalization of
the measure), the fermionic result is just the inverse of the bosonic one. We’ll see various
important consequences of this fact later.
We may also consider the partition function in the presence of sources. Since we want
the action to be bosonic, the source itself must now be fermionic and we denote it by ⌘.
Let
S(✓, ⌘) =
1
2
Aab✓
a✓b + ⌘a✓
b . (2.81)
Completing the square as before gives25
S(✓, ⌘) =
1
2
�
✓a + ⌘c(A
�1)ca
�
Aab
⇣
✓b + ⌘d(A
�1)db
⌘
+
1
2
⌘a(A
�1)ab⌘b , (2.82)
so using the translational invariance of the measure dn✓, the partition function in the
presence of sources is
Z0(⌘) = exp
✓
� 1
2~A
�1(⌘, ⌘)
◆
Z0(0) . (2.83)
As before this allows us to compute correlation functions of the fermion fields. As an
example, the two–point function
h✓a✓bi = ~
2
Z0(0)
@2Z0(⌘)
@⌘a@⌘b
����
⌘=0
= ~ (A�1)ab . (2.84)
which is just the inverse of the kinetic term for the ✓s and plays the role of the ‘propagator’
in this d = 0 theory. Notice that this propagator is the same (not the inverse) of the
propagator we’d obtain in the bosonic theory, except that for fermions the matrix A (and
hence A�1) must be antisymmetric, whereas for bosons M�1 was symmetric.
The fact that functions of a finite number of Grassmann variables can always be
represented as polynomials means that in d = 0, we never need use perturbation theory
24For example, Pfa↵
 
0 a
�a 0
!
= a .
25It’s a good exercise to go through this and check you’re comformtable with all the signs, both here and
in the calculation of the two–point function below.
– 36 –
to evaluate fermionic path integrals: It’s always possible to perform finitely many Berezin
integrations exactly. Nonetheless, for a nonlinear theory such as
S(✓) =
1
2
Aab✓
a✓b +
1
4!
�abcd ✓
a✓b✓c✓d (2.85)
we can, if we choose, construct Feynman diagrams with propagator ~A�1 and vertex
��abcd/~. We then construct Feynman diagrams with these ingredients in just the same
way as for the bosonic theory. In higher dimensional QFT, fermions will be described by
Grassmann valued fields, so we’ll have infinitely many Grassmann variables over which
to integrate (we’ll understand this better later). With infinitely many Grassmann vari-
ables, the situation for fermions is really no di↵erent from bosons, in the sense that in
both cases it is usually necessary to work perturbatively and compute an asymptotic series
approximation to the full path integral.
2.5.2 Supersymmetry and localization
For a generic QFT, the asymptotic series is as good a representation of the partition
function (or correlation functions) as we can hope for, barring numerics. However, if the
action is of a very special type, it may sometimes possible to evaluate the partition function
and even certain correlation functions exactly. There are many mechanisms by which this
might happen; this section gives a toy model of one of them, known as localization in
supersymmetric theories.
Let’s take a theory where that in addition to our bosonic field �, we have two fermionic
fields 1 and 2. With a zero–dimensional space–time, the space of fields is just R1|2. Given
an action S(�, i) the partition function is, as usual,
Z =
Z
d� d 1 d 2p
2⇡
e�S(�, i) (2.86)
where I’ve thrown a factor of 1/
p
2⇡ into the measure for later convenience. Generically,
we’d have to be content with a perturbative evaluation of Z, using Feynman diagrams
formed from edges for the � and i fields, together with vertices from all the di↵erent
vertices that appear in our action. For a complicated action, even low orders of the per-
turbative expansion might be di�cult to compute in general.
However, let’s suppose the action takes the special form
S(�, 1, 2) =
1
2
(@h)2 � 1 2 @2h (2.87)
where h(�) is some (R-valued) polynomial in � and @h is its derivative wrt �. Note that
there can’t be any terms in S involving only one of the fermion fields since this term would
itself be fermionic. There also can’t be higher order terms in the fermion fields since 2i = 0
for a Grassmann variable, so the only thing special about this action is the relation between
the purely bosonic piece and the second term involving 1 2.
Now consider the transformations
�� = ✏1 1 + ✏2 2 , � 1 = ✏2@h , � 2 = �✏1@h (2.88)
– 37 –
where ✏i are fermionic parameters. These are supersymmetry transformations in this zero–
dimensional context; take the Part III Supersymmetry course to meet supersymmetry in
higher dimensions. The most important property of these transformations is that they are
nilpotent26. Under (2.88) the action (2.87) transforms as
�S = @h @2h(✏1 1 + ✏2 2) � (✏2@h) 2 @2h � 1(�✏1@h)@2h = 0 (2.89)
and is thus invariant — this is what the special relation between the bosonic and fermionic
terms in S buys us. (To obtain this result we used the fact that Grassmann variables
anticommute.) It’s also true that the integral measure d� d2 is likewise invariant; I’ll
leave this too as an exercise.
Supersymmetric QFTs are drastically simpler than generic ones, especially in zero
dimensions. Let �O be the supersymmetry variation of some operator O(�, i) and consider
the correlation function h�Oi. Since �S = 0 we have
h�Oi = 1Z0
Z
d� d2 e�S �O = 1Z0
Z
d� d2 �
�
e�SO
�
. (2.90)
The supersymmetry variation here acts on both � and the fermions i in e�SO. But if it
acts on a fermion i then the resulting term does not contain that i and hence cannot
contribute to the integral because
R
d 1 = 0 for Grassmann variables. On the other hand,
if it acts on � then while the resulting term may survive the Grassmann integral, it is a
total derivative in the � field space. Thus, provided O does not disturb the decay of e�S
as |�| ! 1, any such correlation function must vanish, h�Oi = 0.
In particular, if we choose Og = @g 1 for some g(�), then setting the parameters
✏1 = �✏2 = ✏ we have
0 = h�Ogi = ✏h@g @h � @2g 1 2i . (2.91)
The significance of this is that the quantity @g @h � @2g 1 2 is the first–order change in
the action under the deformation h ! h+g, again so long as g does not alter the behaviour
of h as |�| ! 1. The fact that h�Ogi = 0 tells that the partition function Z[h], which
we might think depends on all the couplings in the vertices in the polynomial h, is in
fact largely insensitive to the detailed form of h because we can deform it by any other
polynomial of the same degree or lower. The most important case is if we choose g to be
proportional to h, then our deformation just rescales h ! (1 + �)h and so we see that
Z[h] is independent of the overall scale of h. By iterating this procedure, we can imagine
rescaling h by a large factor so that the bosonic part of the action (@h)2/2 ! ⇤2(@h)2/2.
As ⇤ ! 1, the factor e�S exponentially suppresses any contribution to Z except from an
infinitesimal neighbourhood of thecritical points of h where @h = 0. This phenomenon is
known as localization of the path integral.
It’s now straightforward to work out the partition function. Near any such critical
point �⇤ we have
h(�) = h(�⇤) +
c⇤
2
(�� �⇤)2 + · · · (2.92)
26That is, �21 = 0, �
2
2 = 0 and [�1, �2] = 0, where �1 is the transformation with parameter ✏2 = 0, etc..
You should check this from (2.88) as an exercise!
– 38 –
+1
+1
�1
�
h(�)
�1
Figure 3: The supersymmetric path integral receives contributions just from infinitesimal
neighbourhoods of the critical points of h(�). These alternately contribute ±1 according to
whether they are minima or maxima.
where c⇤ = @2h(�⇤), so the action (2.87) becomes
S(�, i) =
c2⇤
2
(�� �⇤)2 + c⇤ 1 2 + · · · . (2.93)
The higher order terms will be negligible as we focus on an infinitesimal neighbourhood
of �⇤. Expanding the exponential in Grassmann variables the contribution of this critical
point to the partition function is
1p
2⇡
Z
d� d2 e�c
2
⇤(���⇤)2/2 [1 � c⇤ 1 2] =
c⇤p
2⇡
Z
d� e�c
2
⇤(���⇤)2/2
=
c⇤p
c2⇤
= sgn
�
@2h|⇤
�
.
(2.94)
Summing over all the critical points, the full partition function thus becomes
Z[h] =
X
�⇤ : @h|�⇤= 0
sgn
�
@2h|�⇤
�
(2.95)
and, as expected, is largely independent of the detailed form of h. In fact, if h is a
polynomial of odd degree, then @h = 0 must have an even number of roots with @2h being
alternately > 0 and < 0 at each. Thus their contributions to (2.95) cancel pairwise and
Z[hodd] = 0 identically. On the other hand, if h has even degree then it has an odd number
of critical points and we obtain Z[hev] = ±1, with the sign depending on whether h ! ±1
as |�| ! 1. (See figure 3.)
The fact that the partition function is so simple in this class of theories is a really
remarkable result! To reiterate, we’ve found that for any form of polynomial h(�), the
partition function Z[h] is always either 0 or ±1. If we imagined trying to compute Z[h]
perturbatively, then for a non–quadratic h we’d still have to sum infinitely diagrams us-
ing the vertices in the action. In particular, we could certainly draw Feynman graphs �
with arbitrarily high numbers of loops involving both � and i fields, and these graphs
– 39 –
would each contribute to the coe�cient of some power of the coupling constants in the
perturbative expansion. However, by an apparent miracle, we’d find that these graphs al-
ways cancel themselves out; the net coe�cient of each such loop graph would be zero with
the contributions from graphs where either � or 1 2 run around the loop contributing
with opposite sign. The reason for this apparent perturbative miracle is the localization
property of the supersymmetric integral.
In supersymmetric theories in higher dimensions, complications such as spin mean the
cancellation can be less powerful, but it is nonetheless still present and is responsible for
making supersymmetric quantum theories ‘tamer’ than non–supersymmetric ones. As an
important example, diagrams where the Higgs particle of the Standard Model runs around
a loop can have the e↵ect of destabilizing the mass of the Higgs, sending it up to a very
high scale. (We’ll understand this later on.) Until very recently, many physicists believed
in the existence of a hypothesized supersymmetric partner to the Higgs that would cancel
these dangerous loop diagrams, protecting the mass of the Higgs and thereby providing a
rationale why the natural energy scale of the weak interactions is so much lower than the
Planck scale. The ultimate mechanism for this cancellation would be just what we’ve seen
above, though it’s power is filtered through the layers of a much more complicated theory.
Experiment has now shown that supersymmetry – if it is relevant to Nature at all – is not
responsible for looking after the Higgs mass in this way27.
I also want to point out that localization is useful for calculating much more than just
the partition function. For i 2 {1, 2, 3, . . .} suppose that Oi(�, i) is an operator that obeys
�Oi = 0, i.e. each operator is invariant under supersymmetry transformations (2.88). Then
the (unnormalized) correlation function
*
Y
i
Oi
+
=
Z
d� d2 p
2⇡
e�S
Y
i
Oi (2.96)
again localizes to the critical points of h. Once again, this is because deforming h ! h + g
leaves the correlator invariant since the deformation a↵ects the correlation function as
*
Y
i
Oi
+
h!h+g�!
*
�Og
Y
i
Oi
+
=
*
�
 
Og
Y
i
Oi
!+
= 0 (2.97)
which vanishes by the same arguments as before. Here, we used the fact that �Oi = 0 to
write the operator on the rhs as a total derivative.
Of course, if any of the Oi are already of the form �O0, so that this Oi is itself the
supersymmetry transformation of some O0, then h
Q
Oii = 0 which is not very interesting.
The interesting operators are those which are �-closed (�O = 0) but not �-exact (O 6= �O0).
These operators describe the cohomology of the nilpotent operator �. This is the starting–
point for much of the mathematical interest in QFT: we can build supersymmetric QFTs
that compute the cohomology of interesting spaces. For example, Donaldson’s theory of
27Whether anything protects the Higgs mass, or whether it is just fine-tuned, is currently one of the
outstanding mysteries of Beyond the Standard Model physics.
– 40 –
invariants of 4–manifolds that are homeomorphic but not di↵eomorphic, and the Gromov–
Witten generalization of intersection theory can both be understood as examples of (higher–
dimensional) supersymmetric QFTs where the localization / cancellation is precise. In the
absence of experimental evidence for a supersymmetric extension of the Standard Model,
the close connections between supersymmetric QFTs and deep mathematics and the fact
that supersymmetry helps tame otherwise intractable path integrals now provide the main
reasons for studying supersymmetry.
Finally, let me remark that we’ll also meet essentially the same localization idea again
in a slightly di↵erent context later in this course when we study BRST quantization of
gauge theories.
– 41 –
3 QFT in one dimension (= QM)
In one dimension there are two possible compact (connected) manifolds M : the circle S1
and the interval I. We will parametrize the interval by t 2 [0, T ] so that t = 0 and t = T
are the two point–like boundaries, while we will parametrize the circle by t 2 [0, T ) with
the identification t ⇠= t + T .
The most important example of a field on M is a map x : M ! N to a Riemannian
manifold (N, G) which we will take to have dimension n. That is, for each point t on our
‘space–time’ M , x(t) is a point in N . It’s often convenient to describe N using coordinates.
If an open patch U ⇢ N has local co-ordinates xa for a = 1, . . . , n, then we let xa(t) denote
the coordinates of the image point x(t). More precisely, xa(t) are the pullbacks to M of
coordinates on U by the map x.
With these fields, the standard choice of action is
S[x] =
Z
M

1
2
Gab(x)ẋ
aẋb + V (x)
�
dt , (3.1)
where Gab(x) is the pullback to M of the Riemannian metric on N , t is worldline time, and
ẋa = dxa/dt. We’ve also included in the action a choice of function V : N ! R, or more
precisely the pullback of this function to M , which is independent of worldline derivatives
of x. Finally, when writing this action we chose the flat Euclidean metric �tt = 1 on M ;
we’ll examine other choices of metric on M in section 3.4.
Under a small variation �x of x the change in the action is
�S[x] =
Z
M

Gab(x) ẋ
a ˙�x
b
+
1
2
@Gab(x)
@xc
�xc ẋaẋb +
@V (x)
@xc
�xc
�
dt
=
Z
M

� d
dt
(Gac(x)ẋ
a) +
1
2
@Gab(x)
@xc
ẋaẋb +
@V (x)
@xc
�
�xc dt + Gab(x) ẋ
a �xb
���
@M
.
(3.2)
Requiring that the bulk term vanishes for arbitrary �xa(t) gives the Euler–Lagrange equa-
tions
d2xa
dt2
+ �abcẋ
bẋc = Gab(x)
dV
dxb
(3.3)
where �abc =
1
2G
ad (@bGcd + @cGbd � @dGbc) is the Levi–Civita connection on N , again
pulled back to the worldline. If M has boundary,then the boundary term is the sym-
plectic potential on the space of maps, where we note that pa = �L/�ẋa = Gab(x)ẋb is the
momentum of the field.
3.1 Worldline quantum mechanics
The usual interpretation of all this is to image an arbitrary map x(t) describes a possible
trajectory a particle might in principle take as it travels through the space N . (See figure 4.)
In this context, N is called the target space of the theory, while M (or its image x(M) ⇢
N) is known as the worldline of the particle. The field equation (3.3) says that when
V = 0, classically the particle travels along a geodesic in (N, G). V itself is then interpreted
as a (non–gravitational) potential through which this particle moves. The absence of a
– 42 –
(N, G)
0 T
x(t)
��
y
1/(k2 + M2)
x
1/(k2 + m2)
; + ++ + + · · ·
= 1 ++ + + · · ·
=Z/Z0
~�
8m4
� ~
2�2
48m8
~2�2
16m8
~2�2
128m8
and
Figure 4: The theory (3.1) describes a map from an abstract worldline into the Rieman-
nian target space (N, G). The corresponding one–dimensional QFT can be interpreted as
single particle Quantum Mechanics on N .
minus sign on the rhs of (3.3) is probably surprising, but follows from the action (3.1).
This is actually the correct sign with a Euclidean worldsheet, because under the Wick
rotation t ! it back to a Minkowski signature worldline, the lhs of (3.3) acquires a minus
sign. In other words, in Euclidean time F = �ma!
From this perspective, it’s natural to think of the target space N as being the world in
which we live, and computing the path integral for this action will lead us to single particle
Quantum Mechanics, as we’ll see below. However, we’re really using this theory as a further
warm–up towards QFT in higher dimensions, so I also want you to keep in mind the idea
that the worldline M is actually ‘our space–time’ in a one–dimensional context, and the
target space N can be some abstract Riemannian manifold unrelated to the space we see
around us. For example, at physics of low–energy pions is described by a theory of this
general kind, where M is our Universe and N is the coset manifold (SU(2)⇥SU(2))/SU(2).
3.1.1 The quantum transition amplitude
The usual way to do Quantum Mechanics is to pick a Hilbert space H and a Hamiltonian
H, which is a Hermitian operator H : H ! H. In the case relevant above, the Hilbert
space would be L2(N), the space of square–integrable functions on N , and the Hamiltonian
would usually be
H = �~
2
2
�+ V , where � :=
1p
G
@
@xa
✓p
GGab
@
@xb
◆
(3.4)
is the Laplacian acting on functions in L2(N). The amplitude for the particle to travel
from an initial point y0 2 N to a final point y1 2 N in Euclidean time T is given by
KT (y0, y1) = hy1|e�HT/~|y0i , (3.5)
which is known as the heat kernel. (Here I’ve written the rhs in the Heisenberg picture,
which I’ll use below. In the Schrödinger picture where states depend on time we would
instead write KT (y0, y1) = hy1, T |y0, 0i.) The heat kernel is a function on I ⇥N ⇥N which
– 43 –
may be defined to be the solution of the pde28
~ @
@t
Kt(x, y) + HKt(x, y) = 0 (3.6)
subject to the initial condition that K0(x, y) = �(x � y), the Dirac �-function on the
diagonal N ⇢ N ⇥N . (I remind you that we’re in Euclidean worldline time here. Rotating
to Minkowski signature by sending t 7! it, the heat equation becomes
i~ @
@t
Kit(x, y) = HKit(x, y) (3.7)
which we recognize as Schrödinger’s equation with Hamiltonian H.)
In the simplest example of (N, G) ⇠= (Rn, �) with vanishing potential V ⌘ 0, the
Hamiltonian is just
H = �~
2
2
@2
@xa@xa
(3.8)
and the heat kernel takes the familiar form
Kt(x, y) =
1
(2⇡~t)n/2
exp
✓
�kx � yk
2
2~t
◆
(3.9)
where kx � yk is the Euclidean distance between x and y. More generally, while the heat
kernel on a Riemannian manifold (N, G) is typically very complicated, it can be shown
that for small times it always has the asymptotic form
lim
�t!0
K�t(x, y) ⇠
1
(2⇡~�t)n/2
a(x) exp
✓
�d(x, y)
2
2~�t
◆
, (3.10)
where d(x, y) is the distance between x and y measured along a geodesic of the metric G,
and where
a(x) ⇠
p
G(x) [1 + RicG(x) + · · · ] (3.11)
is an expression constructed from the Riemann curvature of G in a way that we won’t need
to be specific about.
Feynman’s intuition was that the amplitude for a particle to be found at y0 at t = 0
and at y1 at t = T could be expressed in terms of the product of the amplitude for it to start
at y0 at t = 0, then be found at some other location x at an intermediate time t 2 (0, T ),
before finally being found at y1 on schedule at t = T . Since we did not measure what
the particle was doing at the intermediate time, we should sum (i.e. integrate) over all
possible intermediate locations x in accordance with the linearity of quantum mechanics.
Iterating this procedure, as in figure 5 we break the time interval [0, T ] into N chunks, each
28Like the factor of 1/2 in front of the Laplacian in (3.4), I’ve included a factor of ~ in this equation for
better agreement with the conventions of quantum mechanics, rather than Brownian motion. If you wish,
you can imagine we’re studying the usual heat equation in a medium with thermal conductivity ~/2.
– 44 –
Figure 5: Feynman’s approach to quantum mechanics starts by breaking the time evolution
of a particle’s state into many chunks, then summing over all possible locations (and any
other quantum numbers) of the particle at intermediate times.
of duration �t = T/N . We then write
hy1|e�HT/~|y0i = hy1|e�H�t/~ e�H�t/~ · · · e�H�t/~|y0i
=
Z
dnx1 · · · dnxN�1 hy1|e�H�t/~|xN�1i · · · hx2|e�H�t/~|x1i hx1|e�H�t/~|y0i
=
Z N�1Y
i=1
dnxi K�t(y1, xN�1) · · · K�t(x2, x1) K�t(x1, y0) .
(3.12)
In the second line here we’ve inserted the identity operator
R
dnxi |xiihxi| on H in between
each evolution operator; in the present context this can be understood as the concatenation
identity
Kt1+t2(x3, x1) =
Z
dnx2 Kt2(x3, x2) Kt1(x2, x1) (3.13)
obeyed by convolutions of the heat kernel.
This more or less takes us to the path integral. The virtue of splitting up the time
interval [0, T ] into many chunks is that we can now use the asymptotic form (3.10) to write
hy1|e�HT/~|y0i ⇠
1
(2⇡~�t)n/2
Z N�1Y
i=1
dnxi
(2⇡~�t)n/2
a(xi) exp
"
�1~
NX
i=0
�t
2
✓
d(xi+1, xi)
�t
◆2#
.
(3.14)
We now consider take the limit that N ! 1 with T fixed (so �t ! 0). We might then
hope that we can define our path integral measure to be
Dx ?:= lim
N!1
✓
1
2⇡~�t
◆nN
2
N�1Y
i=1
dnxi a(xi) (3.15)
as an integral over the values of the fields xa(t) at each time t 2 [0, T ]. Similarly, if
the limiting trajectory is at least once di↵erentiable then as �t ! 0, (d(xi+1, xi)/�t)2
– 45 –
converges to gab ẋaẋb while the sum can be replaced by an integral, so we would have
lim
N!1
"
NX
i=0
�t
2
✓
d(xi+1, xi)
�t
◆2#
?
=
Z T
0
1
2
Gab(x) ẋ
aẋb dt . (3.16)
This recovers the action (3.1), with V = 0. (A more general heat kernel can be used to
incorporate a non–zero potential.) We’ll investigate these limits further below; accepting
them for now, combining (3.15) & (3.16) allows us to represent the heat kernel as an integral
hy1|e�HT/~|y0i = KT (y0, y1) =
Z
CT [y0,y1]
Dx e�S[x]/~ (3.17)
taken over a space CT [y0, y1] of maps x : [0, T ] ! N that are constrained to obey the
boundary conditions x(0) = y0 and x(T ) = y1. A given map is called a path and the
integral over all such paths is the path integral. We’ll investigate exactly what sort of
maps we should allow (smooth? di↵erentiable? continuous?) in more detail below. Note
that from our d = 1 QFT perspective, the path integral gives the amplitude for a field
configuration x = y0 on an initial codimension-1 slice (i.e. the point t = 0) to evolve
through M = [0, T ] and emerge as the field configuration x = y1 on the final codimension-1
slice (i.e. the point T ). Thus, it’s a sort of scattering amplitude y0 ! y1 in our one
dimensional universe. (The name ‘path integral’ is also used in higher dimensionalQFT.)
3.1.2 The partition function
In the operator approach to quantum mechanics, the partition function is defined to be
the trace of the time evolution operator over the Hilbert space:
Z(T ) = TrH(e�TH) . (3.18)
In the case of a single particle moving on Rn, we can take the position eigenstates |yi to
be a (somewhat formal) ‘basis’ of H = L2(Rn, dny), in which case the partition function
becomes
Z(T ) =
Z
dny hy|e�HT |yi =
Z
N
dny
Z
CT [y,y]
Dx e�S (3.19)
where the last equality uses out path integral expression (3.17) for the heat kernel. Because
we’re taking the trace, the path integral here should be taken over maps x : [0, T ] ! N
such that the endpoints are both mapped to the same point y 2 N . We then integrate y
everywhere over N29, erasing the memory of the particular point y. As long as we’re being
vague about the degree of di↵erentiability of our map, this is (plausibly) the same thing as
integrating over maps x : S1 ! N where the worldline has become a circle of circumference
T . This shows that
ZS1 [T ] = TrH(e�TH) =
Z
C
S1
Dx e�S/~ , (3.20)
29In flat space, the heat kernel (3.9) obeys KT (y, y) = KT (0, 0) so is independent of y. Thus if N ⇠= Rn
with a flat metric, this final y integral does not converge. This is an ‘infra-red’ e↵ect that arises because
(Rn, �) is non-compact. The partition function does converge if N is compact, which we can achieve by
imposing that we live in a large box, or on a torus etc., whilst still keeping a flat metric.
– 46 –
which was our earlier definition of the partition function on the compact universe M = S1.
In higher dimensions this formula will be the basis of the relation between QFT and
Statisical Field Theory, and is really the origin of the name ‘partition function’ for Z in
physics.
3.1.3 Operators and correlation functions
As in zero dimensions, we can also use the path integral to compute correlation functions
of operators. A local operator is one which depends on the field only at one point of the
worldline. The simplest type of local operator comes from a function on the target space.
If O : N ! R is a real–valued function on N , let Ô denote the corresponding operator
on H. That is, O depends only on the local coordinates xa and Ô = O(x̂a) is the same
function of the position operator x̂a acting on H. Then for any fixed time t 2 (0, T ) we
have
hy1|e�HT/~ Ô(t)|y0i = hy1|e�H(T�t)/~ Ô e�Ht/~|y0i (3.21)
in the Heisenberg picture. Inserting a complete set of position eigenstates, this is
Z
dnx hy1|e�H(T�t)/~ O(x̂)|xi hx|e�Ht/~|y0i =
Z
dnx O(x) hy1|e�H(T�t)/~|xi hx|e�Ht/~|y0i
=
Z
dnx O(x) KT�t(y1, x) Kt(x, y0) ,
(3.22)
where we note that in the final two expressions O(x) is just a number; the eigenvalue of
the operator Ô acting on the state |xi.
Using (3.17), everything on the rhs of this equation can now be written in terms of
path integrals. We have
hy1|e�H(T�t)/~ Ô e�Ht/~|y0i =
Z
dnxt
"Z
CT�t[y1,xt]
e�S[x]/~ ⇥ O(xt) ⇥
Z
Ct[xt,y0]
e�S[x]/~
#
=
Z
CT [y1,y0]
Dx e�S[x]/~ O(x(t)) ,
(3.23)
where to reach the second line we again note that integrating over all continuous maps
x : [0, t] ! N with endpoint x(t) = xt, then over all continuous maps x : [t, T ] ! N with
initial point x(t) again fixed to xt and finally integrating over all points xt 2 N , is the same
thing as integrating over all continuous maps x : [0, T ] ! N with endpoints y0 and y1.
More generally, we can insert several such operators. If 0 < t1 < t2 < . . . < tn < T
then exactly the same arguments give
hy1|Ôn(tn) · · · Ô2(t2) Ô1(t1)|y0i
= hy1|e�H(T�tn)/~On(x̂) · · · O2(x̂) e�H(t2�t1)/~ O1(x̂) e�Ht1/~|y0i
=
Z
CT [y0,y1]
Dx e�S[x]/~
nY
i=1
Oi(x(ti))
(3.24)
– 47 –
for the n–point correlation function. The hats on the Ôi remind us that the lhs involves
operators acting on the Hilbert space H (which we may or may not choose to describe using
the position representation). By contrast, the objects Oi inside the path integral are just
ordinary functions, evaluated at the points x(ti) 2 N30.
Notice that in order to run our argument, it was very important that the insertion times
ti obeyed ti < ti+1: we would not have been able to interpret the lhs in the Heisenberg
picture had this not been the case31. On the other hand, the insertions Oi(x(ti)) in the
path integral are just functions and have no notion of ordering. Thus the expression on
the right doesn’t have any way to know which insertion times was earliest. For this to be
consistent, for a general set of times {ti} 2 (0, T ) we must actually have
Z
CT [y0,y1]
Dx
 
e�S[x]/~
nY
i=1
Oi(x(ti))
!
= hy1|T {
Y
i
Ôi(ti)}|y0i (3.25)
where the symbol T on the rhs is defined by
T Ô1(t1) := O1(t1) ,
T {Ô1(t1) Ô2(t2)} := ⇥(t2 � t1) Ô2(t2) Ô1(t1) +⇥(t1 � t2) Ô1(t1) Ô2(t2)
(3.26)
and so on, where ⇥(t) is the Heaviside step function and the operators are in the Heisen-
berg picture. By construction, these step functions mean that the rhs is now completely
symmetric with respect to a permutation of the orderings. However, for any given choice
of times ti, only one term on the rhs can be non–zero. In other words, insertions in the
path integral correspond to the time–ordered product of the corresponding operators
in the Heisenberg picture.
The derivative terms in the action play an important role in evaluating these correla-
tion functions. For suppose we’d chosen our action to be just a potential term
R
V (x(t)) dt,
independent of derivatives ẋ(t). Then, regularizing the path integral by dividing M into
many small intervals as before, we’d find that neighbouring points on the worldline com-
pletely decouple: unlike in (3.14) where the geodesic distance d(xi+1, xi)2 in the heat
kernel provides cross–terms linking neighbouring points together, we would obtain simply
a product of independent integrals at each time step. Inserting functions Oi(x(ti)) that
are likewise independent of derivatives of x into such a path integral would not change this
conclusion. Thus, without the derivative terms in the action, we’d find
hO1(t1) O2(t2)i = hO1(t1)i hO2(t2)i (3.27)
for all such insertions. In other words, there would be no possible non–trivial correlations
between objects at di↵erent points of our (one–dimensional) Universe. This would be a
very boring world: without derivatives, the number of people sitting in the lecture theatre
30A more precise statement would be that they are functions on the space of fields CT [y0, y1] obtained
by pullback from a function on N by the evaluation map at time ti.
31It’s a good exercise to check you understand what goes wrong if we try to compute hy1|e
+tH/~
|y1i with
t > 0.
– 48 –
would have nothing at all to do with whether or not a lecture was actually going on, and
what you’re thinking about right now would have nothing to do with what’s written on
this page.
This conclusion is a familiar result in perturbation theory. The kinetic terms in the
action allow us to construct a propagator, and using this in Feynman diagrams enables
us to join together interaction vertices at di↵erent points in space–time. As the name
suggests, we interpret this propagator as a particle traveling between these two space–time
interactions and the ability for particles to move is what allows for non–trivial correlation
functions. Here we’ve obtained the same result directly from the path integral.
So far, we’ve just been considering insertions that are functions of position only. A
wider class of local path integral insertions depend not just on x but also on its worldline
derivatives ẋ, ẍ etc.. In the canonical framework, with Lagrangian L we have
pa =
�L
�ẋa
= Gabẋ
b (3.28)
where the last equality is for our action (3.1). Thus we might imagine replacing a general
operator O(x̂a, p̂b) in the canonical quantization framework by the function O(xa, Gbc(x)ẋc)
of x and its derivative in the path integral. From the path integral perspective, however,
something smells fishy here. Probably the firstthing you learned about QM was that
[x̂a, p̂b] 6= 0. If we replace x̂a and p̂b by xa and Gbcẋc in the path integral, how can these
functions fail to commute, even when Gab = �ab? To understand what’s going on, we’ll
need to look into the definition of our path integral in more detail.
By the way, we should note that there’s an important other side to this story, revealing
ambiguities in the canonical approach to quantum mechanics. Suppose we’re given some
function O(xa, pb) on a classical phase space, corresponding to some observable quantity .
If we wish to ‘quantize’ this classical system, it may not be obvious to decide what operator
to use to represent our observable as the replacement
O(xa, pb) ! O(x̂a, p̂b)
is plagued by ordering ambiguities. For example, if we represent pa by32 �~@/@xa, then
should we replace
xapa ! �xa ~
@
@xa
or should we take
xapa ! �~
@
@xa
xa = �~n � xa ~ @
@xa
or perhaps something else? According to Dirac, if two classical observables f and g have
Poisson bracket {f, g} = h for some other function h, then we should quantize by finding
a Hilbert space on which the corresponding operators f̂ and ĝ i) act irreducibly and ii)
obey [f̂ , ĝ] = i~ ĥ. Unfortunately, even in flat space quantum mechanics with (N, G, V ) =
(Rn, �, 0), the Groenewald–Van Hove theorem states that we cannot generally achieve this,
even for functions that are polynomial in position and momenta, of degree higher than 2.
32The absence of a factor of i on the rhs here is again a consequence of having a Euclidean worldline.
– 49 –
http://www.pims.math.ca/~gotay/r2n.pdf
(We can make progress if some extra structure is present, such as if the operators represent
the action of some finite dimensional Lie group, or if the phase space is provided with
a complex structure, making it a Kähler manifold.) The idea of ‘quantizing’ a classical
system thus remains ambiguous in general.
3.2 The continuum limit
In this section, we’ll take a closer look at the origin of non–commutativity of xa and pb
from the path integral perspective. Doing so will lead to a deeper understanding of the
subtleties involved in taking the naive continuum limit of the path integral measure and
action.
3.2.1 Discretization and non–commutativity
Non–commutativity is present in quantum mechanics right from the beginning, so it will
su�ce to consider the simplest case of a free particle travelling in one dimension. We thus
pick (N, G) = (R, �) and V = 0. Then if 0 < t� < t < t+ < T we have
Z
CT [y0,y1]
Dx e�S[x]/~ x(t) ẋ(t�) = hy1|e�H(T�t)/~ x̂ e�H(t�t�)/~ p̂ e�Ht�/~|y0i , (3.29a)
when the insertion of p̂ is earlier than that of x̂, and
Z
CT [y0,y1]
Dx e�S[x]/~ x(t) ẋ(t+) = hy1|e�H(T�t+)/~ p̂ e�H(t+�t)/~ x̂ e�Ht/~|y0i (3.29b)
when p̂ is inserted at a later time than x̂. Taking the limits t+ ! t from above and t� ! t
from below, the di↵erence between the rhs of (3.29a) & (3.29b) is
hy1|e�H(T�t)/~ [ x̂, p̂ ] e�Ht~|y0i = ~ hy1|e�HT /~|y0i (3.30)
which does not vanish. By contrast, the di↵erence of the lhs seems to be automatically
zero. What have we missed?
In handling the lhs of (3.29a)-(3.29b) we need to be careful. Our arguments allowed
us to be confident of the relation between the canonical and path integral approaches only
when working with some discretization of M = [0, T ]. Taking the continuum limit to obtain
a path integral measure Dx and action S[x] was a formal operator and we did not check
that these limits actually make sense.
To stay on safe ground, let’s regularize the path integrals in (3.29a)-(3.29b) by chopping
[0, T ] into many chunks, each of width �t. With this discretization, we cannot pretend we
are bringing the x and ẋ insertions any closer to each other than �t without also taking
account of the discretization of the whole path integral. Thus we replace
lim
t�" t
[x(t) ẋ(t�)] � lim
t+# t
[x(t) ẋ(t+)]
by the discretized version
xt
xt � xt��t
�t
� xt
xt+�t � xt
�t
(3.31)
– 50 –
where we stop the limiting procedure as soon as x coincides with any part of the discretized
derivative. As always with path integrals, the order of the factors of xt and xt±�t here
doesn’t matter; they’re just ordinary integration variables.
Now consider the integral over xt. Apart from the insertion of (3.31), the only depen-
dence of the discretized path integral on this variable is in the heat kernels K�t(xt+�t, xt)
and K�t(xt, xt��t) that describe the evolution from neighbouring chunks of our discretized
worldline. Using the explicit form (3.9) of these kernels in flat space we have
Z
dxt K�t(xt+�t, xt)
✓
xt
xt � xt��t
�t
� xt
xt+�t � xt
�t
◆
K�t(xt, xt��t)
= �~
Z
dxt xt
@
@xt
✓
K�t(xt+�t, xt) K�t(xt, xt��t)
◆
= ~
Z
dxt K�t(xt+�t, xt) K�t(xt, xt��t) = ~ K2�t(xt+�t, xt��t)
(3.32)
where the first step recognizes the two insertions as being ~ times the xt derivatives of
K�t(xt+�t, xt) and K�t(xt, xt��t), respectively. The second step is a simple integration
by parts and the final equality uses the concatenation property (3.13). The integration over
xt thus removes all the insertions from the path integral, and the remaining integrals can
be done using concatenation as before. We are thus left with ~ KT (y1, y0) = ~ hy1|e�HT |y0i
in agreement with the operator approach.
There’s an important point to notice about this calculation. If we’d assumed that,
in the continuum limit, our path integral included only maps x : [0, T ] ! N whose first
derivative was everywhere continuous, then the limiting value of (3.31) would necessarily
vanish when �t ! 0, contradicting the operator calculation. Non–commutativity arises in
the path integral approach to quantum mechanics precisely because we’re forced to include
non–di↵erentiable paths, i.e. our map x 2 C0(M, N) but x /2 C1(M, N). In fact, since we
want to recover the non–commutativity no matter at which time t we insert x̂ and p̂, we
need path that are nowhere di↵erentiable.
This non–di↵erentiability is the familiar stochastic (‘jittering’) behaviour of a particle
undergoing Brownian motion. It’s closely related to a very famous property of random
walks: that after a time interval t, one has moved through a net distance proportional top
t rather than / t itself. More specifically, averaging with respect to the one–dimensional
heat kernel
Kt(x, y) =
1p
2⇡~t
e�(x�y)
2/2~t ,
in time t, the mean squared displacement is
h(x � y)2i =
Z 1
�1
Kt(x, y) (x � y)2 dx =
Z 1
�1
Kt(u, 0) u
2 du = ~ t (3.33)
so that the rms displacement from the starting point after time t is /
p
t. Similarly, our
regularized path integrals yield a finite result because the average value of
xt+�t
xt+�t � xt
�t
� xt
xt+�t � xt
�t
= �t
✓
xt+�t � xt
�t
◆2
,
– 51 –
Figure 6: Stimulated by work of Einstein and Smoluchowski, Jean–Baptiste Perron made
many careful plots of the locations of hundreds of tiny particles as they underwent Brownian
motion. Understanding their behaviour played a key role in confirming the existence of
atoms. A particle undergoing Brownian motion moves an average (rms) distance of
p
t in
time t, a fact that is responsible for non–trivial commutation relations in the (Euclidean)
path integral approach to Quantum Mechanics.
which for a di↵erentiable path would vanish as �t ! 0, here remains finite.
The importance of nowhere–di↵erentiable paths has a further very important conse-
quence. Since we cannot assign any sensible meaning to
lim
�t!0
xt+�t � xt
�t
,
we cannot sensibly claim that
lim
N!1
exp
"
��t~
NX
i=0
1
2
✓
xti+1 � xti
�t
◆2#
??
= exp

�1~
Z T
0
1
2
ẋ2 dt
�
and thus we do not really have any continuum action. Naively, we might have thought that
the presence of e�S[x]/~ damps out the contribution of wild field configurations. However,
this cannot be the case: nowhere–di↵erentiable paths are essential if we wish our path
integral to know about even basic quantum properties.
3.2.2 The path integral measure
Having realized that we need to include nowhere–di↵erentiablefields, and that the contin-
uum action does not exist — even for a free particle — we now return to consider the limit
of the measure. You probably won’t be surprised to hear that this doesn’t exist either.
First recall that for vector space V of finite dimension D, dµ is a Lebesgue measure
on V if
i) it assigns a strictly positive volume vol(U) =
R
U dµ > 0 to every non–empty open set
U ⇢ V ,
– 52 –
Figure 7: In a D–dimensional vector space, an open hypercube of finite linear dimension
L contains 2D open hypercubes of linear dimension L/2 � ✏ for any L/2 > ✏ > 0. We
choose the side length to be slightly less than half the original length to ensure these smaller
hypercubes are open and non–overlapping.
ii) vol(U 0) = vol(U) whenever U 0 may be obtained from U by translation, and
iii) for every p 2 V there exists at least one open neighbourhood Up, containing p, for
which vol(Up) < 1.
The standard example of a Lebesgue measure is of course dµ = dDx on V = RD.
Now let’s return to consider the path integral measure. To keep things simple, we again
work just with the case that the target space N = Rn with a flat metric. In the continuum,
the space of fields is naturally an infinite dimensional vector space, where addition is given
by pointwise addition of the fields at each t on the worldline. In the previous section
we identified this infinite dimensional space as the space C0(M, Rn) of continuous maps
x : M ! Rn. We certainly want our measure to be strictly positive, since (in Euclidean
signature) it has the interpretation of a probability measure. Also, we used translational
invariance of the measure throughout our discussion in earlier chapters, for example in
completing the square and shifting � ! �̃ = �+ M�1(J, · ) to write the partition function
in the presence of sources as Z(J) = eM�1(J,J)/2~Z(0). So we’d like our measure Dx to be
translationally invariant, too.
But it’s easy to prove that there is no non–trivial Lebesgue measure on an infinite
dimensional vector space. Let Cx(L) denote the open (hyper)cube centered on x and of
side length L. This cube contains 2D smaller cubes Cxn(L/2 � ✏) of side length L/2 � ✏,
all of which are disjoint (see figure 7). Then
vol(Cx(L)) �
2DX
n=1
vol(Cxn(L/2 � ✏)) = 2D vol(Cx(L/2 � ✏)) (3.34)
where the first inequality uses the fact that the measure is positive–definite and the
smaller hypercubes are open and non–overlapping, and the final equality uses transla-
tional invariance. We see that as D ! 1, the only way the rhs can remain finite is if
– 53 –
vol(Cx(L/2 � ✏)) ! 0 for any finite L. So the measure must assign zero volume to any
infinite dimensional hypercube of finite linear size. Finally, provided our vector space V
is of countably infinite dimension (which the discretizes path integral makes plain), we can
cover any open U ⇢ V using at most countably many such cubes, so vol(U) = 0 for any U
and the measure must be identically zero. In particular, the limit
Dx ??= lim
N!1
NY
i=1

dnxi
(2⇡~�t)n/2
�
does not exist, and there is no measure Dx in the continuum limit of the path integral.
In fact, in one dimension, while neither Dx nor e�S[x]/~ themselves have any continuum
meaning, the limit
dµW := lim
N!1
"
NY
i=1
dnxti
(2⇡�t)n/2
exp
"
��t
2
✓
xti+1 � xti
�t
◆2##
(3.35)
of the standard measures dnxti on Rn at each time–step together with the factor e�Si does
exist as a measure on C0(M, Rn). The limit dµW is known as the Wiener measure and,
as you might imagine from our discussion above, it plays a central role in the mathematical
theory of Brownian motion. The presence of the factor e�Si/~ means that this measure
is Gaussian, rather than translationally invariant in the fields, avoiding the above no–go
theorem. However tempting it may be to interpret this as ‘obviously’ the product of a
Gaussian factor and a usual Lebesgue measure, we know from above that this cannot be
true in the continuum limit (though it is true before taking the limit).
Thus far, we’ve considered only the path integral for a free particle travelling in Rn.
Kac was able to show that the Wiener measure could also be used to provide a rigorous
definition of Feynman’s path integral for interacting quantum mechanical models. That
is, suppose our quantum particle feels a potential V : Rn ! R which contributes to its
Hamiltonian. Then, provided V is su�ciently nice33, as a path integral we have
(e�TĤ/~ )(x0) =
Z
Cx0 ([0,T ];Rn)
exp

�1~
Z T
0
V (x(s)) ds
�
 (x(T )) dµW , (3.36)
where Cx0([0, t]; Rn) is the space of continuous maps x : [0, t] ! Rn with x(0) = x0, and
where dµW is the Wiener measure on C([0, t]; Rn). I won’t prove this result here, but if
you’re curious you can consult e.g. B. Simon, Functional Integration and Quantum Physics,
2nd ed, AMS (2005), or B. Hall, Quantum Theory for Mathematicians, Springer (2013),
which also gives a fuller discussion of many of the issues we’ve considered in this section.
Note that, when evaluating an asymptotic series for the path integral using Feynman
graphs, all we ever really needed was the Gaussian measure describing the free theory: all
interaction vertices or operator insertions were treated perturbatively and evaluated using
integration against the path integral measure of the free theory.
33Technically, V must be the sum of a function in L2(Rn, dnx) and a bounded function.
– 54 –
3.3 E↵ective quantum mechanics and locality
We’ve seen that näıve interpretations of path integrals over infinite dimensional spaces can
be very misleading. Rather than try to deal directly with the infinite dimensional space
of continuous maps C0(M, N) (and the even larger, wilder spaces that arise in QFT in
d > 1) it may seem safer to always work with a regularized path integral, delaying taking
the continuum limit until the end of the calculation. However, there are any number of
finite dimensional approximations to an infinite dimensional space, and it’s far from clear
exactly which of these we should choose to define our regularized integral.
Up until now, we’ve reduced the path integral to a finite dimensional integral by
discretising our worldline M , but there many other ways to regularize. For example, even
if our field x(t) is nowhere di↵erentiable, we can represent it as a Fourier series
xa(t) =
X
k2Z
x̃ak e
2⇡ikt/T .
We might choose to regularize by truncating this series to a finite sum with |k|  N . The
(free) action for the truncated field is
SN (x̃k) =
2⇡
T
X
|k|N
k2 �ab x̃
a
k x̃
b
�k (3.37a)
and we can take the path integral over these finitely many Fourier coe�cients with measure
DxN =
NY
k=�N
dnx̃k
(2⇡)n/2
(3.37b)
If we try to include all infinitely many Fourier modes, then the sum (3.37a) will diverge
and the measure (3.37b) ceases to exist. However, with a finite cuto↵ N , we will obtain
perfectly sensible answers.
The problem, of course, is that these answers will depend on the details of how we
chose to regularize. This is not just the question of how they depend on the precise value
of N , or the precise scale of the discretization. Rather, how can we be sure whether the
results we obtain by discretizing our universe are compatible with those we’d obtain by
instead imposing a cut-o↵ on the Fourier modes of the fields? Or with any other way of
regularizing that we might dream up? The answer to this will be the subject of (Wilsonian)
renormalization in the next chapter, but we can get some flavour of it even here in d = 1.
We imagine we have two di↵erent fields x and y on the same worldline M , that I’ll
take to be a circle. We’ll start with the action
S[x, y] =
Z
S1

1
2
ẋ2 +
1
2
ẏ2 + V (x, y)
�
dt (3.38)
where the potential
V (x, y) =
1
2
(m2x2 + M2y2) +
�
4
x2y2 . (3.39)
– 55 –
In terms of the one–dimensional QFT, x and y look like interacting fields with masses m
and M , while from the point of view of the target space R2 you should think of them as
two harmonic oscillators with frequenciesm and M , coupled together in a particular way.
Of course, this coupling has been chosen to mimic what we did in section 2.4.2 in zero
dimensions. If we’re interested in perturbatively computing correlation functions of (local)
operators, then we could proceed by directly using (3.38) to construct Feynman diagrams.
We have the momentum space Feynman rules (with ~ = 1)
��
y
1/(k2 + M2)
x
1/(k2 + m2)
where k is the one–dimensional worldline momentum, which on a circle of circumference T
is quantized in units of 2⇡/T .
However, if we’re interested purely in correlators of operators that depend only on
the field x, such as hx(t2) x(t1)i, then we saw in section 2.4.2 that it’s expendient to first
integrate out the y field, obtaining an e↵ective action for the x that takes the quantum
behaviour of y into account. Let’s repeat that calculation here. As in zero dimensions, we
expect our e↵ective action will contain infinitely many new self–interactions of x. As far
as the path integral over y(t) is concerned, x is just a fixed background field so we have
formally
Z
Dy exp

�1
2
Z
S1
y
✓
� d
2
dt2
+ M2 +
�
2
x2
◆
y
�
=

det
✓
� d
2
dt2
+ M2 +
�
2
x2
◆��1/2
,
(3.40)
where (for fixed x(t)) the determinant of the di↵erential operator can be understood as
Accordingly, the e↵ective action for x is
Se↵ [x] =
Z
S1

1
2
ẋ2 +
m2
2
x2
�
dt +
1
2
ln det
✓
� d
2
dt2
+ M2 +
�
2
x2
◆
. (3.41)
Note the factor of 1/2 in front of the logarithm, which comes because we got a square root
when performing the Gaussian integral over each mode of the real field y(t). Note also
that because the e↵ective action is defined by e�Se↵ [x]/~ =
R
Dy e�S[x,y]/~ the fact that the
square root of the determinant appeared in the denominator after performing the Gaussian
integral over y means that the logarithm contributes positively to the e↵ective action. Had
we integrated out a fermionic field, following the rules of Berezin integration would lead to
a determinant in the numerator, which thus contributes negatively to Se↵ .
Now let’s try to understand the e↵ect of this term. First, using
ln det(AB) = ln(det Adet B) = ln det A + ln det B = tr ln A + tr ln B
we write
ln det
✓
� d
2
dt2
+ M2 +
�
2
x2
◆
= tr ln
✓
� d
2
dt2
+ M2
◆
+ tr ln
 
1 � �
2
✓
d2
dt2
� M2
◆�1
x2
!
.
(3.42)
– 56 –
The first term on the rhs is independent of the field x; it will drop out if we normalize our
calculations by the partition function of the free (� = 0) theory. As in d = 0, this term is
related to the cosmological constant problem and we will consider it further later, but for
now our main interest is in the second, x-dependent term.
To make sense of this second term, let G(t, t0) be the worldline propagator (or Green’s
function), defined by ✓
d2
dt2
� M2
◆
G(t, t0) = �(t � t0) , (3.43)
so that G(t, t0) is the inverse of the free kinetic operator d2/dt2 � M2 on the worldline.
Then ✓
d2
dt2
� M2
◆�1
x2
!
(t) =
Z
S1
G(t, t0) x2(t0) dt0 (3.44)
Explicitly, the Green’s function is
G(t, t0) =
1
2M
X
r2Z
e�M |t�t
0+rT | (3.45)
where the sum over r 2 Z allows the propagator to travel r times around the circle on
its way from t0 to t. With this understanding of the inverse of the di↵erential operator
(d2/dt2 � M2) we can expand the second term in (3.42) as an asymptotic series valid as
� ! 0. From the standard Taylor series of ln(1 + ✏) we have
tr ln
 
1 � �
2
✓
d2
dt2
� M2
◆�1
x2
!
⇠ �
1X
n=1
�n
2nn
Z
(S1)⇥n
dt1 · · · dtn G(tn, t1) x2(t1) G(t1, t2) x2(t2) · · · G(tn�1, tn) x2(tn)
= ��
2
Z
S1
dt G(t, t) x2(t) � �
2
8
Z
S1⇥S1
dt dt0 G(t0, t) x2(t) G(t, t0) x2(t0) + · · ·
(3.46)
As expected, integrating out y has generated both a new contribution to the quadratic
term in x2 and also an infinite series of new interactions, just as it did in d = 0. However,
here there’s a new feature: except for the leading O(�) term, these interactions are now
non–local. They involve the value of the field x integrated over several (or infinitely many)
copies of the worldline.
It’s instructive to see why this non–locality arises. The first two terms in the se-
ries (3.46) represent the Feynman diagrams
�1/M2 �m2 ��
=
1
2
h�2i +
=
1
m2e�
� �4
2m6e�
+ + + +
+�m
2
2
�2 �
�
4M2
�2 +
�2
16M4
�4 � �
3
48M6
�6
=
=
�W(�) · · ·
· · ·
��
2
x2(t) ��
2
x2(t) ��
2
x2(t�)
G(t, t)
G(t, t�)
G(t�, t)
– 57 –
that arise in the perturbative evaluation of the y path integral. (In these diagrams, the
green dot represents the vertex ��x2/2 with x treated as a fixed ‘source’ just as we did in
zero dimensions. Just as before, the second diagram comes with a symmetry factor of 1/2,
as the two propagators are interchangeable.) Unlike the trivial case of zero dimensions,
here the y field is dynamical; the worldline propagator G(t, t0) allows it to move around,
and the insertions of x2 are at independent points in our one–dimensional universe.
Non–locality is generally bad news in physics: the equations of motion we’d obtain
from Se↵ [x] would be integro–di↵erential equations stating that in order to work out the
behaviour of the field x here, we first have to add up what it’s doing everywhere else in the
(one–dimensional) Universe. But we don’t want the results of our experiment in CERN to
depend on what Ming the Merciless may or may not be having for breakfast over on the
far side of the Galaxy.
So how bad is it here? From the explicit form (3.45) of the Green’s function we see
that G(t, t0) decays exponentially quickly when t 6= t0, with a scale set by the inverse mass
M�1 of y. This suggests that the e↵ects of non–locality will be small provided we restrict
attention to fields that vary slowly on scales ⇠ M�1. More specifically, expanding x(t0)
around t0 = t, the second term in (3.46), involving a total of four powers of the field x,
becomes
Z
dt dt0 G(t, t0)2 x2(t) x2(t0)
=
Z
dt dt0 G(t, t0)2 x2(t)
⇥
x2(t) + 2x(t)ẋ(t)(t0 � t) +
�
ẋ2(t) + x(t)ẍ(t)
�
(t0 � t)2 + · · ·
⇤
=
Z
dt

↵
M3
x4(t) +
�
M5
✓
x2ẋ2 +
1
2
x3ẍ
◆
+
�
M7
(four-derivative terms) + · · ·
�
.
(3.47)
In going to the last line we’ve performed the t0 integral, noting that terms that are odd in
(t � t0) will not contribute. The remaining terms are obtained by noting that the Green’s
function G(t, t0) involves an explicit factor of 1/M , and depends on t0 only through the
dimensionless combination u = M(t0 � t). Thus, if we replace the factor (t0 � t)p in the
pth order term in the Taylor expansion by (u/M)p and change variables dt0 = du/M to
integrate over the dimensionless quantity u, the remaining integrals will just yield some
dimensionless numbers ↵,�, �, · · · . (The precise values of these constants don’t matter for
the present discussion.)
The important point is that every new derivative of x in these vertices is suppressed
by a further power of the mass M of the y field. Thus, so long as ẋ, ẍ,
...
x , . . . are all small
in units of M�1, we should have a controllable expansion. Similarly, terms further down
the expansion in (3.46) that involve higher powers of x also come with further powers of
G(ti, tj) and further integrals, and so are again suppressed by higher powers of M . If we
truncate at any finite order both the expansion (3.46) itself and the derivative expansion
of each term in (3.46), we’ll regain an apparently local e↵ective action. This truncation is
justified provided we restrict to processes where the momentum of the x field is ⌧ M .
However, once we start to probe energies ⇠ M something will go badly wrong with
our truncated theory. Assuming the original action (3.38) defined a unitary theory (with
– 58 –
a Minkowski signature worldline M = [0, T ]), simply performing the exact path integral
over y must preserve unitary. This is because we haven’t yet made any approximations,
just taken the first step to performing the full Dx Dy path integral. All the possible states
of the y field are still secretlythere, encoded in the infinite series of non–local interactions
for x. However, the approximation to keep just the first few terms in Se↵ can’t be unitary,
because we’re rejecting by hand various pieces of Feynman diagrams: we’re throwing away
some of the things y might have been doing.
The weak interactions are responsible for many important things, from the formation
of light elements such as deuterium in the early Universe, to powering stars such as our Sun,
to the radioactive �-decay of 14C used in radiocarbon dating. Since the 1960s physicists
have known that these weak interactions are mediated by a field called the W–boson and
in 1983, the UA1 experiment at CERN discovered this field and measured its mass to be
MW ' 80 GeV. Typically, �-decay takes place at much lower energies, so to describe them
it makes sense to integrate out the dynamics of the W boson leaving us with an e↵ective
action for the proton, neutron, electron and neutrino that participate in the interaction.
This e↵ective action contains an infinite series of terms, suppressed by higher and higher
powers of the large mass MW. Truncating this infinite e↵ective action to its first few terms
leads to Fermi’s theory of �-decay which gives excellent results at low energies. However,
if ones extrapolates the results obtained using this truncated action to high energies, one
finds a violation of unitarity. The non–unitarity in Fermi’s theory is what lead physicists
to suspect the existence of the W–boson in the first place.
3.4 Quantum gravity in one dimension
The heat kernels, partition functions and correlation functions we’ve computed depend on
the choices we made in setting up our theory, including in particular the worldline metric
g. So far, we’ve fixed this to be g = �, but it’s interesting to see what happens if we also
allow ourselves to couple to a general worldline metric. Even more
We start by rewriting our original action (3.1) describing maps x : (M, g) ! (N, G) in
a way that makes it invariant under di↵eomorphisms of the worldline M . We have
S[g, x] =
Z
M
p
g

1
2
Gab(x) g
tt(t) @tx
a @tx
b +
1
2
V (x)
�
dt (3.48)
where we’ve emphasized that this action now depends on the wordline metric g. Note that
gtt(t) is a 1 ⇥ 1 positive symmetric matrix, so is specified by just a single positive function
e2 : M ! R>0. We have
p
g = |e| and gtt = e�2. Also, there is no notion of Riemann
curvature since [rt, rt] ⌘ 0 as we only have one direction. Thus there is no analogue of
the Einstein-Hilbert term
R
M
p
g Ric(g) which provides the kinetic terms for the metric, so
gravity in d = 1 is non–dynamical. Varying the action (3.48) with respect to g we obtain
the Einstein equation
Ttt :=
2
p
g
�(
p
gL)
�gtt
=
1
|e|
h
Gab(x) ẋ
aẋb � e2(t)V (x)
i
= 0 (3.49)
– 59 –
which just says that the worldline stress tensor must vanish. In particular, this Einstein
equation fixes the metric to be
gtt(t) = e
2(t) =
1
V (x)
Gab(x) ẋ
aẋb (3.50)
which is positive definite provided V (x) > 0 and (N, G) is Riemannian. Since the metric is
non–dynamical, we can use this equation to eliminate it from the action (3.48) obtaining
S[x] =
p
V0
Z
M
q
Gab(x) ẋaẋb dt (3.51)
in the special case that V (x) = V0 is constant. We recognize this as just the proper length
of the image curve x(M) ⇢ N , which is the most geometrically natural action in d = 1.
With the action (3.48) the momentum conjugate to the field xa is
pa =
�L
�ẋa
=
1
|e|Gab(x)ẋ
b (3.52)
and consequently the Einstein equation (3.49) says that
Gab(x)papb + V (x) = 0 (3.53)
under canonical quantization pa ! �@/@xa this becomes
(�⌘ab@a@b + m2) (x) = 0 (3.54)
which is just the Klein–Gordon equation for a particle of mass m.
Let’s consider the case (N, G) = (Rn�1,1, ⌘) and V (x) = m2, a constant which plays
the role of a cosmological constant in our d = 1 universe. By inserting complete sets of
momentum eigenstates, we have
hy|e�HT/~|xi =
Z
dnp dnq hy|pi hp|e�HT/~|qi hq|xi
=
Z
dnp
(2⇡~)n e
ip·(x�y)/~ e�T (p
2+m2)/2
(3.55)
and so the path integral over the matter fields becomes
Z
CI [x,y]
Dx e�S[x]/~ =
Z
dnp
(2⇡~)n e
ip·(x�y)/~ e�T (p
2+m2)/2 . (3.56)
(An alternative way to obtain the same result is to write the flat space heat kernel (3.9) as
its inverse Fourier transform.)
If we’re doing quantum gravity, we should now integrate this expression over all possible
metrics on M , chosen upto di↵eomorphisms, which plays the role of gauge equivalence in
General Relativity.34 Under a general coordinate transformation t 7! t0(t) the worldline
metric
gtt(t) 7! g0t0t0(t0) =
dt
dt0
dt
dt0
gtt(t) =
✓
dt
dt0
◆2
gtt(t) . (3.57)
34In general for a gauge theory, we always take the path integral over the space of gauge fields considered
upto gauge transformations, as we’ll see in more detail in chapter ??.
– 60 –
Thus we can always find a di↵eomorphism that rescales the value of the metric to anything
we like at each t. In particular, we are always free to choose g = � locally in one dimension.
However, we cannot quite erase all trace of the original metric, because the total volume
T =
Z
I
dt
p
gtt =
Z
I
dt0
p
gt0t0 (3.58)
is unchanged by di↵eomorphisms. (In fact, since there is no notion of curvature, this total
volume is the only invariant of a d = 1 Riemannian manifold.) Consequently, the space
Met(I)/Di↵(I) of metrics upto di↵eomorphism is simply the space of possible total lengths
of our worldline, or in other words all possible values T 2 [0, 1). Rather grandly, this is
known as the moduli space of Riemannian metrics on the interval I and, in this context,
the proper length T is sometimes known as a Schwinger parameter. Integrating the
result (3.56) of the matter path integral over this moduli space thus gives
Z 1
0
dT
Z
dnp
(2⇡~)n e
ip·(x�y)/~ e�T (p
2+m2)/2 = 2
Z
dnp
(2⇡~)n
eip·(x�y)/~
p2 + m2
(3.59)
which we recognise as (twice) the Euclidean space propagator D(x, y) for a scalar field �(x)
of mass m on the target space Rn. In other words, the position space propagator can be
written
D(x, y) =
Z
Met(I)/Di↵(I)
Dg
Z
CI [x,y]
Dx e�S[x,g]/~ (3.60)
as a path integral in worldline quantum gravity. Choosing more elaborate matter content
(e.g. fermions) for our worldline QFT similarly leads to propagators for particles of dif-
ferent spin in the target space (N, G). We’ll meet this way of thinking about field theory
propagators again in chapter 7, where we put it to work calculating propagators in the
presence of background fields.
Feynman realized that one could describe several such particles interacting with one
another if one replaced the worldline I by a worldgraph �. For example, to obtain a
perturbative evaluation of the r–point correlation function
h�(x1)�(x2) . . .�(xr)i
of a massive scalar field �(x) in ��4 theory on Rn, one could start by considering a 1–
dimensional QFT living on a 4–valent graph � with r end–points. This QFT is described
by the action (??), where x is constrained to map each end–point of the graph to a di↵erent
one of the � insertion points xi 2 Rn. We assign independent Schwinger parameters Te to
each edge e of the graph and take the path integral over all maps x : � ! Rn as well as
integrating over all the Schwinger parameters.
Part of what is meant by an ‘integral over all maps x : � ! Rn’ includes an integral
over the location in Rn to which each vertex of � is mapped. When we perform this
integral, the factors of eip·(x�y) in the path integral (3.56) for each edge lead a to target
space momentum conserving �–function at each vertex. As in (3.59), integrating over the
Schwinger parameters generates a propagator 1/(p2+m2) for each edge of the graph. Thus,
– 61 –
after including a factor of (��)|v(�)| and dividing by the symmetry factor of the graph, our
1–dimensional QFT has generated the same expression as we would have obtained from
Feynman rules for ��4 on Rn.
For example, the 4–valent graph with two end–points shown here:T1 T2
T3
x y
z
corresponds to the path integral expression
��
4
Z 1
0
dT1
Z
CT1 [x,z]
Dx e�S ⇥
Z 1
0
dT2
Z
CT2 [y,z]
Dx e�S ⇥
Z 1
0
dT3
Z
CT3 [z,z]
Dx e�S
=
��
4
Z
dnz
dnp
(2⇡)n
dnq
(2⇡)n
dn`
(2⇡)n
eip·(x�z)
p2 + m2
eiq·(y�z)
q2 + m2
ei`·(z�z)
`2 + m2
=
��
4
Z
dnp
(2⇡)n
dn`
(2⇡)n
eip·(x�y)
(p2 + m2)2 (`2 + m2)
.
(3.61)
This is the same order � contribution to the 2–point function h�(x)�(y)i that we’d obtain
from (Fourier transforming) the momentum space Feynman rules for ��4 theory, with the
graph treated as a Feynman graph in Rn rather than a one–dimensional Universe.
To obtain the full perturbative expansion of h�(x1)�(x2) · · ·�(xn)i we now sum over
all graph topologies appropriate to our 4–valent interaction. Thus
h�(x1)�(x2) · · ·�(xn)i =
X
�
(��)|v(�)|
|Aut�|
Z 1
0
d|e(�)|T
Z
C�[x1,x2,...,xn]
D� e�S�[�] , (3.62)
where |e(�)| and |v(�)| are respectively the number of edges and vertices of �.
Thus, the integral over the lengths of all the edges of our graph in (3.62) is best thought
of as an integral over the space of all possible Riemannian metrics on �, up to di↵eomor-
phism invariance. Furthermore, in summing over graphs � we were really summing over
the topological type of our one dimensional Universe. Notice that the vertices of our graphs
are singularities of the one–dimensional Riemannian manifold, so we’re allowing our Uni-
verse to have such wild (even non–Hausdor↵) behaviour. So for fixed lengths Te the path
integral over x(t) is the ‘matter’ QFT on a fixed background space �, while the integral
over the lengths of edges in � together with the sum over graph topologies is Quantum
Gravity.
This worldline approach to perturbative QFT is close to the way Feynman originally
thought about the subject, presenting his diagrams at the Pocono Conference of 1948. The
relation of this approach to higher (four) dimensional QFT as we usually think about it
was worked out by Dyson a year later, long before people used path integrals to compute
anything in higher dimensions. Above, we’ve described just the simplest version of this
picture, relevant for a scalar theory on the target space. There are more elaborate d = 1
QFTs that would allow us to obtain target space Quantum Mechanics for particles with
– 62 –
spin, and we could also allow for more interesting things to happen at the interaction
vertices of our worldgraphs. In this way, one can build up worldline approaches to many
perturbative QFTs. This way of thinking can still be useful in practical calculations today,
and still occasionally throws up conceptual surprises, but we won’t pursue it further in this
course.
This picture is also very close to perturbative String Theory. There, as you’ll learn
if you’re taking the Part III String Theory course, the worldgraph � is replaced by a two
dimensional worldsheet (Riemann surface) ⌃, the d = 1 worldline QFT replaced by a d = 2
worldsheet CFT35. Likewise, the integral over the moduli space of Riemannian metrics on
� becomes an integral over the moduli space of Riemann surfaces, and finally the sum over
graphs is replaced by a sum over the topology of the Riemann surface. We know that the
worldgraph approach to QFT only captures some aspects of perturbation theory, and in
the following chapters we’ll see that deeper insight is provided by QFT proper. Asking
whether there’s a similarly deeper approach to String Theory will take you to the mystic
shores of String Field Theory, about which very little is known.
35CFT = Conformal Field Theory.
– 63 –
5 The Renormalization Group
Even a humble glass of pure water consists of countless H2O molecules, which are made
from atoms that involve many electrons perpetually executing complicated orbits around a
dense nucleus, the nucleus itself is a seething mass of protons and neutrons glued together
by pion exchange, these hadrons are made from the complicated and still poorly understood
quarks and gluons which themselves maybe all we can make out of tiny vibrations of some
string, or modes of a theory yet undreamed of. How then is it possible to understand
anything about water without first solving all the deep mysteries of Quantum Gravity?
In classical physics the explanation is really an aspect of the Principle of Least Action:
if it costs a great deal of energy to excite a degree of freedom of some system, either by
raising it up its potential or by allowing it to whizz around rapidly in space–time, then the
least action configuration will be when that degree of freedom is in its ground state. The
corresponding field will be constant and at a minimum of the potential. This constant is the
zero mode of the field, and plays the role of a Lagrange multiplier for the remaining low–
energy degrees of freedom. You used Lagrange multipliers in mechanics to confine wooden
beads to steel hoops. This is a good description at low energies, but my sledgehammer can
excite degrees of freedom in the hoop that your Lagrange multiplier doesn’t reach.
We must re-examine this question in QFT because we’re no longer constrained to sit
at an extremum of the action. The danger is already apparent in perturbation theory,
for even in a process where all external momenta are small, momentum conservation at
each vertex still allows for very high momenta to circulate around the loop and the value of
these loop integrals would seem to depend on all the details of the high–energy theory. The
Renormalization Group (RG), via the concept of universality, will emerge as our quantum
understanding of why it is possible to understand physics at all.
5.1 Integrating out degrees of freedom
Suppose our QFT is governed by the action
S!0 [!] =
!
ddx
"
1
2
"µ!"µ!+
#
i
!d!di0 gi0Oi(x)
$
. (5.1)
Here we’ve allowed arbitrary local operators Oi(x) of dimension di > 0 to appear in the
action; each Oi can be a Lorentz–invariant monomial involving some number ni powers of
fields and their derivatives, e.g. Oi ! ("!)ri!si with ri + si = ni. For later convenience,
I’ve included explicit factors of some energy scale !0 in the couplings, chosen so as to ensure
that the coupling constants gi0 themselves are dimensionless, but of course the action is
at this point totally general. We’ve simply allowed all possible terms we can include to
appear.
Given this action, we can define a regularized partition function by
Z!0(gi0) =
!
C!(M)"!0
D! e!S!0 [!]/! (5.2)
where the integral is taken over the space C"(M)#!0 of smooth functions on M whose
energy is at most !0. The first thing to note about this integral is that it makes sense:
– 45 –
we’ve explicitly regularized the theory by declaring that we are only allowing momentum
modes up to the cut–o"19 !0. For example, there can be no UV divergences20 in any
perturbative loop integral following from (5.2), because the UV region is simply absent.
Now let’s think what happens as we try to perform the path integral by first integrating
those modes with energy between !0 and ! < !0. The space C"(M)#!0 is naturally a
vector space with addition just being pointwise addition on M . Therefore we can split a
general field !(x) as
!(x) =
!
|p|#!0
ddp
(2#)d
eip·x !̃(p)
=
!
|p|#!
ddp
(2#)d
eip·x !̃(p) +
!
!<|p|#!0
ddp
(2#)d
eip·x !̃(p)
=: $(x) + %(x) ,
(5.3)
where $ " C"(M)#! is the low–energy part of the field, while % " C"(M)(!,!0] has high
energy. The path integral measure on C"(M)#!0 likewise factorizes as
D! = D$ D%
into a product of measures over the low– and high–energy modes. Performing the integral
over the high–energy modes % provides us with an e!ective action at scale !
Se"! [$] := #! log
"!
C!(M)(!,!0]
D% exp (#S!0 [$+ %]/!)
$
(5.4)
involving the low–energy modes only. We call the process of integrating out modes changing
the scale of the theory. We can iterate this process, integrating out further modes and
obtaining a new e"ective action
Se"!# [$] := #! log
"!
C!(M)(!#,!]
D% exp
%
#Se"! [$+ %]/!
&$
(5.5)at a still lower scale !$ < !. For this reason, equation (5.4) is known as the renormalization
group equation for the e"ective action.
Separating out the kinetic part, we write the original action as
S!0 [$+ %] = S
0[$] + S0[%] + Sint!0 [$,%] (5.6)
where S0[%] is the kinetic term
S0[%] =
!
ddx
'
1
2
("%)2 +
1
2
m2%2
(
(5.7)
19In writing S!0 in terms of dimensionless couplings, we used the same energy scale !0 as we chose for
the cut-o". This was purely for convenience.
20On a non–compact space–time manifold M there can be IR divergences. This is a separate issue,
unrelated to renormalization, that we’ll handle later if I get time. If you’re worried, think of the theory as
living in a large box of side L with either periodic or reflecting boundary conditions on all fields. Momentum
is then quantized in units of 2!/L, so the space C!(M)"!0 is finite–dimensional.
– 46 –
for % and S0[$] is similar. Notice that the quadratic terms can contain no cross–terms
! $%, because these modes have di"erent support in momentum space. For the same
reason, the terms in the e"ective interaction Sint!0 [$,%] must be at least cubic in the fields.
Since $ is non–dynamical as far as the % path integral goes, we can bring S0[$] out of
the rhs of (5.4). Observing that the same $ kinetic action already appears on the lhs, we
obtain (! = 1)
Sint! [$] = # log
"!
C!(M)(!,!0]
D% exp
)
#S0[%]# Sint!0 [$,%]
*
$
(5.8)
which is the renormalization group equation for the e"ective interactions.
5.1.1 Running couplings and their &-functions
It should be clear that the partition function
Z!(gi(!)) =
!
C!(M)"!
D! e!Se"! [!]/! (5.9)
obtained from the e"ective action scale ! (or at any lower scale) is exactly the same as the
partition function we started with:
Z!(gi(!)) = Z!0(gi0;!0) (5.10)
because we’re just performing the remaining integrals over the low–energy modes. In
particular, as the scale is lowered infinitesimally (5.10) becomes the di"erential equation
!
dZ!(g)
d!
=
+
!
"
"!
,,,,
gi
+ !
"gi(!)
"!
"
"gi
,,,,
!
-
Z!(g) = 0 . (5.11)
Equation (5.11) is known as the renormalization group equation for the partition function,
and is our first example of a Callan–Symanzik equation. It just says that as change the
scale by integrating out modes, the couplings in the e"ective action Se"! vary to account
for the change in the degrees of freedom over which we take the path integral, so that
the partition function is in fact independent of the scale at which we define our theory,
provided this scale is below our initial cut–o" !0.
The fact that the couplings themselves vary, or ‘run’, as we change the scale is an
important notion. As we saw in zero and one dimensions, it’s quite natural to expect the
couplings to change as we integrate out modes, changing the degrees of freedom we can
access at low scales. However, it seems strange: you’ve learned that the electromagnetic
coupling
' =
e2
4#(0!c
$ 1
137
.
What can it mean for the fine structure constant to depend on the scale? We’ll understand
the answer to such questions later.
– 47 –
With a generic initial action, the e"ective action will also take the general form
Se"! [$] =
!
ddx
"
Z!
2
"µ$"µ$+
#
i
!d!diZni/2! gi(!)Oi(x)
$
, (5.12)
where the wavefunction renormalization factor Z! accounts for the fact that it’s perfectly
possible for the coe#cient of the kinetic term itself to receive quantum corrections as we
integrate out modes. (Z! is not to be confused with the partition function Z!!) At any
given scale, we can of course define a renormalized field
! := Z1/2! $ (5.13)
in terms of which the kinetic term will be canonically normalized. We’ve also included a
power of Z1/2! in the definition of our scale ! couplings so that these powers are removed
once one writes the action in terms of the renormalized field.
Since the running of couplings is so important, we give it a special name and define
the beta–function &i of the coupling gi to be its derivative with respect to the logarithm of
the scale:
&i := !
"gi
"!
. (5.14)
The &-functions for dimensionless couplings take the form
&i(gj(!)) = (di # d) gi(!) + &quanti (gj) (5.15)
where the first term just compensates the variation of the explicit power of ! in front of
the coupling in (5.12). The second term &quanti represents the quantum e"ect of integrating
out the high–energy modes. To actually compute this term requires us to perform the
path integral and so will generically introduce dependence on all the other couplings in the
original action (5.1), so that the &-function for gi is a function of all the couplings &i(gj).
Similarly, although at any given scale we can remove the wavefunction renormalization
factor, moving to a di"erent scale will generically cause it to re-emerge. We define the
anomalous dimension )" of the field $ by
)" := #
1
2
!
" lnZ!
"!
(5.16)
Except for the fact that we’re taken the derivative of the logarithm of Z1/2! , this is just the
&-function for the coupling in front of the kinetic term. Like any &-function, )" depends
on the values of all the couplings in the theory. It gets it’s name for reasons that will be
apparent momentarily. If our theory contained more than one type of field, then we’d have
a wavefunction renormalization factor and anomalous dimension for each field21.
21In fact, in general we’d have a matrix of wavefunction renormalization factors, allowing di"erent fields
(of the same quantum numbers such as spin, charge etc.) to mix their identities as modes are integrated
out.
– 48 –
5.1.2 Anomalous dimensions
Wavefunction renormalization plays an important role in correlation functions. Suppose
we wish to compute the n–point correlator
%$(x1) · · ·$(xn)& :=
1
Z
!
C!(M)"!
D$ e!Se"! [Z
1/2
! "; gi(!0)] $(x1) · · ·$n(xn) (5.17)
of fields inserted at points x1, . . . , xn " M using the scale ! theory, allowing for the
possibility that we hadn’t canonically normalized the field in the action. In terms of the
canonically normalized field ! := Z1/2! $ this is
%$(x1) · · ·$(xn)& = Z!n/2! %!(x1) · · ·!(xn)& (5.18)
since the change in the measure D$ ' D! cancels as we’ve normalized by the partition
function. Upon performing the ! path integral we will (in principle!) evaluate the re-
maining ! correlator as some function $(n)! (x1, . . . , xn; gi(!)) that depends on the scale !
couplings and on the fixed points {xi}.
Now, if the field insertions just involve modes with energies( ! then we should be able
to compute the same correlator using just a lower scale theory — the operator insertions
will be una"ected as we integrate out modes in the range (s!,!] for some fraction s < 1.
Accounting for wavefunction renormalization gives
Z!n/2s! $
(n)
s! (x1, . . . , xn; gi(s!)) = Z
!n/2
! $
(n)
! (x1, . . . , xn; gi(!)) , (5.19)
or equivalently
!
d
d!
$(n)! (x1, . . . , xn; gi(!)) =
.
!
"
"!
+ &i
"
"gi
+ n)"
/
$(n)! (x1, . . . , xn; gi(!)) = 0 (5.20)
infinitesimally. Equation (5.20) is the generalized Callan–Symanzik equation appropriate
for correlation functions. Once again, it simply says that the couplings and wavefunction
renormalization factors change as we lower the scale in such a way that correlation functions
remain unaltered.
In a Poincaré invariant theory, correlation functions depend the distances between
pairs of insertion points, as we saw in section (4.1.1). The typical size of these separations
defines a new scale, quite apart from any choice of !, and we can use this to obtain an
alternative interpretation of renormalization that is often useful. Integrate out modes in
the range (s!,!] as above, but having done so, let’s now change coordinates on our space
by xµ )' x$µ := sx. The kinetic term
0
ddx ("$)2 is invariant under this scaling provided
we take the field to transform as
$(sx) = s(2!d)/2$(x) . (5.21)
The remaining terms in the action are likewise unchanged by the rescaling provided we
also rescale ! ' !/s in the opposite direction tox (as expected for an energy, rather than
length, scale). Thus the energy scale s! is restored to its original value !. It’s important
– 49 –
to realize that these scalings have nothing to do with integrating out degrees of freedom in
the path integral; they’re just scalings.
Under the combined operations we find
$(n)! (x1, . . . , xn; gi(!)) =
'
Z!
Zs!
(n/2
$(n)s! (x1, . . . , xn; gi(s!))
=
'
s2!d
Z!
Zs!
(n/2
$(n)! (sx1, . . . , sxn; gi(s!)) ,
(5.22)
where the first line uses the result (5.19) of integrating out modes, while the second line
shows how correlation functions are related under the rescaling. Notice that the couplings
gi and wavefunction renormalization in the final expression are evaluated at the point s!
appropriate for the low–energy theory: the numerical values of these couplings are not
a"ected by our subsequent rescaling.
Equation (5.22) has an important interpretation. First, notice that if we’d started with
insertions at points xi/s then we could equivalently write
$(n)! (x1/s, . . . , xn/s; gi(!)) =
'
s2!d
Z!
Zs!
(n/2
$(n)! (x1, . . . , xn; gi(s!)) . (5.23)
On the left stands a correlation function computed in the theory with couplings gi(!) where
the separations between operators are |xi # xj |/s. Thus, as s ' 0 this correlator probes
the long distance, or infra–red properties of the theory. We see from the rhs that such
IR correlators may equivalently be obtained by studying a correlation function where all
separations are held constant, but we compute using a theory with di"erent values gi(s!)
for the couplings. This makes perfect sense: the IR properties of the theory are governed
by the low–energy modes that survive as we integrate out more and more high–energy
degrees of freedom.
This equation also allows us to gain insight into the meaning of the anomalous di-
mension )". The power of sn(2!d)/2 on the rhs of (5.22) is the classical scaling behaviour
we’d expect for an object of mass dimension n(d # 2)/2. Equation (5.22) shows that the
net e"ect of integrating out high–energy modes is to modify the expected classical scaling
by a simple factor depending on the wavefunction renormalization. To quantify this, set
s = 1# *s with 0 < *s ( 1. For each insertion of the field, (5.22) gives a factor
'
s2!d
Z!
Zs!
(1/2
= 1 +
'
d# 2
2
+ )"
(
*s+ · · · (5.24)
with )" as in (5.16). We see that the correlation function behaves as if the field scaled
with mass dimension
%" = (d# 2)/2 + )" (5.25)
rather than the classical value (d# 2)/2. %" is known as the scaling dimension of the field
$, and the anomalous dimension )" is the di"erence between this scaling dimension and
the naive classical dimension.
– 50 –
5.2 Renormalization group flow
In this section we’ll build up a general understanding of how theories change as we probe
them in the infra-red. This conceptual understanding, first developed by Kadano" and
Wilson in the context of condensed matter field theory, will stand us in good stead when
we come to renormalize theories perturbatively in later sections. Such calculations are
often rather technical — the general picture of the present section will prevent us from
getting bogged down in the details.
5.2.1 Renormalization group trajectories
To start to understand what happens under renormalization, let’s suppose we start with
a theory where all the &-functions vanish. That is, we consider a special action where the
initial couplings are tuned to particular values gi0 = g%i such that &j |gj=g$i = 0, so that
the couplings for this particular theory in fact do not depend on scale. Such a theory is
known as a critical point of the RG flow. A simple example, called the Gaussian critical
point, is just free theory where g%i = 0 for all terms in the action except for the (massless)
kinetic term. Clearly, the &-functions all vanish at this Gaussian critical point, since the
free theory has no interactions which could be responsible for generating vertices as the
cut–o" is lowered. However, by tuning the initial couplings very carefully, we may be able
to cause non–trivial quantum corrections to cancel precisely the classical rescaling term
in (5.15) so that the beta functions vanish. Thus it may be possible, though di#cult, to
find other critical points beyond the Gaussian one.
The couplings g%i being independent of scale has important implications for correlation
functions. Firstly, note that since it is a dimensionless function of the other couplings, the
anomalous dimension )"(g%i ) := )
%
" is likewise scale independent at a critical point. Then
for the two–point correlation function (5.20) becomes
!
"$(2)! (x, y)
"!
= #2)%" $
(2)
! (x, y) (5.26)
showing that $(2) is a homogeneous function of the scale. By Lorentz invariance it can
depend on the insertion points only through |x# y|, and dimensional analysis shows that
%$(x)$(y)& = !d!2G(!|x # y|, g%i ) for some function of the dimensionless combination
!|x # y| and the dimensionless couplings g%i . Thus, at a critical point the two–point
function must take the form
$(2)! (x, y; g
%
i ) =
!d!2
!2#!
c(g%i )
|x# y|2#!
! c(g
%
i )
|x# y|2#!
(5.27)
in terms of the scaling dimension
%" = (d# 2)/2 + )" (5.28)
of $, and where the constant c(g%i ) is independent of the insertion points. This power–law
behaviour of correlation functions is characteristic of scale–invariant theories. In a theory
where the interactions between the $ insertions was due to some massive state traveling
– 51 –
from x to y, we’d expect the potential to decay as e!m|x!y|/|x # y| where m is the mass
of the intermediate state. As in electromagnetism, the pure power–law we have found for
this correlator is a sign that our states are massless, so that their e"ects are long–range.
Critical theories are certainly very special. The metric appears in the action, so chang-
ing the metric leads to a change in the partition function given by
*gµ#(x)
*
*gµ#(x)
lnZ = #
1
*S
*gµ#(x)
2
= #*gµ#(x) %Tµ#(x)& , (5.29)
the expectation value of the stress tensor Tµ# . If the metric transformation is just a scale
transformation then *gµ# * gµ# , so scale invariance of a theory at a critical point g%i implies
that %Tµµ& = 0. In fact, all known examples of Lorentz–invariant, unitary QFTs that are
scale invariant are actually invariant under the larger group of conformal transformations
and it’s believed that all critical points of RG flows are CFTs22.
Now let’s consider the behaviour of theories near to, but not at, a critical point. Since
by definition the &-functions vanish when gi = g%i , nearby we must have
!
"gi
"!
,,,,
g$j+$gj
= Bij *gj +O(*g2) (5.30)
where *gi = gi#g%i , and where Bij is a constant (infinite dimensional!) matrix. Let +i be an
eigenvector of Bij , and let its eigenvalue be %i#d. Classically, we expected a dimensionless
coupling to scale with a power of ! determined by the explicit powers of ! included in the
action in (5.1), so that we’d have %i = di classically. Just as for the correlation function
in (5.20), the net e"ect of integrating out degrees of freedom is to modify this scaling so
that near a critical point, the couplings really scale with a power of ! determined by the
eigenvalues of the linearized &-function matrix Bij . The di"erence
)i := %i # di (5.31)
is called the anomalous dimension of the operator, mimicking the anomalous dimension
)" of the field itself, while the quantity %i itself is called the scaling dimension of the
operator. If the quantum corrections vanished then the scaling dimension would coincide
with the naive mass dimension of an operator obtained by counting the powers of fields
and derivatives it contains.
Since +i is an eigenvector of B
!
"+i
"!
= (%i # d)+i +O(+2) (5.32)
and so the RG flow for +i is
+i(!) =
.
!
!0
/#i!d
+i(!0) (5.33)
at least to this order in the perturbation away from the critical point.
– 52 –
Figure 5: Theories on the critical surface flow (dashed lines) to a critical point in theIR.
Turning on relevant operators drives the theory away from the critical surface (solid lines),
with flow lines focussing on the (red) trajectory emanating from the critical point.
Now consider starting near a critical point and turning on the coupling to any operator
with %i > d. According to (5.33) this coupling becomes smaller as the scale ! is lowered,
or as we probe the theory in the IR. We say that the corresponding operator is irrelevant
since if we include it in the action then RG flow just makes us flow back to the critical
point g%i . Classically, we can obtain operators with arbitrarily high mass dimension by
including more and more fields and derivatives, so we expect that the critical point g%i sits
on an infinite dimensional surface C such that if we turn on any combination of operators
that move us along C, under RG flow we will end up back at the critical point. C is known
as the critical surface and we can think of the couplings of irrelevant operators as provided
coordinates on C, at least in the neighbourhood of g%i . (See figure 5.)
On the other hand, couplings with %i < d grow as the scale is lowered and so are
called relevant. If our action contains vertices with relevant couplings then RG flow will
drive us away from the critical surface C as we head into the IR. Starting precisely from a
critical point and turning on a relevant operator generates what is known as a renormalized
trajectory: the RG flow emanating from the critical point. As we probe the theory at lower
and lower scales we evolve along the renormalized trajectory either forever or until we
eventually meet another23 critical point g%%i . Since each new field or derivative adds to the
dimension of an operator, in fixed space–time dimension d there will be only finitely many
22It’s a theorem that this is always true in two dimensions. It is believed to be true also in higher
dimensions, but the question is actually a current hot topic of research.
23There are a few exotic examples where the theories flow to a limiting cycle rather than a fixed point.
– 53 –
(and typically only few) relevant operators, so the critical surface has finite codimension.
The remaining possibility is marginal operators, which have vanishing eigenvalues and
so neither increase nor decrease under RG flow. At the Gaussian point, the scaling dimen-
sions of operators are just given by their classical mass dimension, so we expect marginal
operators to have scaling dimension %i = d. Near a critical point, quantum corrections
can bring in a weak (typically logarithmic) dependence on scale to a classically marginal
operator, making it either marginally relevant or marginally irrelevant. Provided the non–
zero eigenvalues of these operators are su#ciently small, the size of such nearly marginal
couplings can be unchanged for long periods of RG flow — although ultimately they will
either be irrelevant or relevant. Such operators play an important role phenomenologically,
as we will see.
A generic QFT will have an action that involves all types of operators and so lies
somewhere o" the critical surface. Under RG evolution, all the many irrelevant operators
are quickly suppressed, while the relevant ones grow just as before. The flow lines of a
generic theory thus strongly focus onto the renormalized trajectory, and so in the IR a
generic QFT will closely resemble a theory emanating from the critical point, where only
relevant operators have been turned on. The fact that many di"erent high energy theories
will flow to look the same in the IR is known as universality. It assures us that the
properties of the theory in the IR are determined not by the infinite set of couplings {gi},
but only by the couplings to a few relevant operators. We say that theories whose RG
flows are all focussed onto the same trajectory emanating from a given critical point are in
the same universality class. Theories in a given universality class could look very di"erent
microscopically, but will all end up looking the same at large distances. In particular, deep
in the IR, these theories will all flow to the second critical point g%%i .This is the reason you
can do physics! To study a problem at a given energy scale you don’t first need to worry
about what the degrees of freedom at much higher energies are doing. They are, quite
literally, irrelevant.
Let me emphasize that eigenvectors +i are generically linear combinations of the naive
couplings in the action. Thus, turning on +i means we perturb away from the critical
point by changing the couplings in front of the corresponding linear combination of our
operators in the action. These RG ‘eigenoperators’ may be very di"erent from any in-
dividual monomial in the fields you choose to include neatly in the e"ective interaction
Se" . A simple–looking individual operator Oi that appears in (5.1) or is explicitly inserted
into a correlation function could actually consist of many RG eigenfunctions. We say
that operators mix, because a given operator transforms under RG flow into its dominant
eigenfunction, which could look very much more complicated.
5.2.2 Counterterms and the continuum limit
So far, we’ve considered a fixed initial theory S!0 [$] with initial couplings gi0, and examined
how these couplings change as we probe the theory at long distances. Our definition of the
– 54 –
Dimension Relevant operators Marginal operators
d = 2 $2k for all k + 0 ("$)2, $2k("$)2 for all k + 0
d = 3 $2k for k = 1, 2 ("$)2, $6
d = 4 $2 for , 3 ("$)2, $4
d > 4 $2 for 0 , k , 3 ("$)2
Table 1: Relevant & marginal operators in a Lorentz invariant theory of a single scalar
field in various dimensions, near the Gaussian critical point where the classical dimensions
of operators are a good guide. Only the operators invariant under $ ' #$ are shown. Note
that the kinetic term ("$)2 is always marginal, and the mass term $2 is always relevant.
low–energy e"ective action as
Se"! [$] := #! log
"!
C!(M)(!,!0]
D% exp (#S!0 [$+ %]/!)
$
(5.34)
ensured that the partition function and correlation functions of low–energy observables were
independent of the scale !. The question remains: what about dependence on the initial
cut–o" !0? In this section we’ll examine this by asking a sort of converse: Suppose we
fix a particular low–energy theory (perhaps motivated by the results of some finite–scale
experiments). How can we remove the high–energy cut–o", sending !0 ' -, without
a"ecting what the theory predicts for low–energy phenomena. We call this taking the
continuum limit of our theory, since sending !0 ' - is allowing the field to fluctuate on
arbitrarily small scales.
The key to achieving this lies with the universality of the renormalization group flow.
First, suppose our initial couplings gi0 happen to lie on the critical surface C, within the
domain of attraction of g%i . Then as we raise the cut–o" !0, the theory we obtain at any
fixed scale ! will be driven to the critical point g%i as all the irrelevant operators become
arbitrarily suppressed by positive powers of !/!0. The critical point is a fixed point of the
RG flow and is scale invariant, so we can happily send !0 ' -. More precisely, whenever
the theory S!0 lives on the critical surface, the limit
lim
!0&"
"!
C!(M)(!,!0]
D% exp (#S!0 [$+ %]/!)
$
, (5.35)
exists, provided we take this limit after computing the path integral. The resulting scale-!
e"ective theory will be a CFT, independent of !. Since C has only finite codimension, we
only have to tune finitely many coe#cients (those of all the relevant operators) in order to
ensure that gi0 " C.
Theories such as Yang–Mills or QCD are not CFTs, but rather have relevant (and
marginally relevant) operators turned on in their actions. How then can we understand
the continuum limit of such theories? Consider a theory whose initial conditions are near,
but not on C. Universality of the RG flow shows that as we head into the IR, such a theory
flowstowards the critical point g%i for a while, but eventually diverges away, focussing on a
– 55 –
renormalized trajectory as in figure 5. Let µ denote the energy scale at which this theory
passes closest to g%i . Since RG flow is determined by the initial conditions, µ depends only
on the theory we started with. On dimensional grounds we must have
µ = !0 f(gi0) (5.36)
where f(gi0) is some function of the dimensionless couplings {gi0} and f = 0 on C, since all
these theories flow to g%i exactly. To obtain a theory with relevant or marginal operators, we
tune the initial couplings {gi0} so that µ remains finite as we take !0 ' -. If codim(C) = r
then this is one condition on r parameters — the coe#cients of the relevant operators in
the initial action. The theory we end up with thus depends on (r#1) parameters, together
with the scale µ.
We achieve this tuning by introducing new counterterms SCT[!,!0] that depend on
the fields $ as well as explicitly on the cut–o" !0, modifying the initial action to
S!0 [!] ' S!0 [!] + SCT[!,!0] . (5.37)
The e"ective actions we considered before already contained all possible monomials in fields
and their derivatives, so in this sense the counterterms add nothing new. However, the
values of the counterterm couplings are to be chosen by hand — varying these couplings
thus changes which initial high–energy theory we’re considering, as opposed to running
a set of couplings under RG flow, which just describes how the same theory appears at
di"erent scales. The counterterms are tuned so that the limit
e!S
e"
! ["]/! = lim
!0&"
"!
C!(M)(!,!0]
D% exp
.
#S[$+ %]! # SCT[$+ %,!0]
/$
(5.38)
exists. Notice again that the limit is taken after performing the path integral. Sending
!0 ' - defines a continuum QFT with finite (or renormalized) relevant couplings at scale
!.
The reason for making SCT explicit, rather than just treating the counterterms as a
modification {gi0}, is that in practice we work perturbatively. To evaluate the path integral
in (5.38), we first compute quantum corrections to 1-loop order using the original action
S. These 1-loop corrections will depend on the cut–o" !0, and will be proportional to
!. In general, they will diverge as !0 ' - reflecting the fact that we lose control of the
original theory if the cut–o" is removed näıvely. However, vertices in SCT provide further
contributions to these quantum corrections. By tuning the values of the couplings in SCT
by hand, we can obtain a finite limit. Notice that SCT comes with one extra power of
! in (5.38) compared to the original action. Thus, quantum corrections to the e"ective
action arising from 1-loop diagrams of the original action should be matched by the tree–
level contributions from SCT. We’ll get plenty of practice in doing this in the following
sections.
There’s one further possibility to consider. Suppose that to explain some experimental
result, be it the scattering of pions and nucleons or the falling of apples, we need our
– 56 –
low–energy theory to contain irrelevant operators. If we really try to take the cut–o"
!0 ' -, such operators will be arbitrarily suppressed at any finite energy scale. So
their presence indicates that our theory cannot be valid up to arbitrarily high energies;
there must be a finite energy scale at which new physics comes in to play. In the case
of pion–nucleon scattering, this scale is ! 217 MeV and indicates the presence of quarks,
gluons and the whole structure of QCD. For radioactive &-decay, the scale is ! 250 GeV
and indicated the electroweak theory, while for gravity the scale is ! 1019 GeV, where
probably the whole notion of QFT itself must give way. Perhaps most interesting of all are
marginally irrelevant operators, like the quartic coupling (&4) of the Higgs in the Standard
Model. Strictly speaking, just like irrelevant operators, marginally irrelevant operators
are arbitrarily suppressed as the cut–o" is removed. However, they typically decay only
logarithmically as !0 is raised, rather than as a power law. Such operators thus a"ord us
a tiny glimpse of new physics at exponentially high energy scales, far beyond the range of
current accelerators.
5.3 Calculating RG evolution
It’s time to think about how to calculate the quantum corrections to &-functions generated
as we integrate out high energy modes. I’ll this section I want to explain this in a way that
I think is conceptually clear, and the natural generalization of what we have already seen
in zero and one dimension. However, I’ll warn you in advance that the techniques here are
not the most convenient way to calculate &-functions.
5.3.1 Polchinski’s equation
In perturbation theory, the rhs of (5.8) may be expanded as an infinite series of connected
Feynman diagrams. If we wish to compute the low–energy e"ective interaction Sint! [$] as an
integral over space–time in the usual way, then we should use the position space Feynman
rules. As in section 3.4, the position space propagator D(%)(x, y) for the high–energy field
% is
D(%)(x, y) =
!
!<|p|#!0
ddp
(2#)d
eip·(x!y)
p2 +m2
(5.39)
where we note the restriction to momenta in the range ! < |p| , !0. As usual, vertices from
the high–energy action Sint!0 [$,%] come with an integration
0
ddx over their location that
imposes momentum conservation at the vertex. Now, diagrams that exclusively involve
vertices which are independent of $ contribute just to a field–independent term on the lhs
of (5.8). This term represents the shift in vacuum energy due to integrating out the %
field; we will henceforth ignore it24. The remaining diagrams use vertices including at least
one $ field, treated as external. Evaluating such a diagram leads to a contribution to the
e"ective interaction Sint! [$] at scale !.
For general scales ! and !0 equation (5.8) is extremely di#cult to handle; the integral
on the right is a full path integral in an interacting theory. To make progress we consider
24This is harmless in a non–gravitational theory, but is really the start of the cosmological constant
problem.
– 57 –
.
. .
.
..
..
.
...
.
. .
+=
n!2!
r=2
gr+1 gn!r+1 gn+2gn
!
d
d!
Figure 6: A schematic representation of the renormalization group equation for the ef-
fective interactions when the scale is lowered infinitesimally. Here the dashed line is a
propagator of the mode with energy ! that is being integrated out, while the external lines
represent the number of low–energy fields at each vertex. All these external fields are eval-
uated at the same point x The total number of fields attached to a vertex is indicated by
the subscripts on the couplings gi.
the case that we lower the scale only infinitesimally, setting ! = !0 # *!. To lowest order
in *!, the % propagator reduces to25
D(%)! (x, y) =
1
(2#)d
!d!1 *!
!2 +m2
!
Sd%1
d' ei!p̂·(x!y) (5.40)
as the range of momenta shrinks down, where d' denotes an integral over a unit (d# 1)-
sphere in momentum space. This is a huge simplification! Since every % propagator comes
with a factor of *!, to lowest order in *! we need only consider diagrams with a single
% propagator. Since $ is treated as an external field, we have only two possible classes of
diagram: either the % propagator links together two separate vertices in Sint[$,%] or else
it joins a single vertex to itself.
This diagrammatic represention of the process of integrating out degrees of freedom
is shown in figure 6. It has a very clear intuitive meaning. The mode % appearing in the
propagator is the highest energy mode in the original scale !0 theory. It thus probes the
shortest distances we can reliably access using Se"!0 . When we integrate this mode out,
we can no longer resolve distances 1/!0 and our view of the ‘local’ interaction vertices is
correspondingly blurred. The graphs on the rhs of figure 6 represent new contributions to
the n–point $ vertex in the lower scale theory coming respectively from two nearby vertices
joined by a % field, or ahigher point vertex with a % loop attached. Below scale !0 we
image that we are unable to resolve the short distance % propagator.
We can write an equation for the change in the e"ective action that captures the
information in the Feynman diagrams in figure 6. It was obtained by Polchinski26, and
is really just the infinitesimal version of Wilson’s renormalization group equation (5.4) for
the e"ective action. Polchinski’s equation is
# !"S
int
! [$]
"!
=
!
ddx ddy
'
*Sint!
*$(x)
D!(x, y)
*Sint!
*$(y)
#D!(x, y)
*2Sint!
*$(x) *$(y)
(
, (5.41)
25To lowest order, it doesn’t matter whether we use !0 or ! in this expression.
26Polchinski actually wrote a slightly more general version of the momentum space version of this equation,
including source terms
!
J" in the e"ective action.
– 58 –
where D!(x, y) is the propagator (5.40) for the mode at energy ! that is being integrated
out. The variations of the e"ective interactions tell us how this propagator joins up the
various vertices. Notice in the second term that since Sint[$] is local, both the */*$
variations must act at the same place if we are to get a non–zero result. On the other
hand, the first term generates non–local contributions to the e"ective action since it links
fields at x to fields at a di"erent point y. In position space we expect a propagator at scale
!2 +m2 to lead to a potential ! e!
'
!2+m2 r/rd!3 so this non–locality is mild and we can
expanding the fields in *Sint/*$(y) as a series in (x# y). This leads to new contributions
to interactions involving derivatives of the fields, just as we saw in section 3.3 in one
dimension. Finally, the minus signs in (5.41) comes from expanding e!S
int["] to obtain the
Feynman diagrams.
It’s convenient to rewrite Polchinski’s equation (5.41) as
"
"t
e!S
int["] = #
!
ddx ddy D!(x, y)
*2
*$(x) *$(y)
e!S
int["] , (5.42)
in which form it reveals itself as a form of heat equation, with renormalization group
‘time’27 t . ln! and ‘Laplacian’
% =
!
ddx ddy D!(x, y)
*2
*$(x) *$(y)
(5.43)
on the space of fields. Heat flow on a Riemannian manifold N is a strongly smoothing
operation: if we expand a function f : N / R>0 ' R as
f(x, t) =
#
k
f̃k(t)uk(x)
in terms of a basis of eigenfunctions uk(x) of the Laplacian on N , then under heat flow
the coe#cients evolve as f̃k(t) = f̃k(0) e!&kt. Consequently, all components f̃k(t) corre-
sponding to positive eigenvalues ,k are quickly damped away, with only the constant piece
(with zero eigenvalue) surviving. This just corresponds to the well–known fact that a heat
spreads out from areas of high concentration (such as a flame) until the whole room is at
constant temperature. On a manifold with a pseudo–Riemannian (rather than Rieman-
nian) manifold, some eigenvalues can be negative. These functions would then be enhanced
under heat flow. Exactly the same thing happens under RG flow. Eigenfunctions of the
Laplacian in (5.42) are combinations of operators in the e"ective interactions. Depending
on the sign of their corresponding eigenvalues, these operators will be either enhanced or
suppressed under the flow.
5.3.2 The local potential approximation
Polchinski’s equation contains exact information about the behaviour of every possible
operator under RG flow. Unfortunately, while it’s structurally simple, actually solving this
equation as it stands is prohibitively di#cult, so we seek a more managable approximation.
27In the AdS/CFT correspondence, this RG time really does turn into an honest direction: into the bulk
of anti–de Sitter space!
– 59 –
To obtain one, observe that except for the kinetic term, operators involving derivatives
are irrelevant28 whenever d > 2. This suggests that we can restrict attention to actions of
the form
Se"! [!] =
!
ddx
'
1
2
"µ!"µ!+ V (!)
(
(5.44)
where the potential
V (!) =
#
k
!d!k(d!2)
g2k
(2k)!
!2k (5.45)
does not involve derivatives of $. For simplicity, we’ve chosen V (#!) = V (!), while the
couplings g2k are dimensionless as before. Neglecting the derivative interactions is known
as the local potential approximation; it’s important because it will tell us the shape of the
e"ective potential experienced by a slowly varying field. Splitting the field ! = $+ % into
its low– and high–energy modes as before, we expand the action as an infinite series
Se"! [$+ %] = S
e"
! [$] +
!
ddx
'
1
2
("%)2 +
1
2
%2 V $$($) +
1
3!
%3 V $$$($) + · · ·
(
. (5.46)
Notice that we have chosen a definition of $ so that it sits at a minimum of the potential,
V $($) = 0. This can always be arranged by adding a constant to $, which is certainly a
low–energy mode.
Now consider integrating out the high–energy modes %. As before, we lower the scale
infinitesimally, setting !$ = ! # *! and working just to first order in *!. In any given
Feynman graph, each % loop comes with an integral of the form
!
!!$!<|p|#!
ddp
(2#)d
(· · · ) = *! !
d!1
(2#)d
!
Sd%1
d' (· · · )
where d' denotes an integral over a unit Sd!1 0 Rd and (· · · ) represents the propagators
and vertex factors involved in this graph. As with Polchinski’s equation, since each loop
integral comes with a factor of *!, to lowest non–trivial order in *! we need consider at
most 1-loop diagrams for %.
Suppose a particular graph involves an number vi vertices containing i powers of %
and arbitrary powers of $. Euler’s identity tells us that a connected graph with e edges
and - loops obeys
e#
#
i
vi = -# 1 , (5.47)
Since we’re only integrating over the high scales modes, % is the only propagating field.
Furthermore, since we’re integrating out % completely, there are no external % lines. Thus
we also have the identity
2e =
#
i
i vi (5.48)
since every % propagator is emitted and absorbed at some (not necessarily distinct) vertex.
Eliminating e from (5.47) gives
- = 1 +
#
i
i# 2
2
vi . (5.49)
28At least near the Gaussian critical point where classical scaling dimensions are a reasonable guide.
– 60 –
.
. .
.
. .
...
..
.
. . .
...
+ + + ...
Figure 7: Diagrams contributing in the local potential approximation to RG flow. The
dashed line represents a % propagator with |p| = !, while the solid lines represent external
$ fields. All vertices are quadratic in %.
We only want to keep track of 1-loop diagrams, so we see that only the vertices with i = 2 %
lines (and arbitrary numbers of $ lines) are important. We can thus truncate the di"erence
Se"! [$+ %]# Se"! [$] in (5.46) to
S(2)[%] =
!
ddx
'
1
2
"%2 +
1
2
V $$($)%2
(
(5.50)
so that % appears only quadratically.
The diagrams that can be constructed from this action are shown in figure 7. If we
make the temporary assumption that the low–energy field $ is actually constant, then in
momentum space the quadratic action S(2) becomes
S(2)[%] =
!
!!$!<|p|#!
ddp
(2#)d
%̃(p)
'
1
2
p2 +
1
2
V $$($)
(
%̃(#p)
=
!d!1*!
2(2#)d
(!2 + V $$($))
!
Sd%1
d' %̃(!p̂) %̃(#!p̂)
(5.51)
using the fact that these modes have energies in a narrow shell of width *!.
Performing the path integral over % is now straightforward. If the narrow shell contains
N momentum modes, then from standard Gaussian integration
e!$!S
e" ["] =
!
D% e!S2[%,"] = C
.
#
!2 + V $$($)
/N/2
. (5.52)
On a non–compact manifold, N is actually infinite. To regularize it, we place our theory
in a box of linear size L and impose periodic boundary conditions. The momentum is
then quantized as pµ = 2#nµ/L for nµ " Z so that there is one mode per (2#)d volume in
Euclidean space–time. The volume of space–time itself is Ld. Thus
N =
Vol(Sd!1)
(2#)d
!d!1*!Ld (5.53)
which diverges as the volume Ld of space–time becomes infinite. However, we can obtain a
(correct) finite answer once we recognize that the cause of this divergence was ou simplifying
– 61 –
assumption that $ was constant. For spatially varying $, we would instead obtain
*!S
e" [$] = a!d!1*!
!
ddx ln
3
!2 + V $$($)
4
(5.54)
where the factor of Ld / ln[!2 + V $$($)] in (5.52) hasbeen replaced by an integral over M .
The constant
a :=
Vol(Sd!1)
2(2#)d
=
1
(4#)d/2 $(d/2)
(5.55)
is proportional to the surface area of a (d# 1)-dimensional unit sphere. Expanding the rhs
of (5.54) in powers of $ leads to a further infinite series of $ vertices which combine with
those present at the classical level in V ($). Once again, integrating out the high–energy
field % has lead to a modification of the couplings in this potential.
We’re now in position to write down the &-functions. Including the contribution from
both the classical action and the quantum correction (5.54), the &-function for the $2k
coupling is
!
dg2k
d!
= [k(d# 2)# d]g2k # a!k(d!2)
"2k
"$2k
ln
3
!2 + V $$($)
4,,,,
"=0
. (5.56)
For instance, the first few terms in this expansion give
!
dg2
d!
= #2g2 #
ag4
1 + g2
!
dg4
d!
= (d# 4)g4 #
ag6
1 + g2
+
3ag24
(1 + g2)2
!
dg8
d!
= (2d# 6)g6 #
ag8
1 + g2
+
15ag4g6
(1 + g2)2
# 30ag
3
4
(1 + g2)3
(5.57)
as &–functions for the mass term, $4 and $6 vertices.
There are several things worth noticing about the expressions in (5.57). Firstly, each
term on the right comes from a particular class of Feynman graph; the first term is the
scaling behaviour of the classical $2k vertex, the second term involves a single % propagator
with both ends joined to the same valence 2k+2 vertex, the third (when present) involves
a pair of % propagators joining two vertices of total valence 2k + 4, etc.. Secondly, we
note that these Feynman diagrams are di"erent to the ones that appeared in (5.41). By
taking the local potential approximation, we have neglected any possible derivative terms
that may have contributed to the running of the couplings in V ($). The e"ect of this is
seen in the higher–order terms that appear on the rhs of (5.57). From the point of view of
the Wilson–Polchinski renormalization group equation, the local potential approximation
e"ectively amounts to solving the &-function equations that follow from (5.41), writing the
derivative couplings in terms of the non–derivative ones, and then substituting these back
into the remaining &-functions for non–derivative couplings to obtain (5.57). The message
is that the price to be paid for ignoring possible couplings in the e"ective action is more
complicated &-functions. We will see this again in chapter ??, where &-functions will no
longer be determined purely at one loop.
– 62 –
Finally, recall that g2 = m2/!2 is the mass of the $ field in units of the cut–o". If this
mass is very large, so g2 1 1, then the quantum corrections to the &-functions in (5.57)
are strongly suppressed. As for correlation functions near to, but not at, a critical point,
this is as we would expect. A particle of mass m leads to a potential V (r) ! e!mr/rd!3 in
position space, so should not a"ect physics on scales r 1 m!1.
5.3.3 The Gaussian critical point
From our discussion in section 5.2, we expect that the limiting values of the couplings in
the deep IR will be a critical point of the RG evolution (5.56). The simplest type of critical
point is the Gaussian fixed–point where g2k = 0 2 k > 1, corresponding to a free theory.
Every one of the Feynman diagrams shown on the right of the Wilson renormalization
group equation in figure 6 involves a vertex containing at least three fields (either % or $),
so if we start from a theory where the couplings to each of these vertices are precisely set to
zero, then no interactions can ever be generated. Indeed, (5.57) shows that the Gaussian
point is indeed a fixed–point of the RG flow, with the mass term &-function &2 = #2g2
simply compensating for the scaling of the explicit power of ! introduced to make the
coupling dimensionless.
Last term you used perturbation theory to study $4 theory in four dimensions. Using
perturbation theory means that you considered this theory in the neighbourhood of the
Gaussian critical point so that the couplings could be treated as ‘small’. Let’s examine this
again using our improved understanding of RG flow. Firstly, to find the behaviour of any
coupling near to the free theory, as in equation (5.30) we should linearize the &-functions
around the critical point. We’ll use our results (5.57) for a theory with an arbitrary
polynomial potential V ($). To linear order in the couplings, only the first two terms on
the rhs of (5.57) contribute, giving
&2k = !
"g2k
"!
= (k(d# 2)# d) g2k # ag2k+2 (5.58)
where *g2k = g2k # g%2k = g2k since g%2k = 0 for the Gaussian critical point. Writing the
linearized &-functions in the form &i = Bijgj we see that the matrix Bij is upper triangular,
so its eigenvalues are simply its diagonal entries k(d # 2) # d. In four dimensions, these
eigenvalues are 2k # 4, which is positive for k + 3. Thus, in four dimensions, deforming
the free action by an operator of the form $2k with k + 3 is an irrelevant perturbation:
turning on any such operator takes us away from the free theory in a direction along the
critical surface, and we are pushed back to the free theory as the cut–o" is lowered.
On the other hand, the mass term g2 is a relevant deformation of the free action.
Turning on even arbitrarily small masses will lead us away from the massless theory as
the cut–o" is lowered. Of course, once g2 is large we cannot trust our linearized approxi-
mation (5.58), and the correct result (5.57) shows that the quantum corrections to g2 are
eventually suppressed as the mass becomes large in units of the cut–o".
The remaining coupling g4 is particularly interesting. We’ve seen that for k + 3, the
$2k interactions are irrelevant in d = 4 near the Gaussian fixed point, so at low energies
– 63 –
we may neglect them. &4 then vanishes to linear order, so that the $4 coupling is marginal
at this order. To study its behaviour, we need to go to higher order. From (5.57) we have
&4 = !
"g4
"!
=
3
16#2
g24 +O(g24g2) (5.59)
to quadratic order, where we’ve again dropped the g6 term. Equation (5.59) is solved by
1
g4(!)
= C # 3
16#2
ln! (5.60a)
where C is an integration constant. Equivalently, we may write
g4(!) =
16#2
3 ln(µ/!)
(5.60b)
in terms of some arbitrary scale µ. Since g4 is the coe#cient of the highest power of $ that
appears in our potential, we must have g4 > 0 if the action is to be bounded as |$| ' -,
so we must choose µ > !.
There are several important things to learn from this result. Firstly, we see that g4(!)
decreases as ! ' 0, ultimately being driven to zero. However, the scale dependence of
g4 is rather mild; instead of power–law behaviour we have only logarithmic dependence
on the cut–o". Thus the $4 coupling, which was marginal at the classical level, because
marginally irrelevant once quantum e"ects are taken into account. In the deep IR, we see
only a free theory.
Secondly, away from the IR we notice that the integration constant µ determines a
scale at which the coupling diverges. If we try to follow the RG trajectories back into the
UV, perturbation theory will certainly break down before we reach ! $ µ. The fact that
the couplings in the action can be traded for energy scales µ at which perturbation theory
breaks down is a ubiquitous phenomenon in QFT known as dimensional transmutation.
We’ll meet it many times in later chapters. The question of whether the $4 coupling really
diverges as we head into the UV or just appears to in perturbation theory is rather subtle.
More sophisticated treatments back up the belief that it does indeed diverge: in the UV
we lose all control of the theory and in fact we do not believe that $4 theory really exists as
a well–defined continuum QFT in four dimensions. This has important phenomenological
implications for the Standard Model, through the quartic coupling of the scalar Higgs
boson; take the Part III Standard Model course if you want to find out more.
The fact that the $4 coupling is not a free constant, but is determined by the scale
and can even diverge at a finite scale ! = µ should be worrying. How canwe ever trust
perturbation theory? The final lesson of (5.60b) is that if we want to use perturbation
theory, we should always try to choose our cut–o" scale so as to make the couplings as
small as possible. In the case of $4 theory this means we should choose ! as low as possible.
In particular, if we want to study physics at a particular length scale -, then our best chance
for a weakly coupled description is to integrate out all degrees of freedom on length scales
shorter than -, so that ! ! -!1.
– 64 –
5.3.4 The Wilson–Fisher critical point
The conclusion at the end of the previous section was that $4 theory does not have a
continuum limit in d = 4. Since the only critical point is the Gaussian free theory we reach
at low energies, four dimensional scalar theory is known as a trivial theory.
It’s interesting to ask whether there are other, non–trivial critical points away from four
dimensions. In general, finding non–trivial critical points is a di#cult problem. Wilson and
Fisher had the idea of introducing a parameter . := 4#d which is treated as ‘small’ so that
one is ‘near’ four dimensions. One then hopes that results obtained via the .–expansion
may remain valid in the physically interesting cases of d = 3 or even d = 2. From the local
potential approximation (5.56) Wilson & Fisher showed that there is a critical point gWFi
where
gWF2 = #
1
6
.+O(.2) , gWF4 =
1
3a
.+O(.2) (5.61)
and gWF2k ! .k for all k > 2. We require . > 0 to ensure that V ($) ' 0 as |$| ' - so that
the theory can be stable.
To find the behaviour of operators near to this critical point, once again we linearize
the &-functions of (5.57) around gWF2k . Truncating to the subspace spanned by (g2, g4) we
have
!
"
"!
+
*g2
*g4
-
=
+
./3# 2 #a(1 + ./6)
0 .
-+
*g2
*g4
-
. (5.62)
The matrix has eigenvalues ./3# 2 and ., with corresponding eigenvectors
+2 =
+
1
0
-
, +4 =
+
#a(3 + ./2)
2(3 + .)
-
(5.63)
respectively. In d = 4# . dimensions we have
a =
1
(4#)d/2
1
$(d/2)
,,,,
d=4!'
=
1
16#2
+
.
32#2
(1# ) + ln 4#) +O(.2) (5.64)
where we have used the recurrence relation $(z + 1) = z $(z) and asymptotic formula
$(#./2) = #2
.
# ) +O(.) (5.65)
for the Gamma function as . ' 0, where ) is the Euler–Mascheroni constant ) $ 0.5772.
Since . is small the first eigenvalue is negative, so the mass term $2 is a relevant perturbation
of the Wilson–Fisher fixed point. On the other hand, the operator#a(3+./2)$2+2(3+.)$4
corresponding to +4 corresponds to an irrelevant perturbation. The projection of RG flows
to the (g2, g4) subspace is shown in figure 8.
Although we’ve seen the existence of the Wilson–Fisher fixed point only for 0 < . ( 1,
more sophisticated techniques can be used to prove its existence in both d = 3 and d = 2
where it in fact corresponds to the Ising Model CFT. As shown in figure 8, both the
Gaussian and Wilson–Fisher fixed–points lie on the critical surface, and a particular RG
trajectory emanating from the Gaussian model corresponding to turning on the operator +4
– 65 –
WF
G
g2
g4
I
II
Figure 8: The RG flow for a scalar theory in three dimensions, projected to the (g2, g4)
subspace. The Wilson–Fisher and Gaussian fixed points are shown. The blue line is the
projection of the critical surface. The arrows point in the direction of RG flow towards the
IR.
ends at the Wilson–Fisher fixed point in the IR. Theories on the line heading vertically out
of the Gaussian fixed–point correspond to massive free theories, while theories in region
I are massless and free in the deep UV, but become interacting and massive in the IR.
These theories are parametrized by the scalar mass and by the strength of the interaction
at any given energy scale. Theories in region II are likewise free and massless in the UV
but interacting in the IR. However, these theories have g2 < 0 so that the mass term is
negative. This implies that the minimum of the potential V ($) lies away from $ = 0, so
for theories in region II, $ will develop a vacuum expectation value, %$& 3= 0. The RG
trajectory obtained by deforming the Wilson–Fisher fixed point by a mass term is shown
in red. All couplings in any theory to the right of this line diverge as we try to follow the
RG back to the UV; these theories do not have well–defined continuum limits.
5.3.5 Zamolodchikov’s C–theorem
Polchinski’s equation showed that renormalization group flow could be understood as a
form of heat flow. It’s natural to ask whether, as for usual heat flow, this can be thought
of as a gradient flow so that there is some real positive function C(gi,!) that decreases
monotonically along the flow. Notice that this implies C = const. at a fixed point g%i , and
that C(g%i ,!) > C(g
%%
i ,!
$) whenever a fixed point g%%i may be reached by perturbing the
theory a fixed point g%i by a relevant operator and flowing to the IR. In 1986, Alexander
Zamolodchikov found such a function C for any unitary, Lorentz invariant QFT in two
– 66 –
dimensions.
Consider a two dimensional QFT whose (improved) energy momemtum tensor is given
by Tµ#(x). This is a symmetric 2/2 matrix, so has three independent components. Intro-
ducing complex coodinates z = x1 + ix2 and z̄ = x1 # ix2, we can group these components
as
Tzz :=
"xµ
"z
"x#
"z
Tµ# =
1
2
(T11 # T22 # iT12)
Tz̄z̄ :=
"xµ
"z̄
"x#
"z̄
Tµ# =
1
2
(T11 # T22 + iT12)
Tzz̄ :=
"xµ
"z
"x#
"z̄
Tµ# =
1
2
(T11 + T22)
(5.66)
where Tz̄z = Tzz̄. This stress tensor is conserved, with the conservation equation being
0 = "µTµ# = "z̄Tzz + "zTz̄z̄ (5.67)
in terms of the complex coordinates. Note that the stress tensor is a smooth function of z
and z̄.
The two–point correlation functions of these stress tensor components are given by
%Tzz(z, z̄)Tzz(0, 0)& =
1
z4
F (|z|2)
%Tzz(z, z̄)Tzz̄(0, 0)& =
4
z3z̄
G(|z|2)
%Tzz̄(z, z̄)Tzz̄(0, 0)& =
16
|z|4H(|z|
2)
(5.68)
where the explicit factors of z and z̄ on the rhs follow from Lorentz invariance, which also
requires that the remaining functions F , G and H depend on position only through |z|.
Like any correlation function, these functions will also depend on the couplings and scale
! used to define the path integral.
The two–point function %Tzz̄(z, z̄)Tzz̄(0)& appearing here satisfies an important posi-
tivity condition. Using canonical quantization, we insert a complete set of QFT states to
find
%Tzz̄(z, z̄)Tzz̄(0)& =
#
n
%0|T̂zz̄(z, z̄) e!H( |n& %n|T̂zz̄(0, 0)|0&
=
#
n
e!En( |%n|T̂zz̄(0, 0)|0&|2
(5.69)
so that this two–point function is positive definite, and it follows thatH(|z|2) is also positive
definite.
Zamolodchikov now used a combination of this positivity condition and the current
conservation equation to construct a certain quantity C(gi,!) that decreases monotonically
along the RG flow. In terms of the two–point functions, current conservation (5.67) for the
energy momentum tensor becomes
4F $ +G$ # 3G = 0 and 4G$ # 4G+H $ # 2H = 0 (5.70)
– 67 –
where F $ = dF (|z|2)/d|z|2 etc. We define the C-function to be
C(|z|2) := 2F #G# 3
8
H (5.71)
which obeys dC/d|z|2 = #3H/4 by the current conservation equations. But by the posi-
tivity of the two–point function %Tzz̄(z, z̄)Tzz̄(0)& this means that
r2
dC
dr2
< 0 (5.72)
so C decreases monotonically under two dimensional scaling transformations, or equiva-
lently under two dimensional RG flow. The value of C at an RG fixed point can be shown
to be the central charge of the CFT.
Ever since it was first proposed, physicists have searched for a generalization of Zamolod-
chikov’s theorem to RG flows in higher dimensions. The two–dimensional quantity Tzz̄ is
just the trace Tµµ of the energy momentum tensor and in 1988 John Cardy proposed that a
certain term – known as “a” — in the expansion of the two–point correlator of Tµµ plays the
role of Zamolodchikov’s C in any even number of dimensions. Cardy’s conjecture was veri-
fied to all orders in perturbation theory the following year by DAMTP’s own Hugh Osborn,
while a complete,non–perturbative proof was finally given in 2011 by Zohar Komargodski
& Adam Schwimmer.
– 68 –
http://arxiv.org/abs/1107.3987
5 Perturbative Renormalization
In this chapter we’ll carry out renormalization of two of the most common theories: ��4
and Quantum Electrodynamics, both working to 1–loop accuracy (i.e. order ~ in the
quantum e↵ective action). We introduce renormalization schemes as a way of fixing the
finite parts of counterterms, and dimensional regularization as a convenient way to control
the asymptotic series expansion of a path integral in terms of Feynman graphs.
5.1 One–loop renormalization of ��4 theory
Consider the scalar theory
S⇤0 [�] =
Z 
1
2
(@�)2 +
1
2
m2�2 +
�
4!
�4
�
d4x (5.1)
with initial couplings m2 and � at scale ⇤0. (The mass coupling is dimensionful here.)
From the analysis of the previous chapter, we expect that near the Gaussian fixed point,
the mass parameter is relevant, while the quartic coupling is marginally irrelevant. Let’s
see how these expectations are borne out in perturbative calculations.
Firstly, the quadratic terms in � — both the kinetic term (@�)2 and the mass term —
receive corrections from graphs with precisely two external � lines. These are computed
by the exact propagator, which in momentum space is the Fourier transform
�(k2) :=
Z
d4x eik·xh�(x)�(0)i (5.2)
of the connected two–point function. If we let ⇧(k2) denote the sum of all 1PI momentum
space Feynman graphs with two external � lines, then we can express the exact propagator
as a geometric series
N0
N1
I+
I�
ı0
ı+
ı�
� �
1PI =
� �
+
� �
+
� � · · ·
+ · · ·= +� � 1PI� � 1PI1PI� ��(k2)
=
1
k2 + m2
+
1
k2 + m2
⇧(k2)
1
k2 + m2
+
1
k2 + m2
⇧(k2)
1
k2 + m2
⇧(k2)
1
k2 + m2
+ · · ·
=
1
k2 + m2 � ⇧(k2) ,
where the leading term in this series
�0(k2) =
1
k2 + m2
(5.3)
is just the propagator in the classical theory.
The corrections to this classical propagator involve ⇧(k2), which is often called the
self–energy of the � field. In perturbation theory, it receives contributions from the loop
diagrams
– 82 –
N0
N1
I+
I�
ı0
ı+
ı�
+ · · ·= +� � 1PI� � 1PI1PI� ��(p2)
= ++
1
p2 + m2
⇧2(p2)
1
p2 + m2
⇧2(p2)
1
p2 + m2
1
p2 + m2
⇧2(p2)
1
p2 + m2
1
p2 + m2
� �
1PI =
� �
+
� �
+
� � · · ·
as well as from counterterms that we’ll consider below. (The dotted lines are intended
to remind us that we amputate the final � propagators in computing the 1PI self-energy
diagrams; the two powers of � are being treated as an external source.) In particular, if
we’re content to work just to 1–loop accuracy, then we only need consider the first of these
Feynman graphs, corresponding to the integral
��
2
Z
|p|⇤0
d4p
(2⇡)4
1
p2 + m2
over the possible values of the momentum running around the loop. Here, we’ve regularised
our theory by only including modes with momenta |p|  ⇤0 and (as always) are working in
Euclidean signature. The factor of 1/2 is the symmetry factor of the 1–loop graph, while
the factor of �� comes from expanding e�S to first order in the coupling. Note also that
this particular integral is independent of the momentum k brought in by the external �
fields. Using the fact that Vol(S3) = 2⇡2, we have
��
2
Z
|p|⇤0
d4p
(2⇡)4
1
p2 + m2
= ��Vol(S
3)
2(2⇡)4
Z ⇤0
0
p3 dp
p2 + m2
= ��m
2
32⇡2
Z ⇤20/m2
0
u du
1 + u
= � �
32⇡2

⇤20 � m2 ln
✓
1 +
⇤20
m2
◆�
.
(5.4)
where we introduced the dimensionless variable u = p2/m2. As expected, this result shows
that the mass parameter is relevant: There’s a quadratic divergence as we try to take the
continuum limit ⇤0 ! 1 (as well as a subleading logarithmic one).
If we wish to obtain finite results in the continuum limit, then we must tune the
coe�cients of each term in our scale-⇤0 action so as to make (5.4) finite in the limit.
Achieving such a tuning by varying (m2,�) directly in (5.4) is complicated, but fortunately,
as discussed above, we don’t need to do this. Instead, we modify the original action by
including counterterms:
S⇤0 [�] ! S⇤0 [�] + ~SCT[�, ⇤0] (5.5)
where in this case the counterterm action is
SCT[�, ⇤0] =
Z 
1
2
�Z(@�)2 +
1
2
�m2�2 +
1
4!
�� �4
�
d4x (5.6)
with (�Z, �m2, ��) representing our freedom to adjust the couplings in the original action
(including the coupling to the kinetic term — or wavefunction renormalization). These
– 83 –
counterterm couplings will depend explicitly on ⇤0, as they represent the tuning that must
be performed starting from the ⇤0 cut–o↵ theory.
The fact that the counterterm action is proportional to ~ means that classical con-
tributions from SCT contribute to the same order in ~ as 1-loop diagrams from S⇤0 [�].
In particular, the two–point function �(k2) receives a correction at order ~ from both
the wavefunction renormalization and mass counterterms, contributing to the self–energy
⇧(k2) as
�1
�2 �3
�4 �1
�2 �3
�4 �1
�2 �3
�4
���
+
��m2�k2�Z
We recall that since the counterterms are being treated as vertices, at first order they
contribute with a minus sign coming from the expansion e�S[�]. We also note that in
momentum space the counterterm �Z(@�)2/2 comes with a factor of k2 since it couples
to the derivatives of the field. The full quantum contribution to the quadratic terms in �
arises from the self–energy terms
⇧1 loop(k2) = �k2�Z � �m2 � �
32⇡2

⇤20 � m2 ln
✓
1 +
⇤20
m2
◆�
(5.7)
to 1–loop accuracy.
If we required higher accuracy, we should include 1PI graphs built from both higher
loops using the propagators and vertices of the original classical action, together with lower
loop graphs that also include one or more insertions of the counterterms as vertices. As
here, each counterterm vertex counts as a loop when considering the order of the graph.
For example, at two loops we have contributions from the 2–loop graphs shown above,
together with the graphs
�1
�2 �3
�4
�1
�2 �3
�4
�1
�2 �3
�4
��� +
��m2�k2�Z
�p2 �Z
�� ��
��m2
���
that include counterterms. Note that the wavefunction renormalization counterterm here
comes with coe�cient p2, since the momentum carried by the propagator on which it is
inserted is p. The result of all these diagrams (which we will not compute) should then be
added to (5.7) to obtain ⇧(k2) to order ~2 accuracy.
5.1.1 The on–shell renormalization scheme
The raison d’etre of counterterms is to ensure (5.7) has a finite continuum limit, so that
they must cancel the part of (5.4) that diverges as ⇤0 ! 1. This still leaves us a lot of
freedom in choosing how much of the finite part of the loop integral can also be absorbed
by the counterterms. There’s no preferred way to do this, and any such choice is called a
renormalization scheme. Ultimately, all physically measurable quantities (such as cross–
sections, branching ratios, particle lifetimes etc.) should be independent of the choice of
renormalization scheme.
– 84 –
Our calculation above involved two independent counterterms, �m2 and �Z, so we’ll
need two independent renormalization conditions to fix them. One way to proceed is known
as the on–shell scheme. In this scheme, we fix (�m2, �Z) by asking that, once we take
the continuum limit, the exact � propagator �(k2) in (5.2) has a simple pole at some
experimentally measured value �k2 = m2phys, and that the residue of this pole is unity.
These requirements are motivated by the fact that the classical propagator (k2 + m2)�1
has a pole when �k2 = m2. Here, m2 is a parameter in the action which would indeed
be interpreted as the mass of the particle according to the classical equations of motion.
However, we’ve seen that the value of this coupling (like all others) is shifted by quantum
corrections, so the true mass of � could well be very di↵erent. Experimentally, we can
measure the true mass of a particle by looking for peaks (resonances) in scattering cross–
sections44 where this particle is exchanged. As you learnt when studying scattering theory
in Quantum Mechanics, these peakscorrespond to poles of the S-matrix in the complex
momentum plane.
Since we want �(k2) to have a simple pole when k2 = �m2phys, the self–energy contri-
butions must obey
⇧(�m2phys) = m2 � m2phys , (5.8a)
and since we want this pole to have unit residue, we also require
@⇧
@k2
����
k2=�m2phys
= 0 . (5.8b)
Most often, it’s convenient to set up our original action so that the parameter m2 in the
action is indeed the experimentally measured value m2phys, so in this case we’d want
⇧(�m2phys) = 0 . (5.8c)
Note that this condition doesn’t mean the counterterms are chosen to identically cancel
quantum corrections to the propagator, just that they cancel at the particular on–shell
value �k2 = m2phys of the external momentum.
To 1-loop accuracy, taking into account both the loop integral and the counterterm,
we found the self–energy (5.7)
⇧1 loop(k2) = �k2�Z � �m2 � �
32⇡2

⇤20 � m2 ln
✓
1 +
⇤20
m2
◆�
. (5.9)
Since the only k2 dependence on the rhs is from the wavefunction renormalization coun-
terterm, condition (5.8b) on the residues gives simply
�Z = 0 (5.10)
at this order. If we wish the Lagrangian parameter m to correspond to the experimentally
measured value of the mass, then condition (5.8c) on the location of the pole fixes
�m2 = � �
32⇡2

⇤20 � m2 ln
✓
1 +
⇤20
m2
◆�
(5.11)
44For short–lived particles, it’s not feasible to get out your kitchen scales.
– 85 –
in the on–shell renormalization scheme.
Equation (5.10) shows that no wavefunction counterterm is necessary in ��4 theory
at 1 loop. This is easy to understand: Consider the Feynman diagram contributions to
⇧(k2) displayed above. In the first two graphs, the momenta flowing around the loops is
independent of the external momentum k, so in space–time they will generate corrections
to the �2 term in the e↵ective action, but not to terms involving derivatives of �2, such
as the kinetic term (@�)2. This is something of a coincidence in ��4 theory at 1–loop and
in a more generic theory, we would expect the two–point function
R
ddx eik·xh�(x)�(0)i to
be sensitive to wavefunction renormalization factors (even at 1–loop) as well as to mass
renormalization.
Indeed, even in ��4 theory, by the time we get to 2–loops there is non–trivial wave-
function renormalization. We can see this from the final Feynman graph shown above.
In this graph, the external momentum k carried by � must flow through at least one of
the internal propagators. Expanding this propagator as an infinite power series in k2, we
see that the corresponding loop integral will contribute an infinite series of position space
derivative terms acting on two powers of �. The term / k2 will correspond to a correc-
tion to the kinetic term (@�)2 and so generates a non–trivial wavefunction renormalization
factor. Subleading terms generate new, higher derivative operators in the e↵ective action,
such as (@2�)2. As we saw in our d = 1 QFT, the infinite series of such operators this
loop graph generates really shows that the e↵ective theory is non-local. However, just as
in d = 1, all these higher derivative operators are irrelevant under RG flow of scalar field
theory in d = 4 and so will vanish when we take ⇤0 ! 1. The resulting theory will be
local.
Notice that we cannot sensibly take the limit ⇤0 ! 1 in either the 1-loop correc-
tion (5.4) or the counterterm (5.11) separately. However, the combined contribution
⇧1�loop(k
2) = ��m2 � �
32⇡2

⇤20 � m2 ln
✓
1 +
⇤20
m2
◆�
= 0 (5.12)
is perfectly well–behaved in the continuum limit. This reflects the fact that neither the
continuum action nor the path integral measure D� over all modes exists: It does not
make sense to take the limit ⇤0 ! 1 before computing the path integral. However, if we
hold ⇤0 fixed at some finite value and compute the path integral to any desired accuracy
in ~, then we can make sense of the final result as ⇤0 ! 1. While our tuning (5.11) is
good to 1–loop accuracy, if we computed the path integral to higher order in perturbation
theory, including the contribution of higher–loop Feynman diagrams, we would have to
make further tunings in (�m2, �Z), proportional to higher powers of the coupling �, so as
to still retain a finite limiting result. In particular, the fact that the combined contribution
to ⇧(k2) vanishes identically (not just when �k2 = m2) is again a 1–loop accident in ��4
theory.
5.1.2 Renormalization of the quartic coupling
The 1-loop correction to the quartic vertex is given by the three Feynman graphs
– 86 –
�1
�2 �3
�4 �1
�2 �3
�4 �1
�2 �3
�4
each with four external � fields. In conventions where each external momentum ki is flowing
into the diagram, these correspond to the integrals
�2
2
Z
d4p
(2⇡)4
1
p2 + m2
1
(p + k1 + k2)2 + m2
+
�2
2
Z
d4p
(2⇡)4
1
p2 + m2
1
(p + k1 + k4)2 + m2
+
�2
2
Z
d4p
(2⇡)4
1
p2 + m2
1
(p + k1 + k3)2 + m2
(5.13)
in momentum space, taken over the region |p|  ⇤0. If the loop momentum has magnitude
much greater than the physical mass m or the momenta ki carried in by the external fields,
then we see that each integral behaves as ⇠
R ⇤0 p3dp/p4 and so diverges logarithmically if
we näıvely try to send ⇤0 ! 1.
Because the external momentum flows through the propagators that make up the loop,
the Feynman graphs above also generate non–local contributions to the e↵ective action.
This is just a momentum space reflection of the fact that, in the position space Feynman
graphs, the two vertices in each graph are at di↵erent points x, x0 2 M . Expanding (5.13)
in powers of the external momenta generates an infinite series of derivative interactions
such as ⇠ @k�2 @k�2, each containing an even number of derivatives acting in some way
on the four external � fields. In the UV region, the appropriate expansion is in powers
of |ki|/|p|. Thus, for every addition power of the external momenta, the loop integral is
suppressed by a further power of 1/|p|. You should check that, just as we expected, all such
derivative terms are irrelevant — although turning on the coupling � in our initial scale
⇤0 theory generates such non–local terms, they all vanish in the continuum limit ⇤0 ! 1
as we are driven to focus on the renormalized trajectory.
The pure correction to the �4 coupling comes just from the ki-independent part of
each of the three loop diagrams, all of which are equal. The �4 coupling thus receives a
1–loop correction
3
�2
2
Z
d4p
(2⇡)4
1
(p2 + m2)2
=
3�2
16⇡2
Z ⇤0
0
p3dp
(p2 + m2)2
=
3�2
32⇡2
Z ⇤20/m2
0
u du
(1 + u)2
=
3�2
32⇡2

ln
✓
1 +
⇤20
m2
◆
� ⇤
2
0
⇤20 + m
2
� (5.14)
which is indeed logarithmically divergent. To obtain a finite continuum limit we tune our
initial coupling using the counterterm contribution
– 87 –
�1
�2 �3
�4 �1
�2 �3
�4 �1
�2 �3
�4
���
Again, the counterterm must remove the logarithmic divergence in (5.21), and a choice of
renormalization scheme is a choice of how to fix the finite parts ��. A standard choice
would be to take
�� =
3�2
32⇡2

ln
⇤20
m2
� 1
�
(5.15)
in which case the pure �4 coupling in the quantum e↵ective action is
�e↵ = ��
3~�2
32⇡2

ln
✓
1 +
m2
⇤20
◆
+
m2
m2 + ⇤20
�
(5.16)
which is again perfectly finite as ⇤0 ! 1. In particular, with this renormalization scheme,
in the continuum limit we find �e↵ = � so that the pure �4 coupling in the quantum
e↵ective action again coincides with the parameter � in the classical action.
5.1.3 Irrelevant interactions and the quantum e↵ective action
To uncover something more interesting, let’s reconsider the e↵ect of the external momenta
in the previous calculation. As we said above, these will mean that the quantum e↵ective
action contains an infinite number of quartic interactions including ever more derivatives
of the fields. To compute them, we must evaluate the loop integrals in (5.13) at ki 6= 0.
Let’s now do this, concentrating on the first integral.
We first note Feynman’s trick
Z 1
0
dx
[xA + (1 � x)B]2 =
1
B � A

1
xA + (1� x)B
�1
0
=
1
AB
(5.17)
which allows us to combine the two propagators as e.g.
1
(p + k12)2 + m2
1
p2 + m2
=
Z 1
0
dx
[x((p + k12)2 + m2) + (1 � x)(p2 + m2)]2
=
Z 1
0
dx
[p2 + m2 � 2xp · k12 + xk212]2
=
Z 1
0
dx
[(p � xk12)2 + m2 + x(1 � x)k212]2
(5.18)
in terms of k212 = (k1 + k2)
2. Defining ` = p � xk12 we have the loop integral
�2
2
Z
d4p
(2⇡)4
1
p2 + m2
1
(p + k1 + k2)2 + m2
=
�2
2
Z
d4`
(2⇡)4
Z 1
0
dx
[`2 + m2 + x(1 � x)k212]2
.
(5.19)
We’re really supposed to integrate this over the region |p|  ⇤0. However, since our interest
is ultimately in the ⇤0 ! 1 continuum limit, we can instead integrate over the region
– 88 –
|`|  ⇤0 which di↵ers from the original region by terms of order |k12|/⇤0.45 Interchanging
the order of the integrals, which is allowed as they are both absolutely convergent whenever
⇤0 is finite, we have
Z
d4`
Z 1
0
dx
[`2 + m2 + x(1 � x)k212]2
= Vol(S3)
Z 1
0
dx
Z ⇤0
0
`3d`
[`2 + m2 + x(1 � x)k212]2
= ⇡2
Z 1
0
dx
Z ⇤20
0
`2 d(`2)
[`2 + m2 + x(1 � x)k212]2
= ⇡2
Z 1
0

ln
✓
⇤20 + m
2 + x(1 � x)k212
m2 + x(1 � x)k212
◆
+
m2 + x(1 � x)k212
⇤20 + m
2 + x(1 � x)k212
� 1
�
dx .
(5.20)
Including the contributions from the remaining two diagrams and keeping only terms which
do not vanish as ⇤0 ! 1 (since these are the only parts we have reliably computed after
our change of integration region), these three loop diagrams actually yield
�2
32⇡2
Z 1
0

ln
⇤20
m2 � x(1 � x)s + ln
⇤20
m2 � x(1 � x)t + ln
⇤20
m2 � x(1 � x)u � 3
�
dx , (5.21)
where we’ve written the external momenta in terms of the Mandelstam variables
s = �(k1 + k2)2 , t = �(k1 + k4)2 , u = �(k1 + k3)2 . (5.22)
These terms are present in the quantum e↵ective action, written in momentum space, with
the factors of s, t and u indicating the presence of derivatives acting on the fields in the
position space quantum e↵ective action.
Combined with the classical action and counterterm as before, the total coe�cient of
�4 in the e↵ective action is thus
�+ ~ ��� ~�
2
32⇡2
Z 1
0

ln
⇤20
m2 � x(1 � x)s + ln
⇤20
m2 � x(1 � x)t + ln
⇤20
m2 � x(1 � x)u � 3
�
dx
to order ~. With our earlier choice
�� =
3�2
32⇡2

ln
⇤20
m2
� 1
�
(5.23)
for the counterterm, this is
A(ki) = ��
~�2
32⇡2
Z 1
0

ln
m2
m2 � x(1 � x)s + ln
m2
m2 � x(1 � x)t + ln
m2
m2 � x(1 � x)u
�
dx
= �e↵ +
~�2e↵
32⇡2
Z 1
0
"
ln
 
1 � x(1 � x)s
m2phys
!
+ ln
 
1 � x(1 � x)t
m2phys
!
+ ln
 
1 � x(1 � x)u
m2phys
!#
dx .
(5.24)
where we can replace (m2,�) 7! (m2phys,�e↵) to this order in ~. (For the second term
in (5.24), which has an explicit factor of ~ in front, at order ~ this is true no matter what
45Note that the two regions are certainly the same if the integral was over all of R4 as the measure
d4p = d4` is translationally invariant. Thus the di↵erence between the two regions vanishes as ⇤0 ! 1.
– 89 –
renormalization scheme we use, since any scheme dependence would have to come with a
further factor of ~.)
The first thing to notice about (5.24) is that it is completely finite in the continuum
limit. We only needed to tune the initial values of m2 and � – the relevant and (classically)
marginal couplings – in order to obtain this result, even though this represents an infinite
series of higher–derivative interactions in the quantum action. Put di↵erently, although
quantum e↵ects do indeed turn on this infinite series of irrelevant higher derivative oper-
ators (together with others we’ll consider momentarily), the coe�cients of these operators
are completely fixed in terms of (m2phys,�e↵). We can understand this result in terms of
our previous picture46
of RG flow as follows. In the continuum limit, we’ve focussed in on some renormalized
trajectory, but by tuning our initial couplings (m2,�) we’re only finitely far along this
trajectory even in the continuum limit. There’s no claim that the trajectory always stays
‘perpendicular’ or ‘normal’ to the critical surface; travelling along the trajectory may well
turn on other, irrelevant operators. However, the presence of these operators is simply
a consequence of being on the renormalized trajectory, so they do not come with inde-
pendent coe�cients. At least in principle, the relation between the coe�cients of such
irrelevant operators and (m2phys,�e↵) can indeed be understood as an equation describing
our renormalized trajectory in the space of all possible couplings.
The quantum e↵ective action also contains further irrelevant operators with more
powers of the field (and again arbitrarily many derivatives), coming from 1PI loop diagrams
of the general form
N0
N1
I+
I�
ı0
ı+
ı�
� �
1PI =
� �
+
� �
+
� � · · ·
+ · · ·= +� � 1PI� � 1PI1PI� ��(k2)
...
46It took me ages to construct this figure, so you can bet I’m going to use it as often as possible!
– 90 –
These loop diagrams, containing e > 2 propagators, behave in the UV like
R ⇤0 d4p/p2e and
so are finite in the continuum limit.47 Again, the coe�cients of these operators in �[�] are
completely fixed in terms of our location along the renormalized trajectory, parametrized
by (m2phys,�e↵).
The whole point of the 1PI quantum e↵ective action �[�] is that its vertices incorporate
all possible quantum e↵ects already. Thus, if we want to compute a scattering amplitude it
is su�cient to compute with tree–level diagrams built from the vertices and propagator of
the quantum e↵ective action. Beginning with just a ��4 interaction in the classical action,
�[�] contains no interactions with fewer than four fields48, so if in particular we wish to
compute a four–particle scattering amplitude, the only tree diagram we can consider is
�1
�2 �3
�4
�1
�2 �3
�4
�1
�2 �3
�4
��� +
��m2�k2�Z
�p2 �Z
�� ��
��m2
���
� �
4�(�)
��4
����
�=0
where the vertex corresponds to all possible terms in �[�] with exactly four fields, including
all the irrelevant, higher derivative terms. We already know the answer: if the fields we’re
scattering correspond to momentum eigenstates with momenta (k1, k2, k3, k4), it’s
�4
 
4X
i=1
ki
!
A(ki)
where A(ki) is given in equation (5.24).
Of course, the work in computing all the quartic terms in the quantum e↵ective action
is precisely the same work we have to do to compute this one–loop amplitude, and indeed
the calculation is usually presented without mentioning �[�] just as a way of getting a
result for this amplitude, correct to order ~. Exactly as we saw earlier in low dimensions,
the virtue of knowing the e↵ective action is that we can put the same result to work in
computing higher point scattering amplitudes. For example, the six particle amplitude is
given by the tree diagrams
�1
�2 �3
�4
�1
�2 �3
�4
�1
�2 �3
�4
��� +
��m2�k2�Z
�p2 �Z
�� ��
��m2
���
� �
4�(�)
��4
����
�=0
using the vertex �6�[�]/��6|�=0 that incorporates the e↵ect of the 1-loop diagram above, to-
gether with tree diagrams built from the vertex �4�[�]/��4|�=0 and propagator
�
�2�[�]/��2
��1
.
47The expression
R ⇤0 d4p/p2e is only a good approximation to our integral in the UV region where
p
2 � m2, k2i . Thus, while both the integral and the actual loop integral receive vanishing contributions
from this UV region, the true loop integral has a finite contribution from the low energy region where the
masses and external momenta are non–negligible.
48This result is obvious at 1-loop, as there are no 1PI 1–loop diagrams with 3 external legs. We’ll prove
it more generally in the next chapter.
– 91 –
It’s also interesting to consider what happens if we start from a more general clas-
sical action, including couplings to higher powers of � and its derivatives to begin with.
According to our picture of RG flow, we expecting these higher couplings to be irrelevant
and thus have negligible e↵ect at any finite energy if we start with fixed numerical values
for the dimensionless couplings gi and take the continuum limit. Let’s briefly see how this
works, choosingan action
S⇤0 [�] =
Z 
1
2
(@�)2 +
1
2
m2�2 + V (�)
�
d4x , (5.25)
where for simplicity we assume the potential
V (�) =
X
k�2
g2k ⇤
4�2k
0 �
2k (5.26)
includes only even powers of � and no derivatives of �.
To compute the coupling to �2m in the quantum e↵ective action, we need to consider
all 1PI Feynman diagrams we can build using these vertices. To 1-loop accuracy, the only
such graphs are of the generic type
..
.
..
.
... ..
.
. . .
...
+ + + · · ·
r = 0 r = a
u = const
u
r
A A
� �
6.
4.
1
N
ai
ve
Fe
yn
m
an
R
ul
es
W
e
wa
nt
to
de
te
rm
in
e
th
e
Fe
yn
m
an
ru
les
fo
r t
hi
s
th
eo
ry
.
Fo
r f
er
m
ion
s,
th
e
ru
les
ar
e
th
e
sa
m
e
as
th
os
e
gi
ve
n
in
Se
ct
io
n
5.
Th
e
ne
w
pi
ec
es
ar
e:
•
W
e
de
no
te
th
e
ph
ot
on
by
a
wa
vy
lin
e.
Ea
ch
en
d
of
th
e
lin
e
co
m
es
wi
th
an
i,
j =
1,
2,
3
in
de
x
te
lli
ng
us
th
e c
om
po
ne
nt
of
�A.
W
e
ca
lcu
la
te
d
th
e t
ra
ns
ve
rs
e
ph
ot
on
pr
op
ag
at
or
in
(6
.3
3)
: i
t i
s
an
d
co
nt
rib
ut
es
D
tr
ij
=
i
p
2 +
i�
✓ � ij
�
p i
p j
|�p|
2
◆
•
Th
e
ve
rt
ex
co
nt
rib
ut
es
�i
e�
i .
Th
e
in
de
x
on
�
i c
on
tr
ac
ts
wi
th
th
e
in
de
x
on
th
e
ph
ot
on
lin
e.
•
Th
e
no
n-
lo
ca
l i
nt
er
ac
tio
n
wh
ich
, i
n
po
sit
io
n
sp
ac
e,
is
gi
ve
n
by
x
y
co
nt
rib
ut
es
a
fa
ct
or
of
i(e
�
0 )
2 �(
x
0 �
y
0 )
4�
|�x
�
�y|
Th
es
e
Fe
yn
m
an
ru
les
ar
e
ra
th
er
m
es
sy
.
Th
is
is
th
e
pr
ice
we
’ve
pa
id
fo
r
wo
rk
in
g
in
Co
ul
om
b
ga
ug
e.
W
e’l
l n
ow
sh
ow
th
at
we
ca
n
m
as
sa
ge
th
es
e e
xp
re
ss
ion
s i
nt
o
so
m
et
hi
ng
m
uc
h
m
or
e
sim
pl
e
an
d
Lo
re
nt
z
in
va
ria
nt
. L
et
’s
st
ar
t w
ith
th
e
o�
en
di
ng
in
st
an
ta
ne
ou
s
in
te
ra
ct
io
n.
Si
nc
e
it
co
m
es
fro
m
th
e
A 0
co
m
po
ne
nt
of
th
e
ga
ug
e
fie
ld
, w
e
co
ul
d
tr
y
to
re
de
fin
e
th
e
pr
op
ag
at
or
to
in
clu
de
a
D 0
0
pi
ec
e
wh
ich
wi
ll
ca
pt
ur
e
th
is
te
rm
. I
n
fa
ct
, i
t
fit
s q
ui
te
ni
ce
ly
in
th
is
fo
rm
: i
f w
e
lo
ok
in
m
om
en
tu
m
sp
ac
e,
we
ha
ve
�(
x
0 �
y
0 )
4�
|�x
�
�y|
=
Z
d
4 p
(2
�)
4
e
ip
·(x
�y
)
|�p|
2
(6
.8
3)
so
we
ca
n
co
m
bi
ne
th
e n
on
-lo
ca
l i
nt
er
ac
tio
n
wi
th
th
e t
ra
ns
ve
rs
e
ph
ot
on
pr
op
ag
at
or
by
de
fin
in
g
a
ne
w
ph
ot
on
pr
op
ag
at
or
D µ
⌫
(p
) =
������
�����
+
i
|�p|
2
µ,
�
=
0
i
p
2 +
i�
✓ � ij
�
p i
p j
|�p|
2
◆
µ
=
i 6=
0,
�
=
j 6=
0
0
ot
he
rw
ise
(6
.8
4)
W
ith
th
is
pr
op
ag
at
or
, t
he
wa
vy
ph
ot
on
lin
e
no
w
ca
rr
ies
a
µ,
�
=
0,
1,
2,
3
in
de
x,
wi
th
th
e e
xt
ra
µ
=
0
co
m
po
ne
nt
ta
ki
ng
ca
re
of
th
e i
ns
ta
nt
an
eo
us
in
te
ra
ct
io
n.
W
e n
ow
ne
ed
to
ch
an
ge
ou
r v
er
te
x
sli
gh
tly
: t
he
�i
e�
i ab
ov
e
ge
ts
re
pl
ac
ed
by
�i
e�
µ w
hi
ch
co
rr
ec
tly
ac
co
un
ts
fo
r t
he
(e
�
0 )
2 p
iec
e
in
th
e
in
st
an
ta
ne
ou
s i
nt
er
ac
tio
n.
–
14
1
–
6.
4.
1
N
ai
ve
Fe
yn
m
an
R
ul
es
W
e
wa
nt
to
de
te
rm
in
e
th
e
Fe
yn
m
an
ru
les
fo
rt
hi
s
th
eo
ry
.
Fo
rf
er
m
ion
s,
th
e
ru
les
ar
e
th
e
sa
m
e
as
th
os
e
gi
ve
n
in
Se
ct
io
n
5.
Th
e
ne
w
pi
ec
es
ar
e:
•
W
e
de
no
te
th
e
ph
ot
on
by
a
wa
vy
lin
e.
Ea
ch
en
d
of
th
e
lin
e
co
m
es
wi
th
an
i,
j=
1,
2,
3
in
de
x
te
lli
ng
us
th
ec
om
po
ne
nt
of
�A.
W
e
ca
lcu
la
te
d
th
et
ra
ns
ve
rs
e
ph
ot
on
pr
op
ag
at
or
in
(6
.3
3)
:i
ti
s
an
d
co
nt
rib
ut
es
D
tr
ij
=
i
p
2+
i�
✓�ij
�
pi
pj
|�p|
2
◆
•
Th
e
ve
rt
ex
co
nt
rib
ut
es
�i
e�
i.
Th
e
in
de
x
on
�
ic
on
tr
ac
ts
wi
th
th
e
in
de
x
on
th
e
ph
ot
on
lin
e.
•
Th
e
no
n-
lo
ca
li
nt
er
ac
tio
n
wh
ich
,i
n
po
sit
io
n
sp
ac
e,
is
gi
ve
n
by
x
y
co
nt
rib
ut
es
a
fa
ct
or
of
i(e
�
0)
2�(
x
0�
y
0)
4�
|�x
�
�y|
Th
es
e
Fe
yn
m
an
ru
les
ar
e
ra
th
er
m
es
sy
.
Th
is
is
th
e
pr
ice
we
’ve
pa
id
fo
r
wo
rk
in
g
in
Co
ul
om
b
ga
ug
e.
W
e’l
ln
ow
sh
ow
th
at
we
ca
n
m
as
sa
ge
th
es
ee
xp
re
ss
ion
si
nt
o
so
m
et
hi
ng
m
uc
h
m
or
e
sim
pl
e
an
d
Lo
re
nt
z
in
va
ria
nt
.L
et
’s
st
ar
tw
ith
th
e
o�
en
di
ng
in
st
an
ta
ne
ou
s
in
te
ra
ct
io
n.
Si
nc
e
it
co
m
es
fro
m
th
e
A0
co
m
po
ne
nt
of
th
e
ga
ug
e
fie
ld
,w
e
co
ul
d
tr
y
to
re
de
fin
e
th
e
pr
op
ag
at
or
to
in
clu
de
a
D0
0
pi
ec
e
wh
ich
wi
ll
ca
pt
ur
e
th
is
te
rm
.I
n
fa
ct
,i
t
fit
sq
ui
te
ni
ce
ly
in
th
is
fo
rm
:i
fw
e
lo
ok
in
m
om
en
tu
m
sp
ac
e,
we
ha
ve
�(
x
0�
y
0)
4�
|�x
�
�y|
=
Z
d
4p
(2
�)
4
e
ip
·(x
�y
)
|�p|
2
(6
.8
3)
so
we
ca
n
co
m
bi
ne
th
en
on
-lo
ca
li
nt
er
ac
tio
n
wi
th
th
et
ra
ns
ve
rs
e
ph
ot
on
pr
op
ag
at
or
by
de
fin
in
g
a
ne
w
ph
ot
on
pr
op
ag
at
or
Dµ
⌫
(p
)=
������
�����
+
i
|�p|
2
µ,
�
=
0
i
p
2+
i�
✓�ij
�
pi
pj
|�p|
2
◆
µ
=
i6=
0,
�
=
j6=
0
0
ot
he
rw
ise
(6
.8
4)
W
ith
th
is
pr
op
ag
at
or
,t
he
wa
vy
ph
ot
on
lin
e
no
w
ca
rr
ies
a
µ,
�
=
0,
1,
2,
3
in
de
x,
wi
th
th
ee
xt
ra
µ
=
0
co
m
po
ne
nt
ta
ki
ng
ca
re
of
th
ei
ns
ta
nt
an
eo
us
in
te
ra
ct
io
n.
W
en
ow
ne
ed
to
ch
an
ge
ou
rv
er
te
x
sli
gh
tly
:t
he
�i
e�
iab
ov
e
ge
ts
re
pl
ac
ed
by
�i
e�
µw
hi
ch
co
rr
ec
tly
ac
co
un
ts
fo
rt
he
(e
�
0)
2p
iec
e
in
th
e
in
st
an
ta
ne
ou
si
nt
er
ac
tio
n.
–
14
1
–
⇠ |k|
2g2
⇤3
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesDtrij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�i.Theindexon�icontractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenbyxy
contributesafactorof
i(e�0)2�(x0�y0)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x0�y0)
4�|�x��y|=
Z
d4p
(2�)4
eip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
�����
�����
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�iabovegetsreplacedby�ie�µwhichcorrectly
accountsforthe(e�0)2pieceintheinstantaneousinteraction.
–141–
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesDtrij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�i.Theindexon�icontractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenbyxy
contributesafactorof
i(e�0)2�(x0�y0)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave�(x0�y0)
4�|�x��y|=
Z
d4p
(2�)4
eip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
�����
�����
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�iabovegetsreplacedby�ie�µwhichcorrectly
accountsforthe(e�0)2pieceintheinstantaneousinteraction.
–141–
A� A�
q
where each graph has a total of 2m (amputated) external legs. These graphs are of course
just a four dimensional version of the graphs we already considered in our e↵ective theories
in zero and one dimension, and indeed they can be understood as the contribution to the
quantum e↵ective action from the determinant det(�r2 + m2 + V 00(�))�1/2 obtained by
integrating out quadratic fluctuations around a background value of �. We saw that the
order ~ contribution to the asymptotic series of the quantum e↵ective action can always
be understood in terms of such determinants.
A 1-loop graph with a total of e propagators contributes an amount
/
Z
d4p
(2⇡)4
eY
j=1
1
(p + Kj)2 + m2
to the �2m coupling in the momentum space quantum e↵ective action49, where p + Kj is
the total momentum carried by the jth propagator. If the cut–o↵ ⇤0 is much larger than
the mass or typical scale of the incoming momenta, then this behaves as
Z ⇤0 p3 dp
p2v
⇠
(
⇤20 when e = 1
ln ⇤0 when e = 2
(5.27)
49As is by now familiar, diagrams with more than one vertex are non–local and so also generate and
infinite series of derivative interactions. These terms are suppressed in the UV by powers of |Kj |/⇤0
compared to the non–derivative terms.
– 92 –
in the UV region, and is UV finite if there are more than two propagators. On the other
hand, for every vertex �2k+2 in V (�) that is involved in such a graph, as well as the
dimensionless coupling g2k+2 we have a factor of ⇤
4�2k�2
0 = ⇤
2(1�k)
0 , which is a suppression
at large ⇤0 if k > 1.
Since the above loop integrals with more than two propagators (e > 2) are all UV
finite, every such diagram that involves one or more �6 or higher vertex is suppressed by
a positive power of ⇤0 and so vanishes in the continuum limit. This also applies to the
logarithmically divergent loop integrals with e = 2, and any quadratically divergent loop
integral with e = 1 provided if its vertex is �8 or higher.50
The only divergent loop diagrams are thus the mass and quartic vertex diagrams
N0
N1
I+
I�
ı0
ı+
ı�
� �
1PI =
� �
+
� �
+
� � · · ·
+ · · ·= +� � 1PI� � 1PI1PI� ��(k2)
...we’ve already considered. There is also a finite contribution to the quartic coupling coming
from the diagram
N0
N1
I+
I�
ı0
ı+
ı�
� �
1PI =
� �
+
� �
+
� � · · ·
+ · · ·= +� � 1PI� � 1PI1PI� ��(k2)
...
involving a single propagator and a �6 vertex. However, since we in any case have to tune
the initial coupling of �4 using a counterterm, we can always incorporate the e↵ects of this
finite contribution in a choice of renormalisation scheme. These terms are thus nothing
beyond what we’ve already seen.
Thus, even starting from our generic potential V (�), the only contributions to �2m
with m > 2 that do not vanish in the continuum limit come from 1-loop integrals that are
built exclusively using the �4 vertex
N0
N1
I+
I�
ı0
ı+
ı�
� �
1PI =
� �
+
� �
+
� � · · ·
+ · · ·= +� � 1PI� � 1PI1PI� ��(k2)
...
as before. There must be exactly m propagators in this loop, so these contributions to the
quantum e↵ective action are finite. Thus, even allowing for arbitrary irrelevant operators
in the original classical action, when we take the continuum limit the strength of the in-
teractions in the quantum e↵ective action �[�] is completely fixed by our location along the
renormalized trajectory. The irrelevant couplings in the classical action play no role.
50I caution you that the extension of this argument to higher loops is much more involved. Nonetheless,
the WIlsonian picture of couplings evolving with scale assures us it must hold.
– 93 –
5.1.4 Dimensional regularization and the MS scheme
While the idea of integrating out momenta only up to a cut–o↵ ⇤0 is very intuitive, in more
complicated examples it becomes very cumbersome to perform the loop integrals over |p|
with a finite upper limit. More seriously, in a gauge theory, simply imposing a cut–o↵ in
momentum space is incompatible with gauge invariance
We saw that whether a coupling was relevant, marginal or irrelevant is largely deter-
mined just by dimensional analysis, and that the the dimension of an operator or coupling
depends on the dimension of the space on which the QFT lives. This suggests that we
can regularize our loop integrals by computing in some generic number of dimensions d.
Since couplings may be relevant or irrelevant as d changes, they can’t diverge if we keep
d arbitrary. Having computed our loop integrals in a generic number of dimensions, we
include counterterms to tune the initial couplings in our d–dimensional theory so as to
obtain a finite limit as we approach the physically relevant dimension (usually, d = 4).
Unlike a lattice regularization, or choosing to integrate only finitely many momentum
modes in the path integral, dimensional regularization is only a perturbative regularization
scheme: whilst it does allow us to regulate individual loop integrals over the full range
|p| 2 [0, 1) (as we shall see), it does not provide any definition of a finite–dimensional
path integral measure. Furthermore, in order to ‘approach’ the physical dimension, we
need to analytically continue the results of our generic d–dimensional theory through non–
integer values of d. I stress that this is purely a convenient device for regularizing loop
integrals — there is no suggestion that Nature ‘really’ lives in non–integer dimensions.
These conceptual shortcomings are compensated by its practical convenience: dimensional
regularization is very easy to implement, whilst integrating complicated integrals over a
finite range |p| 2 [0, ⇤0) is often prohibitively di�cult.
To see how this works, let’s repeat our previous calculation of the 1–loop corrections to
⇧(k2) in ��4 theory, now working in some generic space–time dimension d. In d dimensions,
the quartic coupling � has non–zero mass dimension [�] = 4 � d, so we write
� = µ4�dg(µ) (5.28)
where the new coupling g(µ) = �/µ4�d is dimensionless. We see that µ is an arbitrary
mass scale, such as a kilogram, and g(µ) is the value of our quartic coupling in units of
this scale. I stress that µ is not a cut–o↵; it’s just an arbitrary scale introduced so to allow
us to use dimensionless couplings. We will usually choose µ to be of the typical scale of
the experiment we’re interested in performing: if you’re working with subatomic particles,
it’s not likely that you’ll want to express your answers in terms of kilotonnes, or inverse
lightyears.
In terms of this scale, we obtain the 1-loop correction to the mass
� 1
2
g(µ) µ4�d
Z
ddp
(2⇡)d
1
p2 + m2
= �g(µ) µ4�d Vol(S
d�1)
2(2⇡)d
Z 1
0
pd�1 dp
p2 + m2
(5.29)
– 94 –
in dimensional regularization. To compute the area of a unit sphere in d dimensions, we
use the following trick. For any d 2 N, comparing Cartesian and polar coordinates gives us
(
p
⇡)d =
Z
Rd
dY
i=1
e�x
2
i dxi = Vol(S
d�1)
Z 1
0
e�r
2
rd�1 dr =
1
2
Vol(Sd�1) �(d/2) , (5.30)
where we recognize the radial integral as (half) a gamma function. Thus, when d 2 N we
have
Vol(Sd�1) =
2⇡
d
2
� (d/2)
. (5.31)
We now define this to be Vol(Sd�1) for any d 2 C by analytic continuation.
The remaining integral in (5.29) is
µ4�d
Z 1
0
pd�1 dp
p2 + m2
=
1
2
µ4�d
Z 1
0
(p2)d/2�1 d(p2)
p2 + m2
=
m2
2
⇣ µ
m
⌘4�d Z 1
0
(1 � u)
d
2�1 u�
d
2 du
=
m2
2
⇣ µ
m
⌘4�d �(d2) �(1 �
d
2)
�(1)
,
(5.32)
where u = m2/(p2 + m2), and in thelast line we recognized the integral form of the Euler
beta–function
B(s, t) =
Z 1
0
us�1(1 � u)t�1 du = �(s)�(t)
�(s + t)
, (5.33)
again writing this in terms of gamma functions. We’ll see that the presence of such gamma
functions is a ubiquitous feature of dimensional regularization.
Combining the pieces, the 1-loop contribution to ⇧(k2) is
⇧1 loop(k
2) = �g(µ) m
2
2(2⇡)d
2⇡d/2
�(d2)
⇣ µ
m
⌘4�d �(d2) �(1 �
d
2)
�(1)
= � g(µ) m
2
2(4⇡)d/2
⇣ µ
m
⌘4�d
�
✓
1 � d
2
◆
(5.34)
in dimensional regularization. �(z) has a pole whenever z is a non–positive integer, so our
loop integral diverges in d = 4 as expected. More precisely, the �–function the asymptotic
expansion
�(✏) ⇠ 1
✏
� � + O(✏) as ✏ ! 0+ , (5.35)
where � ⇡ 0.577 is known as the Euler–Mascheroni constant. Thus, if we set d = 4 � ✏,
our 1–loop diagram is asymptotic to
g(µ) m2
32⇡2

2
✏
� � + ln
✓
4⇡µ2
m2
◆
+ 1
�
+ O(✏) (5.36)
as ✏ ! 0+. The divergence we saw as ⇤0 ! 1 in the cut–o↵ regularization has become
a pole in d = 4 in dimensional regularization. (The logarithm of µ2/m2 is perfectly finite,
because µ is not a cut–o↵ so we have no wish to send µ ! 1.)
– 95 –
We now use counterterms in the d–dimensional theory to tune our couplings so as
to obtain a finite limit as d ! 4. This is the analogue of choosing the counterterms to
give a finite limit as ⇤0 ! 1 in the cut–o↵ regularization above. Once again, there’s a
lot of arbitrariness in how we accomplish this. It’s of course perfectly possible to use the
same, on–shell renormalization scheme as we did when regularizing our integrals with a
cut–o↵. However, in dimensional regularization other choices of scheme turn out to be
more convenient.
The simplest choice of renormalization scheme available in conjunction with dimen-
sional regularization is called minimal subtraction (MS). Here, one simply chooses the
counterterm
�m2 = �g(µ) m
2
16⇡2✏
(MS scheme) (5.37)
so as to remove the purely divergent part of the loop diagram (5.36). However, it’s often
even more convenient to use modified minimal subtraction (MS), where the countert-
erm is chosen to also remove the Euler–Mascheroni constant and the log 4⇡ term,
�m2 = �g(µ) m
2
32⇡2
✓
2
✏
� � + ln 4⇡
◆
(MS scheme) . (5.38)
In this scheme, we’d find
⇧1 loop(k
2) =
g(µ) m2
32⇡2

ln
µ2
m2
� 1
�
, (5.39)
which is perfectly finite in d = 4. Note also that in d = 4, the dimensionless coupling g
coincides with the original coupling �.
Let’s also take a look at the renormalization of the quartic coupling in dimensional
regularization. Like � itself, the loop integrals (5.13) have mass dimension
2[�] + [ddp/p4] = 2(4 � d) + d � 4 = 4 � d
in d-dimensions. In dimensional regularization it’ll be more convenient to study instead the
correction to the dimensionless coupling g(µ) = �µd�4, which is given by the dimensionless
integrals
g2µ4�d
2
Z
ddp
(2⇡)d
1
p2 + m2
1
(p + k1 + k2)2 + m2
+ other channels , (5.40)
corresponding to the 1-loop Feynman graphs we saw earlier. As before, setting ki = 0 to
extract the correction to the pure �4 coupling in the quantum e↵ective action, we have
µ4�d
Z 1
0
pd�1 dp
(p2 + m2)2
=
µ4�d
2
Z 1
0
(p2)(d�2)/2
(p2 + m2)2
d(p2)
=
1
2
⇣ µ
m
⌘4�d Z 1
0
u1�
d
2 (1 � u)
d
2�1 du
=
1
2
⇣ µ
m
⌘4�d �
�
2 � d2
�
�
�
d
2
�
�(2)
.
(5.41)
– 96 –
Thus, using our result (5.31) for Vol(Sd�1), we have a total 1-loop contribution
3g2
Vol(Sd�1)
2(2⇡)d
1
2
⇣ µ
m
⌘4�d �
�
2 � d2
�
�
�
d
2
�
�(2)
=
3g2
2(4⇡)d/2
⇣ µ
m
⌘4�d
�
✓
2 � d
2
◆
⇠ 3g
2
32⇡2
✓
2
✏
� � + log 4⇡µ
2
m2
◆
+ O(✏) ,
(5.42)
as ✏ = 4 � d ! 0+. Just like the mass correction (5.36), we’ve found a pole when d = 4,
reflecting the divergence of the original d = 4 loop integral.
Yet again, to obtain a finite limit we tune our d–dimensional coupling using the coun-
terterm ��. In the MS scheme we choose this to be
�� =
3g2
32⇡2
✓
2
✏
� � + log 4⇡
◆
(5.43)
removing both the pole as ✏ ! 0 and the � � log 4⇡ terms. Altogether, restoring ~, in the
MS scheme we’ve found that in the quantum e↵ective action the scalar field has a quartic
coupling with strength
ge↵(µ) = g(µ) �
3~g2(µ)
32⇡2
log
µ2
m2
+ O(~2) . (5.44)
Here, the leading term is the classical coupling (which may be thought of as a 1PI tree
diagram with four amputated external legs), while the order ~ term is the result of our
1–loop calculation combined with the contribution of the MS counterterm. Once again,
this result is perfectly finite in d = 4.
In exactly four dimensions, the �4 coupling is naturally dimensionless, i.e. it’s just a
number. The coupling we actually measure is ge↵ in the quantum e↵ective action, since
this is the one that incorporates all the quantum e↵ects of the original theory, which are
certainly present in our experiment. The numerical value of ge↵ cannot depend on the
choice of scale µ, which was nothing more than a choice of units in which to write the
d dimensional coupling g(µ) in the classical action. For this to be compatible with the
result (5.44), we must have
0 = µ
@
@µ
ge↵(µ) = µ
@
@µ

g(µ) � 3~g
2(µ)
32⇡2
log
µ2
m2
+ O(~)
�
, (5.45)
at least to 1-loop accuracy. Consequently, the coupling g(µ) in the original action must
vary with µ, having �-function
�(g) = µ
@g
@µ
=
3~g2
16⇡2
+ O(~2) . (5.46)
Notice that this �–function is proportional to ~: at the classical level we had
g(µ)
classical
= �µd�4 ,
so classically g(µ) would have been independent of µ in d = 4, which just says that the
quartic coupling was classically marginal in four dimensions. The fact that the �-function is
– 97 –
positive even when d = 4 shows that the classical assessment of g as marginal was incorrect
and in fact the quartic coupling is actually marginally irrelevant in d = 4, at least in a
neighbourhood of the Gaussian critical point.
Solving (5.46) for the coupling gives
1
g(µ0)
=
1
g(µ)
+
3~
16⇡2
ln
µ
µ0
+ O(~2) , (5.47)
relating the coupling g(µ) at scale µ to the coupling g(µ0) at a di↵erent scale µ0. How
should we understand this running coupling? For example, since any real calculation using
the original action will inevitably have to be carried out using perturbation theory, should
we do perturbation theory in g(µ) or in
g(µ0) =
g(µ)
1 + 3~ g(µ)16⇡2 ln
µ
µ0
⇡ g(µ) � 3~g
2(µ)
16⇡2
ln
µ
µ0
+ · · · ?
If we were able to calculate the path integral exactly then, by definition, the result would
be independent of µ. However, in real life we can only compute finitely many Feynman
diagrams and then perturbation theory in g(µ) will certainly be di↵erent from perturbation
theory in g(µ0). Can we choose the coupling g(µ) to be arbitrarily small in the hope of
achieving excellent perturbative results?
Remarkably, the answer is no! In fact, we have no freedom at all to choose the coupling
because it is totally fixed in terms of an energy scale ⇤�4 that is inherently present in the
theory as the scale at which the coupling diverges. That is 1/g(µ0) = 0 at µ0 = ⇤�4 .
Using this in (5.47), at any scale µ (such as the scale of our experiments) the coupling is
determined in terms of ⇤�4 by
g(µ) =
16⇡2
3~
1
ln(⇤�4/µ)
, (5.48)
(at least as long as out one–loop result for the running may be considered accurate). Note
that the scale ⇤�4 can be given invariant meaning by measuring it in terms of the physical
mass mphys of the particle. We simply have to hope that g(µ) ⌧ 1 at the scales we are
interested in so that perturbation theory can stand a chance of being reliable, and (5.48)
shows that this will be the case only for experimental energies far below ⇤�4 . The fact
that we’ve traded a dimensionless coupling g for an energy scale ⇤�4 is an example of
dimensional transmutation at work.
However, something is not quite right here. To define a continuum QFT with finite
values of relevant and marginal couplings at our experimental scale µ we tuned our initial
couplings to be closer and closer to the critical surface, so asto keep the scale at which the
RG trajectory passed closest to the (Gaussian) critical point finite. Our treatment of �4
theory – including a counterterm �� – was based on the classical picture that this quartic
coupling was marginal. We now realize that in fact it’s (marginally) irrelevant. Like the
irrelevant �6 and higher couplings we considered earlier, from a Wilsonian point of view, it
is in principle possible to generate an irrelevant quartic coupling in the quantum e↵ective
– 98 –
action but it’s value should be fixed in terms of the genuinely relevant couplings which
give coordinates along the RG trajectory. However, in this scalar theory, the only such
relevant ‘couplings’ are the mass and kinetic terms. We can solve the theory of a massive
free scalar exactly, and we know that no non–trivial quartic coupling is generated. Thus
the only value of the quartic coupling that really exists in the continuum is � = 0 and we
say the theory is trivial.
Our perturbative treatment is not powerful enough to say what really occurs as we
head into the deep UV, where the coupling becomes large — perturbation theory certainly
breaks down in this region. More sophisticated treatments indeed show that ��4 does not
exist as a continuum QFT.
5.2 One–loop renormalization of QED
We next turn to Quantum Electrodynamics, the theory of a (massive) charged Dirac spinor
coupled to the electromagnetic field. The classical action for this theory is
SQED[A, ] =
Z
ddx

1
4e2
Fµ⌫Fµ⌫ + ̄( /r + m) 
�
(5.49)
where the covariant derivative in the fermion kinetic term is /r = �µ(@µ � iAµ) , and
the Dirac matrices �µ obey {�µ, �⌫} = +2�µ⌫ . I’m working in conventions where these �
matrices are each anti-Hermitian (�µ)† = ��µ, and then the generators Sµ⌫ = i4 [�
µ, �⌫ ]
of the compact rotation group SO(d) are all Hermitian (unlike the boost generators for
the non–compact group SO(d � 1, 1).) In Euclidean signature, it’s also natural to define
 ̄ = ( )† without the factor of �0, which plays no distinguished role. You should check
that the action (5.49) is then real. In order for the covariant derivative rµ = @µ � iAµ to
make sense, the gauge field must have mass dimension 1 even in d dimensions. Thus, the
electric charge e has mass dimension (4 � d)/2, so at the classical level we expect that e is
relevant when d < 4, irrelevant in d > 4 and marginal in d = 4.
To do perturbation theory, we’d like the kinetic terms to be canonically normalized,
so we introduce a rescaled photon field Anewµ = e
�1Aoldµ . In terms of this rescaled field the
action becomes
SQED[A
new, ] =
Z
ddx

1
4
Fµ⌫Fµ⌫ + ̄(/@ + m) � ie ̄ /A 
�
(5.50)
with e appearing only in the electron–photon vertex, as befits a coupling. Notice that the
rescaled photon field has mass dimension
[Anew] = [Aold] � [e] = d � 2
2
, (5.51)
just like a scalar field in d dimensions. To obtain the classical photon propagator, we write
the Maxwell term in this canonically normalized action as
1
4
Z
Fµ⌫Fµ⌫ d
dx =
1
4
Z
�i
⇣
kµÃ⌫(�k) � k⌫õ(�k)
⌘
i
⇣
kµÃ⌫(k) � k⌫õ(k)
⌘
ddk
=
1
2
Z
k2
✓
�µ⌫ � k
µk⌫
k2
◆
õ(�k)Ã⌫(k) ddk ,
(5.52)
– 99 –
in terms of an integral over momentum space. Therefore, as you learned last term, in
Lorenz (or Landau) gauge the tree–level photon propagator is
�0µ⌫(k) =
1
k2
✓
�µ⌫ �
kµk⌫
k2
◆
(5.53)
in momentum space. The Lorenz gauge condition @µAµ(x) = 0 is reflected in the fact that
the propagator obeys kµ�0µ⌫(k) = 0, a condition you’ll also meet when studying Ward
identities in the problem sheets. This condition ensures that only transverse polarizations
propagate.
5.2.1 Vacuum polarization: loop calculation
In quantum theory, the exact Lorenz gauge photon propagator
�µ⌫(k) :=
Z
ddx eik·xhAµ(x)A⌫(0)i (5.54)
di↵ers from the classical expression (5.53) because of the photon’s interaction with the
charged electron field; the exact propagator accounts for the e↵ect quantum fluctuations of
the electron field have on the photon’s propagation. In exactly the same way as we did for
scalar field theory earlier (or even earlier in d = 0 and d = 1), to incorporate these e↵ects
we write
= �0µ� �
0
µ�⇧
�
��
0�
� + �
0
µ�⇧
�
��
0�
�⇧
�
��
0�
� · · ·� �
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesDtrij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�i.Theindexon�icontractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenbyxy
contributesafactorof
i(e�0)2�(x0�y0)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x0�y0)
4�|�x��y|=
Z
d4p
(2�)4
eip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
�����
�����
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�iabovegetsreplacedby�ie�µwhichcorrectly
accountsforthe(e�0)2pieceintheinstantaneousinteraction.
–141–
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesDtrij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�i.Theindexon�icontractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenbyxy
contributesafactorof
i(e�0)2�(x0�y0)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x0�y0)
4�|�x��y|=
Z
d4p
(2�)4
eip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
�����
�����
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�iabovegetsreplacedby�ie�µwhichcorrectly
accountsforthe(e�0)2pieceintheinstantaneousinteraction.
–141–
1PI
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesDtrij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�i.Theindexon�icontractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenbyxy
contributesafactorof
i(e�0)2�(x0�y0)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtrytoredefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x0�y0)
4�|�x��y|=
Z
d4p
(2�)4
eip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
�����
�����
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�iabovegetsreplacedby�ie�µwhichcorrectly
accountsforthe(e�0)2pieceintheinstantaneousinteraction.
–141–
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesD
tr
ij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�
i
.Theindexon�
i
contractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenby
xy
contributesafactorof
i(e�
0
)
2
�(x
0
�y
0
)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x
0
�y
0
)
4�|�x��y|
=
Zd4p
(2�)4
e
ip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
���
��
���
��
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�
i
abovegetsreplacedby�ie�
µ
whichcorrectly
accountsforthe(e�
0
)
2
pieceintheinstantaneousinteraction.
–141–
1PI1PI+ · · ·=�µ� � �
· · ·++= S0(k)⌃(/k)S0(k)⌃(/k)S0(k)S0(k)⌃(/k)S0(k)S0(k)
+ · · ·= +S(k) ̄ 1PI1PI ̄ 1PI ̄ 
= �0µ⌫(k) + �
0
µ⇢(k)⇧
⇢
�(k)�
0�
⌫(k) + �
0
µ⇢(k)⇧
⇢
�(k)�
0�
(k)⇧

�(k)�
0�
⌫(k) + · · ·
where the photon self–energy ⇧⇢�(k) is the sum of all 1PI graphs with two external
photon lines. This is again similar to what we saw for the scalar, with the di↵erence that
because the photon field Aµ carries a target space index, the photon self–energy is a tensor.
Below, we’ll show that ⇧⇢�(k) takes the form
⇧⇢�(k) = k
2
✓
�⇢� �
k⇢k�
k2
◆
⇡(k2) (5.55)
in terms of a dimensionless scalar function ⇡(k2). Accepting this for the moment, the factor
P ⇢� = �
⇢
� �
k⇢k�
k2
is the same factor as we saw in the tree–level photon propagator and projects onto polar-
ization states transverse to k�. As for any projection operator, this factor is idempotent:
P ⇢� P
�
 = P
⇢
 . (5.56)
The fact that the self–energy diagrams and tree–level propagator both involve the same
projection operator allows us to simplify our expression for the exact photon propagator
– 100 –
to
�µ⌫(k) = �
0
µ⌫ + �
0
µ⇢ ⇧
⇢
� �
0�
⌫ + �
0
µ⇢ ⇧
⇢
� �
0�
 ⇧

� �
0�
⌫ + · · ·
= �0µ⌫
�
1 + ⇡(k2) + ⇡2(k2) + ⇡3(k2) · · ·
�
=
�0µ⌫(k)
1 � ⇡(k2) ,
(5.57)
again summing the geometric series.
Just as the classical propagator �0µ⌫(k) was the inverse of the Maxwell kinetic term
1
4
Z
Fµ⌫Fµ⌫ d
dx =
1
2
Z
k2
✓
�µ⌫ � k
µk⌫
k2
◆
õ(�k)Ã⌫(k) ddk (5.58)
in momentum space, so too the exact photon propagator �µ⌫(k) can be thought of as
resulting from the term
S(2)e↵ [Ã] =
1
2
Z
[1 � ⇡(k2)] k2
✓
�µ⌫ � k
µk⌫
k2
◆
õ(�k)Ã⌫(k) ddk (5.59)
in the momentum space quantum e↵ective action that is quadratic in the photon field. We
interpret the factor of [1�⇡(k2)] as representing the e↵ects of quantum fluctuations in the
electron and photon fields on the photon’s propagation. In particular, if we expand ⇡(k2)
as a power series in k2, the part ⇡(0) that is independent of k2 just provides an overall
factor multiplying the classical Maxwell action. In other words, in position space this term
is
S(2)e↵ [A] =
1 � ⇡(0)
4
Z
Fµ⌫(x)Fµ⌫(x) d
dx (5.60)
and so contributes to wavefunction renormalization for the photon. As always, the re-
maining, k2-dependent terms in ⇡(k2) correspond to an infinite series of higher derivative
interactions of the schematic form @nFµ⌫ @nFµ⌫ . This infinite series again means that the
e↵ective theory is non-local, but the higher derivative couplings are irrelevant in d = 4 so
at low energies non–locality is strongly suppressed at length scales larger than the inverse
electron mass.
After these general considerations, let’s now compute the leading contribution to
⇧⇢�(k). We’ll make use of dimensional regularization, so it will be convenient to work
with a dimensionless coupling g(µ), introduced into the classical action via
e2 = µ4�d g2(µ) (5.61)
in terms of an arbitrary mass scale µ. Thus the electron–photon vertex is written as
�iµ2�d/2g(µ)
R
 ̄ /A ddx. At sub–leading order in ~, the unique loop graph we need to
consider is
= �0µ� �
0
µ�⇧
�
��
0�
� + �
0
µ�⇧
�
��
0�
�⇧
�
��
0�
� · · ·� �
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesDtrij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�i.Theindexon�icontractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenbyxy
contributesafactorof
i(e�0)2�(x0�y0)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x0�y0)
4�|�x��y|=
Z
d4p
(2�)4
eip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
�����
�����
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�iabovegetsreplacedby�ie�µwhichcorrectly
accountsforthe(e�0)2pieceintheinstantaneousinteraction.
–141–
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesDtrij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�i.Theindexon�icontractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenbyxy
contributesafactorof
i(e�0)2�(x0�y0)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x0�y0)
4�|�x��y|=
Z
d4p
(2�)4
eip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
������
�����
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�iabovegetsreplacedby�ie�µwhichcorrectly
accountsforthe(e�0)2pieceintheinstantaneousinteraction.
–141–
1PI
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesDtrij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�i.Theindexon�icontractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenbyxy
contributesafactorof
i(e�0)2�(x0�y0)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x0�y0)
4�|�x��y|=
Z
d4p
(2�)4
eip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
�����
�����
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�iabovegetsreplacedby�ie�µwhichcorrectly
accountsforthe(e�0)2pieceintheinstantaneousinteraction.
–141–
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesD
tr
ij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�
i
.Theindexon�
i
contractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenby
xy
contributesafactorof
i(e�
0
)
2
�(x
0
�y
0
)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x
0
�y
0
)
4�|�x��y|
=
Zd4p
(2�)4
e
ip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
���
��
���
��
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�
i
abovegetsreplacedby�ie�
µ
whichcorrectly
accountsforthe(e�
0
)
2
pieceintheinstantaneousinteraction.
–141–
1PI1PI+ · · ·=�µ� � �
· · ·++= S0(k)⌃(/k)S0(k)⌃(/k)S0(k)S0(k)⌃(/k)S0(k)S0(k)
+ · · ·= +S(k) ̄ 1PI1PI ̄ 1PI ̄ 
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesDtrij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�i.Theindexon�icontractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenbyxy
contributesafactorof
i(e�0)2�(x0�y0)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x0�y0)
4�|�x��y|=
Z
d4p
(2�)4
eip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
�����
�����
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�iabovegetsreplacedby�ie�µwhichcorrectly
accountsforthe(e�0)2pieceintheinstantaneousinteraction.
–141–
��Z3
6.4.1 Naive Feynman Rules
We want to determine the Feynman rules for this theory. For fermions, the rules are
the same as those given in Section 5. The new pieces are:
• We denote the photon by a wavy line. Each end of the line comes with an i, j =
1, 2, 3 index telling us the component of �A. We calculated the transverse photon
propagator in (6.33): it is and contributes D
tr
ij =
i
p2 + i�
✓
�ij �
pipj
|�p|2
◆
• The vertex contributes �ie�
i
. The index on �
i
contracts with the
index on the photon line.
• The non-local interaction which, in position space, is given by
x y
contributes a factor of
i(e�
0
)
2
�(x
0
� y
0
)
4�|�x � �y|
These Feynman rules are rather messy. This is the price we’ve paid for working in
Coulomb gauge. We’ll now show that we can massage these expressions into something
much more simple and Lorentz invariant. Let’s start with the o�ending instantaneous
interaction. Since it comes from the A0 component of the gauge field, we could try to
redefine the propagator to include a D00 piece which will capture this term. In fact, it
fits quite nicely in this form: if we look in momentum space, we have
�(x
0
� y
0
)
4�|�x � �y|
=
Z d4p
(2�)4
e
ip·(x�y)
|�p|2 (6.83)
so we can combine the non-local interaction with the transverse photon propagator by
defining a new photon propagator
Dµ⌫(p) =
�
���
��
���
��
+
i
|�p|2 µ, � = 0
i
p2 + i�
✓
�ij �
pipj
|�p|2
◆
µ = i 6= 0, � = j 6= 0
0 otherwise
(6.84)
With this propagator, the wavy photon line now carries a µ, � = 0, 1, 2, 3 index, with
the extra µ = 0 component taking care of the instantaneous interaction. We now need
to change our vertex slightly: the �ie�
i
above gets replaced by �ie�
µ
which correctly
accounts for the (e�
0
)
2
piece in the instantaneous interaction.
– 141 –
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesD
tr
ij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�
i
.Theindexon�
i
contractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenby
xy
contributesafactorof
i(e�
0
)
2
�(x
0
�y
0
)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x
0
�y
0
)
4�|�x��y|
=
Zd4p
(2�)4
e
ip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
���
��
���
��
+
i
|�p|2µ,�=0i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�
i
abovegetsreplacedby�ie�
µ
whichcorrectly
accountsforthe(e�
0
)
2
pieceintheinstantaneousinteraction.
–141–
A� A�
k
where a virtual e+e� pair is formed and then reabsorbed. (There is no symmetry factor
here because, unlike the propagator for a real scalar field, the electron–positron propagator
– 101 –
is oriented.) As always, in computing the self–energy contribution the external photon
propagators are amputated. The Feynman rules following from (5.50) give
⇧⇢�1 loop(k) = �µ
4�d(ig)2
Z
ddp
(2⇡)d
tr
✓
1
i/p + m
�⇢
1
i(/p � /k) + m
��
◆
= µ4�dg2
Z
ddp
(2⇡)d
tr
�
(�i/p + m)�⇢(�i(/p � /k) + m)��
�
(p2 + m2)((p � k)2 + m2) .
(5.62)
Note the overall sign: we obtain a factor of (�)2(�ig)2 expanding e�SQED/~ to second order
in the electron–photon vertex, and an additional factor of �1 from the fermionic loop. To
understand this additional sign, suppose the electron propagator is
S�↵(x, y) = h �(x) ̄↵(y)i (5.63)
in position space, where ↵, � are Dirac spinor indices. Expanding e�SQED[A, ]/~ to second
order in the electron–photon vertex, we meet a term
(�ig)2µ4�d
⌧Z
 ̄ /A (x) ddx
Z
 ̄ /A (y) ddy
�
= (�ig)2µ4�d
Z
ddx ddy /A
↵
�(x) /A
�
�(y) h ̄↵(x) �(x) ̄�(y) �(y)i
� (�ig)2µ4�d
Z
ddx ddy /A
↵
�(x) /A
�
�(y) h �(x) ̄�(y)i h ̄↵(x) �(y)i ,
(5.64)
where in the last line we’ve paired up the electron fields to give two propagators, being
careful to move an even number of fermions through one another. The term h �(x) ̄�(y)i
is precisely the propagator S��(x, y) representing the electron travelling from one vertex
to the next. However, the term h ̄↵(x) �(y)i = �h �(y) ̄↵(x)i is minus the propaga-
tor S�↵(y, x) representing propagation back to the first vertex, because the two fermions
naturally appeared in the opposite order.
More generally, in pairing a string of ̄ ’s from the action into propagators, we note
h ̄↵1 �1(x1) ̄↵2 �2(x2) · · · ̄↵n �n(xn)i
� h �1(x1) ̄↵2(x2)i h �2(x2) ̄↵3(x3)i · · · h �n�1(xn�1) ̄↵ni h ̄↵1(x1) �n(xn)i
= �S�1↵2(x1, x2) S
�2
↵3(x2, x3) · · · S
�n�1
↵n (xn�1, xn) S
�n
↵1(xn, x1)
(5.65)
with all but one propagator naturally in the correct order. You should check as an exercise
that the same result holds no matter which order we choose to join up the fermions. Thus,
we always obtain an additional minus sign, beyond those in the individual propagators and
vertices, for every fermion loop. At one loop, this minus sign can equivalently be understood
as coming from the fact that the path integral over the Grassmann valued electron field gives
det( /r+m) in the numerator of the remaining path integral, whereas had the electron been
a (complex) boson we would have obtained this factor in the denominator. Exponentiating
this determinant into the e↵ective action e��[A]/~ also shows we get an extra minus sign
for the fermion loop. (We’ll consider this determinant further below.)
– 102 –
Having understood its constituents, we’re now ready to evaluate ⇧⇢�1 loop(k) in (5.62).
The loop integral is somewhat complicated, and we’ll need to work hard to evaluate it.51
We begin by recalling the Feynman trick (5.17) that allows us to can combine the two
propagators in (5.62) as
Z 1
0
dx
[(p2 + m2)(1 � x) + ((p � k)2 + m2)x]2
=
Z 1
0
dx
[p2 + m2 � 2xp · k + k2x]2
=
Z 1
0
dx
[(p � kx)2 + m2 + k2x(1 � x)]2 .
(5.66)
We change variables p ! p0 = p + kx, whereupon (5.62) becomes (dropping the prime)
⇧⇢�1 loop(k) = µ
4�dg2
Z
ddp
(2⇡)d
Z 1
0
dx
tr
�
(�i(/p + /kx) + m)�⇢(�i(/p � /k(1 � x)) + m)��
�
[p2 + �]2
,
(5.67)
where we’ve introduced � = m2 + k2x(1 � x) as shorthand.
The next step is to perform the traces over the Dirac matrices. We’ll do this treating
the Dirac spinors as having 4 components52 as appropriate for our final goal of d = 4. Thus
tr(�⇢��) = 4 �⇢�
tr(�µ�⇢�⌫��) = 4(�µ⇢�⌫� � �µ⌫�⇢� + �µ��⌫⇢)
(5.68)
in Euclidean signature, so that
tr
�
(�i(/p + /kx) + m)�⇢(�i(/p � /k(1 � x)) + m)��
�
= 4 [�(p + kx)⇢(p � k(1 � x))� + (p + kx) · (p � k(1 � x))�⇢�
�(p + kx)�(p � k(1 � x))⇢ + m2�⇢�
⇤
.
(5.69)
Using this in the above loop integral, we obtain
⇧⇢�1 loop(k) = 4µ
4�dg2
Z
ddp
(2⇡)d
Z 1
0
dx
1
[p2 + �]2
⇥ [�(p + kx)⇢(p � k(1 � x))� + (p + kx) · (p � k(1 � x))�⇢�
�(p + kx)�(p � k(1 � x))⇢ + m2�⇢�
⇤
,
(5.70)
which would be quadratically divergent in d = 4.
We’re now ready to perform the loop integral. Observing that whenever d 2 N, any
term involving an odd number of powers of momentum would vanish, we drop these terms.
For the same reason, we replace
pµp⌫ ! 1
d
�µ⌫p2 and pµp⌫p⇢p� ! (p
2)2
d(d + 2)
[�µ⌫ �⇢� + �µ⇢ �⌫� + �µ� �⌫⇢]
51Don’t worry: asking you to reproduce this laborious calculation is not the sort of thing I’m keen on for
exams.
52In certain supersymmetric theories, it is often convenient to work instead with d–dimensional spinors,
which is known as dimensional reduction, rather than dimensional regularization. Here, we’ll stick with
the simpler idea of naively working with a spinor representation of SO(4), rather than SO(d).
– 103 –
where the tensor structure is fixed by SO(d) invariance and permutation symmetry, and
the numerical factors are determined by contracting both sides with d-dimensional metrics.
After these replacements the integrand depends only on p2, so the angular integrals may
be performed trivially to obtain
ddp
(2⇡)d
= Vol(Sd�1)
pd�1 dp
(2⇡)d
=
1
(4⇡)d/2 �(d/2)
(p2)
d
2�1 d(p2) (5.71)
as in section 5.1.4. Thus (5.70) becomes
⇧⇢�1 loop(k) = 4µ
4�d g
2
(4⇡)
d
2 �(d2)
⇥
Z 1
0
dx
Z 1
0
d(p2) (p2)
d
2�1
"
p2(1 � 2d) �
⇢� + (2k⇢k� � k2�⇢�)x(1 � x) + m2�⇢�
(p2 + �)2
#
.
(5.72)
To go further, we use the integrals
Z 1
0
d(p2)
(p2)
d
2�1
(p2 + �)2
=
✓
1
�
◆2� d2 �(2 � d2) �(
d
2)
�(2)
Z 1
0
d(p)2
(p2)
d
2
(p2 + �)2
=
✓
1
�
◆1� d2 �(1 + d2) �(1 �
d
2)
�(2)
(5.73)
that can be evaluated using the substitution u = �/(p2 + �) and the definition of the
Euler B–function.
Putting everything together, one finds that the 1-loop contribution to vacuum polar-
ization is given by
⇧⇢�1 loop(k) =
4g2µ4�d
(4⇡)d/2
�
✓
2 � d
2
◆
⇥
Z 1
0
dx

�⇢�(m2 � x(1 � x)k2) � �⇢�(m2 + x(1 � x)k2) + 2x(1 � x)k⇢k�
�2�
d
2
�
=: (k2 �⇢� � k⇢k�)⇡1 loop(q2) ,
(5.74a)
where in the last line we have defined ⇡1 loop(k2) to be the dimensionless quantity
⇡1 loop(k
2) = �
8g2(µ) �
�
2 � d2
�
(4⇡)d/2
Z 1
0
dx x(1 � x)
✓
µ2
�
◆2�d/2
. (5.74b)
and we recall that � = m2 + k2x(1 � x).
As promised, the loop integral has come out proportional to the operator �⇢��k⇢k�/k2
projecting onto polarizations transverse to k. Gauge invariance is also maintained in lattice
regularization, but would fail if one simply imposed a cut–o↵ ⇤0, because the requirement
that fields only contain Fourier modes with |p|  ⇤0 is not preserved under the gauge
– 104 –
transformation ! ei�(x) , even if it is true of and � separately.53 The desire to
maintain manifest gauge invariance was one of the main motivation to use dimensional
regularization in the first place, and our result (5.74a) vindicates this decision: we see that
the 1-loop correction ⇧⇢�1 loop(k
2) is proportional to (k2�⇢� � k⇢k�), so
k⇢ ⇧
⇢�
1 loop(k) = 0 . (5.75)
This signifies that the quantum e↵ective action is also gauge invariant (at least to order ~
accuracy, but in fact it holds in general). We’ll explore this important point further in the
following chapters.
5.2.2 Counterterms in QED
The first thing to notice about our result (5.74a) is that it, if all couplings remain constant,
it will diverge in the physically interesting dimension d = 4 where �(2 � d2) has a pole. To
obtain a finite result in d = 4 we must tune the initial couplings in the action, which as
always we do by introducing counterterms. For QED the counterterms are
SCT[A, ,✏] =
Z
ddx

�Z3
1
4
Fµ⌫Fµ⌫ + �Z2 ̄ /r + �m ̄ 
�
. (5.76)
Adding these to the classical action (5.50) allows us to tune the initial values of the pho-
ton and electron wavefunction kinetic terms, and the electron mass. The labels (�Z3, �Z2)
for the photon and electron wavefunction renormalization counterterms are conventional
(and the wavefunction renormalization factors themselves are likewise called (Z3, Z2), re-
spectively). The fact that the entire kinetic term for the electron, including the gauge
covariant derivative operator /r = /@� ie /A, receives only one counterterm assumes that the
regularized path integral preserves gauge invariance: our dimensionally regularized loop
integral is indeed gauge invariant, so /@ and i /A cannot appear independently. We’ll
study this further in section 6.3.1.
To fix the counterterm �Z3 we set d = 4 � ✏ and find the asymptotic expression
⇡1 loop(k
2) ⇠ �g
2(µ)
2⇡2
Z 1
0
dx x(1 � x)
✓
2
✏
� � + ln 4⇡µ
2
�
◆
+ O(✏) (5.77)
as ✏ ! 0+, where again � is the Euler–Mascheroni constant. The contribution
= �0µ� �
0
µ�⇧
�
��
0�
� + �
0
µ�⇧
�
��
0�
�⇧
�
��
0�
� · · ·� �
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesDtrij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�i.Theindexon�icontractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenbyxy
contributesafactorof
i(e�0)2�(x0�y0)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x0�y0)
4�|�x��y|=
Z
d4p
(2�)4
eip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
�����
�����
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�iabovegetsreplacedby�ie�µwhichcorrectly
accountsforthe(e�0)2pieceintheinstantaneousinteraction.
–141–
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesDtrij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�i.Theindexon�icontractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenbyxy
contributesafactorof
i(e�0)2�(x0�y0)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x0�y0)
4�|�x��y|=
Z
d4p
(2�)4
eip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
�����
�����
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�iabovegetsreplacedby�ie�µwhichcorrectly
accountsforthe(e�0)2pieceintheinstantaneousinteraction.
–141–
1PI
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesDtrij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�i.Theindexon�icontractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenbyxy
contributesafactorof
i(e�0)2�(x0�y0)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x0�y0)
4�|�x��y|=
Z
d4p
(2�)4
eip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
�����
�����
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�iabovegetsreplacedby�ie�µwhichcorrectly
accountsforthe(e�0)2pieceintheinstantaneousinteraction.
–141–
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesD
tr
ij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�
i
.Theindexon�
i
contractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenby
xy
contributesafactorof
i(e�
0
)
2
�(x
0
�y
0
)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x
0
�y
0
)
4�|�x��y|
=
Zd4p
(2�)4
e
ip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
���
��
���
��
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�
i
abovegetsreplacedby�ie�
µ
whichcorrectly
accountsforthe(e�
0
)
2
pieceintheinstantaneousinteraction.
–141–
1PI1PI+ · · ·=�µ� � �
· · ·++= S0(k)⌃(/k)S0(k)⌃(/k)S0(k)S0(k)⌃(/k)S0(k)S0(k)
+ · · ·= +S(k) ̄ 1PI1PI ̄ 1PI ̄ 
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesDtrij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�i.Theindexon�icontractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenbyxy
contributesafactorof
i(e�0)2�(x0�y0)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneousinteraction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x0�y0)
4�|�x��y|=
Z
d4p
(2�)4
eip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
�����
�����
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�iabovegetsreplacedby�ie�µwhichcorrectly
accountsforthe(e�0)2pieceintheinstantaneousinteraction.
–141–
��Z3
from the counterterm 14�Z3 F
⇢�F⇢� must remove this pole, and in the MS scheme we’d set
�Z3 = �
g2(µ)
12⇡2
✓
2
✏
� � + ln 4⇡
◆
(5.78)
53In fact, the conceptually simple idea of integrating over modes only up to a cut–o↵ can be done in
gauge theory, but requires the introduction of a fair amount of technology beyond the scope of this course;
see e.g. K. Costello’s book cited in the introduction.
– 105 –
so as also to remove the contribution / (��+ln 4⇡). (To check that this counterterm does
indeed cancel the pole, note that
R 1
0 dx x(1 � x) =
1
6 .) Thus the total contribution to the
e↵ective photon self–energy at O(~) is
⇧⇢�1 loop(k) =
�
k2�⇢� � k⇢k�
�
⇡1 loop(k
2) (5.79)
where
⇡1 loop(k
2) =
g2(µ)
2⇡2
Z 1
0
dx x(1 � x) ln

m2 + x(1 � x)k2
µ2
�
(5.80)
in the MS scheme.
Strikingly, the loop correction to the photon propagator has created the logarithm54
ln
⇥
m2 + x(1 � x) k2
⇤
in momentum space. This is quite unlike anything you’ve seen at tree–level, where Feynman
diagrams are always rational functions of momenta, but it is very similar to the logarithms
we obtained from integrating out fields in lower dimensional examples. In the present case,
the logarithm has a branch cut in the region m2 + x(1 � x)k2  0. For x 2 [0, 1] we have
0  x(1 � x)  1/4, so the branch point is inaccessible for real Euclidean q. However, in
Lorentz signature with k0 = iE we have
x(1 � x) (E2 � k2) � m2 . (5.81)
in Lorentzian signature, so the smallest value energy at which the branch point can be
reached is
E2 = (2m)2 . (5.82)
This is precisely the threshold energy for the creation of a real (as opposed to virtual)
electron–positron pair.
5.2.3 The �-function of QED
Knowing the e↵ective action allows us to read o↵ the �-function for the electric charge. To
relate the photon kinetic term – involving wavefunction renormalization – to the � function
for the electromagnetic coupling e, we first undo our rescaling Aoldµ = eA
new
µ and work back
in terms of the original gauge field Aoldµ . Then (strictly in d = 4) the quadratic term (5.60)
in the e↵ective action becomes
S(2)e↵ [A
old] =
1
4g2e↵
Z
Fµ⌫Fµ⌫ d
4z
=
1 � ⇡(0)
4g2(µ)
Z
Fµ⌫Fµ⌫ d
4z
=
1
4

1
g2(µ)
� ~
12⇡2
ln
m2
µ2
+ O(~2)
� Z
Fµ⌫Fµ⌫ d
4z .
(5.83)
54Actually, it has produced a certain integral of a logarithm involving x(1 � x). This integral can be
explicitly computed in terms of dilogarithms, but we won’t need to know the result.
– 106 –
where the final expression uses our 1-loop result (5.80). In this way, we can view the
vacuum polarization as a quantum correction to the value of the coupling 1/g2(µ) in front
of the classical kinetic term.
Arguing again that the numerical value of the dimensionless coupling ge↵ in the quan-
tum e↵ective action quantity cannot depend on our choice of scale µ, we have
0 = µ
@
@µ

1
g2(µ)
� ~
12⇡2
ln
m2
µ2
�
= � 2
g3(µ)
�(g) +
~
6⇡2
(5.84)
so that the � function for g(µ) is
�(g) =
~g3(µ)
12⇡2
+ O(~2) . (5.85)
Thus the coupling at scale µ0 is related to that at scale µ by
1
g2(µ0)
=
1
g2(µ)
+
~
6⇡2
ln
µ
µ0
+ O(~2) , (5.86)
to this order.
Just as for �4 theory, the running coupling defines a natural scale inherent in our
theory. We define ⇤QED to be the scale at which the coupling diverges. Then at any other
scale µ the value of the coupling cannot be chosen freely, but is fixed in terms of ⇤QED by
g2(µ) =
6⇡2
~
1
ln(⇤QED/µ)
. (5.87)
Since the positive �-function (5.85) shows that (like the quartic scalar coupling) the QED
coupling is in fact marginally irrelevant, we expect its value in the continuum theory should
be fixed by our location along the renormalized trajectory parametrized purely in terms
of the relevant and truly marginal couplings. But for QED the only such couplings are
the electron mass and electron and photon kinetic terms. Once again, for any value of
these masses and kinetic coe�cients, we have a free theory and no interaction terms are
generated at any point along the RG trajectory. Consequently, it is believed that pure
QED does not exist as a continuum QFT in four dimensions!55
In practical terms, as for the quartic scalar theory, this result is of no great importance.
We can treat QED as a low–energy e↵ective theory, valid provided the energy scale µ of
our experiments remains well below ⇤QED. Experimentalists tell us that at the scale µ =
me ⇡ 511 keV of the electron mass, the dimensionless coupling ↵ = g2(me)/4⇡ ⇡ 1/137.
From this data point, equation (5.87) gives
⇤QED ⇡ 10286 GeV (!)
which is an extremely high energy, and certainly well beyond any point at which we claim
to even vaguely trust QFT as a description of Nature. In Nature, QED unifies with the
weak interactions at around ⇠ 100 GeV, where the physics of non–Abelian gauge theories
comes into play, modifying our conclusions here.
55Once again, to be sure one should really go further than this 1–loop calculation, but what we can be
sure of already is that any continuum version of interacting QED would necessarily be strongly coupled.
More sophisticated treatments support the view that in fact, no such continuum interacting theory exists.
– 107 –
5.2.4 Decoupling: on-shell schemes vs MS
Before moving on, I wish to point out a small peculiarity inherent in the MS renormaliza-
tion scheme. If we perform experiments at low energies, we do not expect to have large
corrections coming from particles whose masses are much higher than the energy scale of
our experiments: for example, you did not worry about corrections from the muon or tau
particle when studying classical electrodynamics.
Indeed, this is just what we’d find if, instead of the MS scheme, we’d used a renormal-
ization scheme that fixes the value of the counterterm �Z3 in terms of ⇡1 loop(k2) at some
definite scale. For example, in the on–shell scheme we fix the counterterm by requiring
that (just like the classical propagator) the exact photon propagator �0µ⌫(k)/[1 � ⇡(k2)]
has a simple pole with unit residue when the photon’s momentum obeys k2 = 0. Thus, in
the on–shell scheme, we’d fix �Z3 by demanding ⇡(0) = 0 so that the quantum corrections
(including both the loop integral and counterm) vanish at the on–shell point k2 = 0.
For the purposes of extracting the �–function, it’s more convenient to ask instead that
the quantum corrections vanish at our arbitrary scale k2 = µ2. That is, we set
�Z3 = �
~g2(µ)
2⇡2
Z 1
0
dx x(1 � x)
✓
2
✏
� � + ln 4⇡µ
2
m2 + x(1 � x)µ2
◆
(5.88)
cancelling the value of ⇡1 loop(µ2). The total contribution to ⇡(k2) in such a scheme is then
⇡(k2) =
~g2(µ)
2⇡2
Z 1
0
dx x(1 � x) ln
✓
m2 + x(1 � x)k2
m2 + x(1 � x)µ2
◆
+ O(~2) . (5.89)
As in equation (5.84), asking that the total contribution to the e↵ective action is inde-
pendent of the scale µ means that the coupling must run, and in this scheme we find the
�-function
�(g) =
g3(µ)
2⇡2
Z 1
0
dx
x2(1 � x)2µ2
m2 + x(1 � x)µ2 , (5.90)
as you should check. This does indeed approach zero when µ ⌧ m, so that the coupling
automatically stops running at scales far below the electron mass.
However, in the MS scheme, the �-function
�MS =
g3(µ)
12⇡2
+ O(~2)
of (5.85) shows no suppression at any scale and the coupling (5.86)still runs even at scales
µ ⌧ me much lower than the mass of the lightest (indeed the only) charged particle. There’s
nothing wrong with this in principle: as µ ! 0 there’s a balance in the e↵ective action (5.83)
between g2(µ) ! 0 and the growing e↵ect of the loop contribution / ln(1/µ2) ! 1 so
that the actual observable physics does remain constant. Nonetheless, it’s strange to have
loop e↵ects dominating the classical ones, so for some purposes it’s better to find a way to
mimic the decoupling of the electron in MS.
To do so, we consider two di↵erent theories. One includes the electron and is valid at
scales µ > me, while the second contains no electron field and is valid at scales µ < me.
– 108 –
Physical quantities in the two theories are matched at µ = me. The e↵ects of the electron
(or other heavy particle) are then manually frozen out as we continue on to our other
theory at lower scales. In particular, since pure Maxwell theory is free, for any µ < me
the fine structure constant ↵(µ) = g2(µ)/4⇡ will remain frozen at its value ⇡ 1/137 at the
electron mass.
This matching procedure, discarding the e↵ects particles whose masses are � µ by
hand, is certainly rather cumbersome. However, for most purposes (particularly in more
complicated theories such as Yang–Mills theory, or the full Standard Model) the MS scheme
is so convenient that most physicists view this as a price worth paying.
5.3 The Euler–Heisenberg e↵ective action
The electron field appears purely quadratically in the QED action (5.50)56
SQED[A, ] =
Z 
1
4
Fµ⌫Fµ⌫ + ̄( /r + m) 
�
d4x , (5.91)
so we can perform its path integral exactly. The electron is fermionic, so its path integral
yields det( /r+m) in the numerator. The remaining photon path integral thus involves the
e↵ective action
�e↵ [A] =
1
4
Z
Fµ⌫Fµ⌫ d
4x � ~ ln det( /r + m) (5.92)
for the photon that takes into account quantum e↵ects due to the electron. (As always,
this expression is somewhat formal: if we don’t regulate the path integral measure then
the e↵ective action will diverge. We’ll consider a convenient regularization below.)
The functional determinant is non–polynomial in the photon field Aµ. Expanding it
as an inifinite series around Aµ = 0 gives
ln det( /r + m) = ln det(/@ + m) + tr ln(1 � ie(/@ + m)�1 /A)
= ln det(/@ + m) +
1X
n=1
(�ie)n
n
Z nY
i=1
d4xi Tr(S(xn, x1) /A(x1)S(x1, x2) /A(x2) · · · S(xn�1, xn) /A(xn))
(5.93)
where S(xi, xi+1) is the Dirac propagator (/@+m)�1 and Tr indicates a trace over the Dirac
gamma matrices. The terms on the rhs correspond to the 1PI Feynman diagrams57
�a �b ~ (M�1)ab==h�a�biJ �J/~
+
=
+
+
⇠ · · ·
· · ·
~
m2e�
�4~2
2m6e�
�
h�2i = 1Z
Z
d� e�W(�)/~ �2
· · ·
6
.4
.1
N
a
iv
e
F
ey
n
m
a
n
R
u
les
W
e
w
ant
to
d
eterm
in
e
th
e
F
eyn
m
an
ru
les
for
th
is
th
eory.
F
or
ferm
ion
s,
th
e
ru
les
are
th
e
sam
e
as
th
ose
given
in
S
ection
5.
T
h
e
n
ew
p
ieces
are:
•
W
e
d
en
ote
th
e
p
h
oton
by
a
w
avy
lin
e.
E
ach
en
d
of
th
e
lin
e
com
es
w
ith
an
i,j
=
1,2,3
in
d
ex
tellin
g
u
s
th
e
com
p
on
ent
of
� A
.
W
e
calcu
lated
th
e
tran
sverse
p
h
oton
p
rop
agator
in
(6.33):
it
is
an
d
contrib
u
tes
D
tr ij
=
i
p
2
+
i�✓
�
ij�
p
ip
j
|�p|2◆
•
T
h
e
vertex
contrib
u
tes
�
ie�
i .
T
h
e
in
d
ex
on
�
i
contracts
w
ith
th
e
in
d
ex
on
th
e
p
h
oton
lin
e.
•
T
h
e
n
on
-local
interaction
w
h
ich
,
in
p
osition
sp
ace,
is
given
by
x
y
contrib
u
tes
a
factor
of
i(e�
0 )
2 �(x
0 �
y
0 )
4�|�x
�
�y|
T
h
ese
F
eyn
m
an
ru
les
are
rath
er
m
essy.
T
h
is
is
th
e
p
rice
w
e’ve
p
aid
for
w
orkin
g
in
C
ou
lom
b
gau
ge.
W
e’ll
n
ow
sh
ow
th
at
w
e
can
m
assage
th
ese
exp
ression
s
into
som
eth
in
g
m
u
ch
m
ore
sim
p
le
an
d
L
orentz
invariant.
L
et’s
start
w
ith
th
e
o�
en
d
ing
in
stantan
eou
s
interaction
.
S
in
ce
it
com
es
from
th
e
A
0
com
p
on
ent
of
th
e
gau
ge
fi
eld
,
w
e
cou
ld
try
to
red
efi
n
e
th
e
p
rop
agator
to
in
clu
d
e
a
D
00
p
iece
w
h
ich
w
ill
cap
tu
re
th
is
term
.
In
fact,
it
fi
ts
qu
ite
n
icely
in
th
is
form
:
if
w
e
look
in
m
om
entu
m
sp
ace,
w
e
h
ave
�(x
0 �
y
0 )
4�|�x
�
�y|
=
Z
d
4 p
(2�
)
4
e
ip·(x�
y
)
|�p|2
(6.83)
so
w
e
can
com
b
in
e
th
e
n
on
-local
interaction
w
ith
th
e
tran
sverse
p
hoton
p
rop
agator
by
d
efi
n
in
g
a
n
ew
p
h
oton
p
rop
agator
D
µ
⌫(p)
=
� � � � � � � � � � �
+
i
|�p|2
µ
,�
=
0
i
p
2
+
i�✓
�
ij�
p
ip
j
|�p|2◆
µ
=
i6=
0,�
=
j
6=
0
0
oth
erw
ise
(6.84)
W
ith
th
is
p
rop
agator,
th
e
w
avy
p
h
oton
lin
e
n
ow
carries
a
µ
,�
=
0,1,2,3
in
d
ex,
w
ith
th
e
extra
µ
=
0
com
p
on
ent
takin
g
care
of
th
e
in
stantan
eou
s
interaction
.
W
e
n
ow
n
eed
to
ch
an
ge
ou
r
vertex
slightly:
th
e
�
ie�
i
ab
ove
gets
rep
laced
by
�
ie�
µ
w
h
ich
correctly
accou
nts
for
th
e
(e�
0 )
2
p
iece
in
th
e
in
stantan
eou
s
interaction
.
–
141
–
6
.4
.1
N
a
iv
e
F
ey
n
m
a
n
R
u
le
s
W
e
w
an
t
to
d
et
er
m
in
e
th
e
F
ey
n
m
an
ru
le
s
fo
r
th
is
th
eo
ry
.
F
or
fe
rm
io
n
s,
th
e
ru
le
s
ar
e
th
e
sa
m
e
as
th
os
e
gi
ve
n
in
S
ec
ti
on
5.
T
h
e
n
ew
p
ie
ce
s
ar
e:
•
W
e
d
en
ot
e
th
e
p
h
ot
on
by
a
w
av
y
li
n
e.
E
ac
h
en
d
of
th
e
li
n
e
co
m
es
w
it
h
an
i,
j
=
1,
2,
3
in
d
ex
te
ll
in
g
u
s
th
e
co
m
p
on
en
t
of
� A
.
W
e
ca
lc
u
la
te
d
th
e
tr
an
sv
er
se
p
h
ot
on
p
ro
p
ag
at
or
in
(6
.3
3)
:
it
is
an
d
co
nt
ri
b
u
te
s
D
tr ij
=
i
p2
+
i�
✓
� i
j
�
p i
p j |�p
|2
◆
•
T
h
e
ve
rt
ex
co
nt
ri
b
u
te
s
�
ie
�
i .
T
h
e
in
d
ex
on
�
i
co
nt
ra
ct
s
w
it
h
th
e
in
d
ex
on
th
e
p
h
ot
on
li
n
e.
•
T
h
e
n
on
-l
oc
al
in
te
ra
ct
io
n
w
h
ic
h
,
in
p
os
it
io
n
sp
ac
e,
is
gi
ve
n
by
x
y
co
nt
ri
b
u
te
s
a
fa
ct
or
of
i(
e�
0
)2
�(
x
0
�
y0
)
4�
|�x
�
�y|
T
h
es
e
F
ey
n
m
an
ru
le
s
ar
e
ra
th
er
m
es
sy
.
T
h
is
is
th
e
p
ri
ce
w
e’
ve
p
ai
d
fo
r
w
or
ki
n
g
in
C
ou
lo
m
b
ga
u
ge
.
W
e’
ll
n
ow
sh
ow
th
at
w
e
ca
n
m
as
sa
ge
th
es
e
ex
p
re
ss
io
n
s
in
to
so
m
et
h
in
g
m
u
ch
m
or
e
si
m
p
le
an
d
L
or
en
tz
in
va
ri
an
t.
L
et
’s
st
ar
t
w
it
h
th
e
o�
en
d
in
g
in
st
an
ta
n
eo
u
s
in
te
ra
ct
io
n
.
S
in
ce
it
co
m
es
fr
om
th
e
A
0
co
m
p
on
en
t
of
th
e
ga
u
ge
fi
el
d
,
w
e
co
u
ld
tr
y
to
re
d
efi
n
e
th
e
p
ro
p
ag
at
or
to
in
cl
u
d
e
a
D
00
p
ie
ce
w
h
ic
h
w
il
l
ca
p
tu
re
th
is
te
rm
.
In
fa
ct
,
it
fi
ts
qu
it
e
n
ic
el
y
in
th
is
fo
rm
:
if
w
e
lo
ok
in
m
om
en
tu
m
sp
ac
e,
w
e
h
av
e
�(
x
0
�
y0
)
4�
|�x
�
�y|
=
Z
d4
p
(2
�
)4
ei
p
·(
x
�
y
)
|�p
|2
(6
.8
3)
so
w
e
ca
n
co
m
b
in
e
th
e
n
on
-l
oc
al
in
te
ra
ct
io
n
w
it
h
th
e
tr
an
sv
er
se
p
ho
to
n
p
ro
p
ag
at
or
by
d
efi
n
in
g
a
n
ew
p
h
ot
on
p
ro
p
ag
at
or
D
µ
⌫
(p
)
=
� � � � � � � � � � �
+
i |�p
|2
µ
,�
=
0
i
p2
+
i�
✓
� i
j
�
p i
p j |�p
|2
◆
µ
=
i
6=
0,
�
=
j
6=
0
0
ot
h
er
w
is
e
(6
.8
4)
W
it
h
th
is
p
ro
p
ag
at
or
,
th
e
w
av
y
p
h
ot
on
li
n
e
n
ow
ca
rr
ie
s
a
µ
,�
=
0,
1,
2,
3
in
d
ex
,
w
it
h
th
e
ex
tr
a
µ
=
0
co
m
p
on
en
t
ta
ki
n
g
ca
re
of
th
e
in
st
an
ta
n
eo
u
s
in
te
ra
ct
io
n
.
W
e
n
ow
n
ee
d
to
ch
an
ge
ou
r
ve
rt
ex
sl
ig
ht
ly
:
th
e
�
ie
�
i
ab
ov
e
ge
ts
re
p
la
ce
d
by
�
ie
�
µ
w
h
ic
h
co
rr
ec
tl
y
ac
co
u
nt
s
fo
r
th
e
(e
�
0
)2
p
ie
ce
in
th
e
in
st
an
ta
n
eo
u
s
in
te
ra
ct
io
n
.
–
14
1
–
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesDtrij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�i.Theindexon�icontractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenbyxycontributesafactorof
i(e�0)2�(x0�y0)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x0�y0)
4�|�x��y|=
Z
d4p
(2�)4
eip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
�����
�����
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�iabovegetsreplacedby�ie�µwhichcorrectly
accountsforthe(e�0)2pieceintheinstantaneousinteraction.
–141–
6.4.1 Naive Feynman Rules
We want to determine the Feynman rules for this theory. For fermions, the rules are
the same as those given in Section 5. The new pieces are:
• We denote the photon by a wavy line. Each end of the line comes with an i, j =
1, 2, 3 index telling us the component of �A. We calculated the transverse photon
propagator in (6.33): it is and contributes Dtrij =
i
p2 + i�
✓
�ij �
pipj
|�p|2
◆
• The vertex contributes �ie�i. The index on �i contracts with the
index on the photon line.
• The non-local interaction which, in position space, is given by x y
contributes a factor of
i(e�0)2�(x0 � y0)
4�|�x � �y|
These Feynman rules are rather messy. This is the price we’ve paid for working in
Coulomb gauge. We’ll now show that we can massage these expressions into something
much more simple and Lorentz invariant. Let’s start with the o�ending instantaneous
interaction. Since it comes from the A0 component of the gauge field, we could try to
redefine the propagator to include a D00 piece which will capture this term. In fact, it
fits quite nicely in this form: if we look in momentum space, we have
�(x0 � y0)
4�|�x � �y| =
Z
d4p
(2�)4
eip·(x�y)
|�p|2 (6.83)
so we can combine the non-local interaction with the transverse photon propagator by
defining a new photon propagator
Dµ⌫(p) =
�
�����
�����
+
i
|�p|2 µ, � = 0
i
p2 + i�
✓
�ij �
pipj
|�p|2
◆
µ = i 6= 0, � = j 6= 0
0 otherwise
(6.84)
With this propagator, the wavy photon line now carries a µ, � = 0, 1, 2, 3 index, with
the extra µ = 0 component taking care of the instantaneous interaction. We now need
to change our vertex slightly: the �ie�i above gets replaced by �ie�µ which correctly
accounts for the (e�0)2 piece in the instantaneous interaction.
– 141 –
6
.4
.1
N
a
iv
e
F
ey
n
m
a
n
R
u
le
s
W
e
w
an
t
to
d
et
er
m
in
e
th
e
F
ey
n
m
an
ru
le
s
fo
r
th
is
th
eo
ry
.
F
or
fe
rm
io
n
s,
th
e
ru
le
s
ar
e
th
e
sa
m
e
as
th
os
e
gi
ve
n
in
S
ec
ti
on
5.
T
h
e
n
ew
p
ie
ce
s
ar
e:
•
W
e
d
en
ot
e
th
e
p
h
ot
on
by
a
w
av
y
lin
e.
E
ac
h
en
d
of
th
e
lin
e
co
m
es
w
it
h
an
i,
j
=
1,
2,
3
in
d
ex
te
lli
n
g
u
s
th
e
co
m
p
on
en
t
of
� A
.
W
e
ca
lc
u
la
te
d
th
e
tr
an
sv
er
se
p
h
ot
on
p
ro
p
ag
at
or
in
(6
.3
3)
:
it
is
an
d
co
nt
ri
b
u
te
s
D
tr ij
=
i
p2
+
i�
✓
� i
j
�
p i
p j |�p
|2
◆
•
T
h
e
ve
rt
ex
co
nt
ri
b
u
te
s
�
ie
�
i .
T
h
e
in
d
ex
on
�
i
co
nt
ra
ct
s
w
it
h
th
e
in
d
ex
on
th
e
p
h
ot
on
lin
e.
•
T
h
e
n
on
-l
oc
al
in
te
ra
ct
io
n
w
h
ic
h
,
in
p
os
it
io
n
sp
ac
e,
is
gi
ve
n
by
x
y
co
nt
ri
b
u
te
s
a
fa
ct
or
of
i(
e�
0
)2
�(
x
0
�
y0
)
4�
|�x
�
�y|
T
h
es
e
F
ey
n
m
an
ru
le
s
ar
e
ra
th
er
m
es
sy
.
T
h
is
is
th
e
p
ri
ce
w
e’
ve
p
ai
d
fo
r
w
or
ki
n
g
in
C
ou
lo
m
b
ga
u
ge
.
W
e’
ll
n
ow
sh
ow
th
at
w
e
ca
n
m
as
sa
ge
th
es
e
ex
p
re
ss
io
n
s
in
to
so
m
et
h
in
g
m
u
ch
m
or
e
si
m
p
le
an
d
L
or
en
tz
in
va
ri
an
t.
L
et
’s
st
ar
t
w
it
h
th
e
o�
en
d
in
g
in
st
an
ta
n
eo
u
s
in
te
ra
ct
io
n
.
S
in
ce
it
co
m
es
fr
om
th
e
A
0
co
m
p
on
en
t
of
th
e
ga
u
ge
fi
el
d
,
w
e
co
u
ld
tr
y
to
re
d
efi
n
e
th
e
p
ro
p
ag
at
or
to
in
cl
u
d
e
a
D
00
p
ie
ce
w
h
ic
h
w
ill
ca
p
tu
re
th
is
te
rm
.
In
fa
ct
,
it
fi
ts
qu
it
e
n
ic
el
y
in
th
is
fo
rm
:
if
w
e
lo
ok
in
m
om
en
tu
m
sp
ac
e,
w
e
h
av
e
�(
x
0
�
y0
)
4�
|�x
�
�y|
=
Z
d4
p
(2
�
)4
ei
p
·(
x
�
y
)
|�p
|2
(6
.8
3)
so
w
e
ca
n
co
m
b
in
e
th
e
n
on
-l
oc
al
in
te
ra
ct
io
n
w
it
h
th
e
tr
an
sv
er
se
p
ho
to
n
p
ro
p
ag
at
or
by
d
efi
n
in
g
a
n
ew
p
h
ot
on
p
ro
p
ag
at
or
D
µ
⌫
(p
)
=
� � � � � � � � � � �
+
i |�p
|2
µ
,�
=
0
i
p2
+
i�
✓
� i
j
�
p i
p j |�p
|2
◆
µ
=
i
6=
0,
�
=
j
6=
0
0
ot
h
er
w
is
e
(6
.8
4)
W
it
h
th
is
p
ro
p
ag
at
or
,
th
e
w
av
y
p
h
ot
on
lin
e
n
ow
ca
rr
ie
s
a
µ
,�
=
0,
1,
2,
3
in
d
ex
,
w
it
h
th
e
ex
tr
a
µ
=
0
co
m
p
on
en
t
ta
ki
n
g
ca
re
of
th
e
in
st
an
ta
n
eo
u
s
in
te
ra
ct
io
n
.
W
e
n
ow
n
ee
d
to
ch
an
ge
ou
r
ve
rt
ex
sl
ig
ht
ly
:
th
e
�
ie
�
i
ab
ov
e
ge
ts
re
p
la
ce
d
by
�
ie
�
µ
w
h
ic
h
co
rr
ec
tl
y
ac
co
u
nt
s
fo
r
th
e
(e
�
0
)2
p
ie
ce
in
th
e
in
st
an
ta
n
eo
u
s
in
te
ra
ct
io
n
.
–
14
1
–
6
.4
.1
N
a
iv
e
F
ey
n
m
a
n
R
u
les
W
e
w
ant
to
d
eterm
in
e
th
e
F
eyn
m
an
ru
les
for
th
is
th
eory.
F
or
ferm
ion
s,
th
e
ru
les
are
th
e
sam
e
as
th
ose
given
in
S
ection
5.
T
h
e
n
ew
p
ieces
are:
•
W
e
d
en
ote
th
e
p
h
oton
by
a
w
avy
lin
e.
E
ach
en
d
of
th
e
lin
e
com
es
w
ith
an
i,j
=
1,2,3
in
d
ex
tellin
g
u
s
th
e
com
p
on
ent
of
� A
.
W
e
calcu
lated
th
e
tran
sverse
p
h
oton
p
rop
agator
in
(6.33):
it
is
an
d
contrib
u
tes
D
tr ij
=
i
p
2
+
i�✓
�
ij�
p
ip
j
|�p|2◆
•
T
h
e
vertex
contrib
u
tes
�
ie�
i .
T
h
e
in
d
ex
on
�
i
contracts
w
ith
th
e
in
d
ex
on
th
e
p
h
oton
lin
e.
•
T
h
e
n
on
-local
interaction
w
h
ich
,
in
p
osition
sp
ace,
is
given
by
x
y
contrib
u
tes
a
factor
of
i(e�
0 )
2 �(x
0 �
y
0 )
4�|�x
�
�y|
T
h
ese
F
eyn
m
an
ru
les
are
rath
er
m
essy.
T
h
is
is
th
e
p
rice
w
e’ve
p
aid
for
w
orkin
g
in
C
ou
lom
b
gau
ge.
W
e’ll
n
ow
sh
ow
th
at
w
e
can
m
assage
th
ese
exp
ression
s
into
som
eth
in
g
m
u
ch
m
ore
sim
p
le
an
d
L
orentz
invariant.
L
et’s
start
w
ith
th
e
o�
en
d
ing
in
stantan
eou
s
interaction
.
S
in
ce
it
com
es
from
th
e
A
0
com
p
on
ent
of
th
e
gau
ge
fi
eld
,
w
e
cou
ld
try
to
red
efi
n
e
th
e
p
rop
agator
to
in
clu
d
e
a
D
00
p
iece
w
h
ich
w
ill
cap
tu
re
th
is
term
.
In
fact,
it
fi
ts
qu
ite
n
icely
in
th
is
form
:
if
w
e
look
in
m
om
entu
m
sp
ace,
w
e
h
ave
�(x
0 �
y
0 )
4�|�x
�
�y|
=
Z
d
4 p
(2�
)
4
e
ip·(x�
y
)
|�p|2
(6.83)
so
w
e
can
com
b
in
e
th
e
n
on
-local
interaction
w
ith
th
e
tran
sverse
p
hoton
p
rop
agator
by
d
efi
n
in
g
a
n
ew
p
h
oton
p
rop
agator
D
µ
⌫(p)
=
� � � � � � � � � � �
+
i
|�p|2
µ
,�
=
0
i
p
2
+
i�✓
�
ij�
p
ip
j
|�p|2◆
µ
=
i6=
0,�
=
j
6=
0
0
oth
erw
ise
(6.84)
W
ith
th
is
p
rop
agator,
th
e
w
avy
p
h
oton
lin
e
n
ow
carries
a
µ
,�
=
0,1,2,3
in
d
ex,
w
ith
th
e
extra
µ
=
0
com
p
on
ent
takin
g
care
of
th
e
in
stantan
eou
s
interaction
.
W
e
n
ow
n
eed
to
ch
an
ge
ou
r
vertex
slightly:
th
e
�
ie�
i
ab
ove
gets
rep
laced
by
�
ie�
µ
w
h
ich
correctly
accou
nts
for
th
e
(e�
0 )
2
p
iece
in
th
e
in
stantan
eou
s
interaction
.
–
141
–
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesDtrij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�i.Theindexon�icontractswiththe
indexonthephotonline.•Thenon-localinteractionwhich,inpositionspace,isgivenbyxy
contributesafactorof
i(e�0)2�(x0�y0)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x0�y0)
4�|�x��y|=
Z
d4p
(2�)4
eip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
�����
�����
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�iabovegetsreplacedby�ie�µwhichcorrectly
accountsforthe(e�0)2pieceintheinstantaneousinteraction.
–141–
6.4.1 Naive Feynman Rules
We want to determine the Feynman rules for this theory. For fermions, the rules are
the same as those given in Section 5. The new pieces are:
• We denote the photon by a wavy line. Each end of the line comes with an i, j =
1, 2, 3 index telling us the component of �A. We calculated the transverse photon
propagator in (6.33): it is and contributes Dtrij =
i
p2 + i�
✓
�ij �
pipj
|�p|2
◆
• The vertex contributes �ie�i. The index on �i contracts with the
index on the photon line.
• The non-local interaction which, in position space, is given by x y
contributes a factor of
i(e�0)2�(x0 � y0)
4�|�x � �y|
These Feynman rules are rather messy. This is the price we’ve paid for working in
Coulomb gauge. We’ll now show that we can massage these expressions into something
much more simple and Lorentz invariant. Let’s start with the o�ending instantaneous
interaction. Since it comes from the A0 component of the gauge field, we could try to
redefine the propagator to include a D00 piece which will capture this term. In fact, it
fits quite nicely in this form: if we look in momentum space, we have
�(x0 � y0)
4�|�x � �y| =
Z
d4p
(2�)4
eip·(x�y)
|�p|2 (6.83)
so we can combine the non-local interaction with the transverse photon propagator by
defining a new photon propagator
Dµ⌫(p) =
�
�����
�����
+
i
|�p|2 µ, � = 0
i
p2 + i�
✓
�ij �
pipj
|�p|2
◆
µ = i 6= 0, � = j 6= 0
0 otherwise
(6.84)
With this propagator, the wavy photon line now carries a µ, � = 0, 1, 2, 3 index, with
the extra µ = 0 component taking care of the instantaneous interaction. We now need
to change our vertex slightly: the �ie�i above gets replaced by �ie�µ which correctly
accounts for the (e�0)2 piece in the instantaneous interaction.
– 141 –
6.4.1
N
aive
Feyn
m
an
R
u
les
W
e
w
ant
to
determ
ine
the
Feynm
an
rules
for
this
theory.
For
ferm
ions,
the
rules
are
the
sam
e
as
those
given
in
Section
5.
T
he
new
pieces
are:
•
W
e
denote
the
photon
by
a
w
avy
line.
E
ach
end
of
the
line
com
es
w
ith
an
i,j
=
1,2,3
index
telling
us
the
com
ponent
of
�A
.
W
e
calculated
the
transverse
photon
propagator
in
(6.33):
it
is
and
contributes
D
tr ij=
i p2
+
i�✓
�
ij�
p
ip
j
|�p|2◆
•
T
he
vertex
contributes
�
ie�i .
T
he
index
on
�i
contracts
w
ith
the
index
on
the
photon
line.
•
T
he
non-local
interaction
w
hich,
in
position
space,
is
given
by
x
y
contributes
a
factor
ofi(e�0)2�(x0
�
y0)
4�|�x�
�y|
T
hese
Feynm
an
rules
are
rather
m
essy.
T
his
is
the
price
w
e’ve
paid
for
w
orking
in
C
oulom
b
gauge.
W
e’llnow
show
that
w
e
can
m
assage
these
expressions
into
som
ething
m
uch
m
ore
sim
ple
and
Lorentz
invariant.
Let’s
start
w
ith
the
o�ending
instantaneous
interaction.
Since
it
com
es
from
the
A
0
com
ponent
of
the
gauge
field,
w
e
could
try
to
redefine
the
propagator
to
include
a
D
00
piece
w
hich
w
illcapture
this
term
.
In
fact,it
fits
quite
nicely
in
this
form
:
if
w
e
look
in
m
om
entum
space,w
e
have
�(x0
�
y0)
4�|�x�
�y|=Z
d4p (2�)4eip·(x�
y)
|�p|2
(6.83)
so
w
e
can
com
bine
the
non-localinteraction
w
ith
the
transverse
photon
propagator
by
defining
a
new
photon
propagator
D
µ
⌫(p)
=
� �����
�����
+
i |�p|2
µ,�
=
0
i p2
+
i�✓
�
ij�
p
ip
j
|�p|2◆
µ
=
i6=
0,�
=
j6=
0
0
otherw
ise
(6.84)
W
ith
this
propagator,
the
w
avy
photon
line
now
carries
a
µ,�
=
0,1,2,3
index,
w
ith
the
extra
µ
=
0
com
ponent
taking
care
ofthe
instantaneous
interaction.
W
e
now
need
to
change
our
vertex
slightly:
the�
ie�i
above
gets
replaced
by
�
ie�µ
w
hich
correctly
accounts
for
the
(e�0)2
piece
in
the
instantaneous
interaction.
–
141
–
6.
4.
1
N
ai
ve
Fe
yn
m
an
R
u
le
s
W
e
w
an
t
to
de
te
rm
in
e
th
e
Fe
yn
m
an
ru
le
s
fo
r
th
is
th
eo
ry
.
Fo
r
fe
rm
io
ns
,
th
e
ru
le
s
ar
e
th
e
sa
m
e
as
th
os
e
gi
ve
n
in
Se
ct
io
n
5.
T
he
ne
w
pi
ec
es
ar
e:
•
W
e
de
no
te
th
e
ph
ot
on
by
a
w
av
y
lin
e.
E
ac
h
en
d
of
th
e
lin
e
co
m
es
w
it
h
an
i,
j
=
1,
2,
3
in
de
x
te
lli
ng
us
th
e
co
m
po
ne
nt
of
�A
.
W
e
ca
lc
ul
at
ed
th
e
tr
an
sv
er
se
ph
ot
on
pr
op
ag
at
or
in
(6
.3
3)
:
it
is
an
d
co
nt
ri
bu
te
s
D
tr ij
=
i
p
2
+
i�
✓ � ij
�
p i
p j
|�p|
2
◆
•
T
he
ve
rt
ex
co
nt
ri
bu
te
s
�i
e�
i .
T
he
in
de
x
on
�
i
co
nt
ra
ct
s
w
it
h
th
e
in
de
x
on
th
e
ph
ot
on
lin
e.
•
T
he
no
n-
lo
ca
l
in
te
ra
ct
io
n
w
hi
ch
,
in
po
si
ti
on
sp
ac
e,
is
gi
ve
n
by
x
y
co
nt
ri
bu
te
s
a
fa
ct
or
of
i(
e�
0 )
2 �
(x
0
�
y
0 )
4�
|�x
�
�y|
T
he
se
Fe
yn
m
an
ru
le
s
ar
e
ra
th
er
m
es
sy
.
T
hi
s
is
th
e
pr
ic
e
w
e’
ve
pa
id
fo
r
w
or
ki
ng
in
C
ou
lo
m
b
ga
ug
e.
W
e’
ll
no
w
sh
ow
th
at
w
e
ca
n
m
as
sa
ge
th
es
e
ex
pr
es
si
on
s
in
to
so
m
et
hi
ng
m
uc
h
m
or
e
si
m
pl
e
an
d
Lo
re
nt
z
in
va
ri
an
t.
Le
t’
s
st
ar
t
w
it
h
th
e
o�
en
di
ng
in
st
an
ta
ne
ou
s
in
te
ra
ct
io
n.
Si
nc
e
it
co
m
es
fr
om
th
e
A
0
co
m
po
ne
nt
of
th
e
ga
ug
e
fie
ld
,
w
e
co
ul
d
tr
y
to
re
de
fin
e
th
e
pr
op
ag
at
or
to
in
cl
ud
e
a
D
00
pi
ec
e
w
hi
ch
w
ill
ca
pt
ur
e
th
is
te
rm
.
In
fa
ct
, i
t
fit
s
qu
it
e
ni
ce
ly
in
th
is
fo
rm
:
if
w
e
lo
ok
in
m
om
en
tu
m
sp
ac
e,
w
e
ha
ve
�(
x
0
�
y
0 )
4�
|�x
�
�y|
=
Z
d
4 p
(2
�)
4
e
ip
·(x
�
y)
|�p|
2
(6
.8
3)
so
w
e
ca
n
co
m
bi
ne
th
e
no
n-
lo
ca
l i
nt
er
ac
ti
on
w
it
h
th
e
tr
an
sv
er
se
ph
ot
on
pr
op
ag
at
or
by
de
fin
in
g
a
ne
w
ph
ot
on
pr
op
ag
at
or
D
µ
⌫
(p
)
=
� �����
�����
+
i
|�p|
2
µ,
�
=
0
i
p
2
+
i�
✓ � ij
�
p i
p j
|�p|
2
◆
µ
=
i 6=
0,
�
=
j
6=
0
0
ot
he
rw
is
e
(6
.8
4)
W
it
h
th
is
pr
op
ag
at
or
,
th
e
w
av
y
ph
ot
on
lin
e
no
w
ca
rr
ie
s
a
µ,
�
=
0,
1,
2,
3
in
de
x,
w
it
h
th
e
ex
tr
a
µ
=
0
co
m
po
ne
nt
ta
ki
ng
ca
re
of
th
e
in
st
an
ta
ne
ou
s
in
te
ra
ct
io
n.
W
e
no
w
ne
ed
to
ch
an
ge
ou
r
ve
rt
ex
sl
ig
ht
ly
:
th
e
�i
e�
i
ab
ov
e
ge
ts
re
pl
ac
ed
by
�i
e�
µ
w
hi
ch
co
rr
ec
tl
y
ac
co
un
ts
fo
r
th
e
(e
�
0 )
2
pi
ec
e
in
th
e
in
st
an
ta
ne
ou
s
in
te
ra
ct
io
n.
–
14
1
–
6.4.1
N
aive
Feyn
m
an
R
u
les
W
e
w
ant
to
determ
ine
the
Feynm
an
rules
for
this
theory.
For
ferm
ions,
the
rules
are
the
sam
e
as
those
given
in
Section
5.
T
he
new
pieces
are:
•
W
e
denote
the
photon
by
a
w
avy
line.
E
ach
end
of
the
line
com
es
w
ith
an
i, j
=
1, 2, 3
index
telling
us
the
com
ponent
of
�A.
W
e
calculated
the
transverse
photon
propagator
in
(6.33):
it
is
and
contributes
D
tr
ij
=
i
p
2 +
i�
✓ � ij�
p ip j
|�p|
2
◆
•
T
he
vertex
contributes
�
ie�
i .
T
he
index
on
�
i contracts
w
ith
the
indexon
the
photon
line.
•
T
he
non-local
interaction
w
hich,
in
position
space,
is
given
by
x
y
contributes
a
factor
of
i(e�
0 )
2 �(x
0 �
y
0 )
4�|�x�
�y|
T
hese
Feynm
an
rules
are
rather
m
essy.
T
his
is
the
price
w
e’ve
paid
for
w
orking
in
C
oulom
b
gauge.
W
e’ll now
show
that
w
e
can
m
assage
these
expressions
into
som
ething
m
uch
m
ore
sim
ple
and
Lorentz
invariant.
Let’s
start
w
ith
the
o�ending
instantaneous
interaction.
Since
it
com
es
from
the
A
0
com
ponent
of
the
gauge
field,
w
e
could
try
to
redefine
the
propagator
to
include
a
D
00
piece
w
hich
w
ill capture
this
term
.
In
fact, it
fits
quite
nicely
in
this
form
:
if
w
e
look
in
m
om
entum
space, w
e
have
�(x
0 �
y
0 )
4�|�x�
�y|
=
Z
d
4 p
(2�)
4
e
ip·(x�
y)
|�p|
2
(6.83)
so
w
e
can
com
bine
the
non-local interaction
w
ith
the
transverse
photon
propagator
by
defining
a
new
photon
propagator
D
µ
⌫(p)
=
� ��
���
�����
+
i
|�p|
2
µ, �
=
0
i
p
2 +
i�
✓ � ij�
p ip j
|�p|
2
◆
µ
=
i 6=
0, �
=
j 6=
0
0
otherw
ise
(6.84)
W
ith
this
propagator,
the
w
avy
photon
line
now
carries
a
µ, �
=
0, 1, 2, 3
index,
w
ith
the
extra
µ
=
0
com
ponent
taking
care
of the
instantaneous
interaction.
W
e
now
need
to
change
our
vertex
slightly:
the �
ie�
i above
gets
replaced
by
�
ie�
µ w
hich
correctly
accounts
for
the
(e�
0 )
2 piece
in
the
instantaneous
interaction.
–
141
–
6.
4.
1
N
ai
ve
Fe
yn
m
an
R
u
le
s
W
e
w
an
t
to
de
te
rm
in
e
th
e
Fe
yn
m
an
ru
le
s
fo
r
th
is
th
eo
ry
.
Fo
r
fe
rm
io
ns
,
th
e
ru
le
s
ar
e
th
e
sa
m
e
as
th
os
e
gi
ve
n
in
Se
ct
io
n
5.
T
he
ne
w
pi
ec
es
ar
e:
•
W
e
de
no
te
th
e
ph
ot
on
by
a
w
av
y
lin
e.
E
ac
h
en
d
of
th
e
lin
e
co
m
es
w
it
h
an
i,
j
=
1,
2,
3
in
de
x
te
lli
ng
us
th
e
co
m
po
ne
nt
of
�A
.
W
e
ca
lc
ul
at
ed
th
e
tr
an
sv
er
se
ph
ot
on
pr
op
ag
at
or
in
(6
.3
3)
:
it
is
an
d
co
nt
ri
bu
te
s
Dt
r ij
=
i p2
+
i�✓
�i
j
�p
ip
j |�p|
2◆
•
T
he
ve
rt
ex
co
nt
ri
bu
te
s
�i
e�
i
.
T
he
in
de
x
on
�i
co
nt
ra
ct
s
w
it
h
th
e
in
de
x
on
th
e
ph
ot
on
lin
e.
•
T
he
no
n-
lo
ca
l
in
te
ra
ct
io
n
w
hi
ch
,
in
po
si
ti
on
sp
ac
e,
is
gi
ve
n
by
x
y
co
nt
ri
bu
te
s
a
fa
ct
or
of
i(
e�
0
)2
�(
x0
�
y0
)
4�
|�x
�
�y|
T
he
se
Fe
yn
m
an
ru
le
s
ar
e
ra
th
er
m
es
sy
.
T
hi
s
is
th
e
pr
ic
e
w
e’
ve
pa
id
fo
r
w
or
ki
ng
in
C
ou
lo
m
b
ga
ug
e.
W
e’
ll
no
w
sh
ow
th
at
w
e
ca
n
m
as
sa
ge
th
es
e
ex
pr
es
si
on
s
in
to
so
m
et
hi
ng
m
uc
h
m
or
e
si
m
pl
e
an
d
Lo
re
nt
z
in
va
ri
an
t.
Le
t’
s
st
ar
t
w
it
h
th
e
o�
en
di
ng
in
st
an
ta
ne
ou
s
in
te
ra
ct
io
n.
Si
nc
e
it
co
m
es
fr
om
th
e
A
0
co
m
po
ne
nt
of
th
e
ga
ug
e
fie
ld
,
w
e
co
ul
d
tr
y
to
re
de
fin
e
th
e
pr
op
ag
at
or
to
in
cl
ud
e
a
D
00
pi
ec
e
w
hi
ch
w
ill
ca
pt
ur
e
th
is
te
rm
.
In
fa
ct
,i
t
fit
s
qu
it
e
ni
ce
ly
in
th
is
fo
rm
:
if
w
e
lo
ok
in
m
om
en
tu
m
sp
ac
e,
w
e
ha
ve
�(
x0
�
y0
)
4�
|�x
�
�y|
=Z
d4
p (2
�)
4ei
p·
(x
�
y) |�p|
2
(6
.8
3)
so
w
e
ca
n
co
m
bi
ne
th
e
no
n-
lo
ca
li
nt
er
ac
ti
on
w
it
h
th
e
tr
an
sv
er
se
ph
ot
on
pr
op
ag
at
or
by
de
fin
in
g
a
ne
w
ph
ot
on
pr
op
ag
at
or
D
µ
⌫(
p)
=
� ��
���
�����
+
i |�p
|2
µ,
�
=
0
i p2
+
i�✓
�i
j
�p
ip
j |�p|
2◆
µ
=
i6=
0,
�
=
j
6=
0
0
ot
he
rw
is
e
(6
.8
4)
W
it
h
th
is
pr
op
ag
at
or
,
th
e
w
av
y
ph
ot
on
lin
e
no
w
ca
rr
ie
s
a
µ,
�
=
0,
1,
2,
3
in
de
x,
w
it
h
th
e
ex
tr
a
µ
=
0
co
m
po
ne
nt
ta
ki
ng
ca
re
of
th
e
in
st
an
ta
ne
ou
s
in
te
ra
ct
io
n.
W
e
no
w
ne
ed
to
ch
an
ge
ou
r
ve
rt
ex
sl
ig
ht
ly
:
th
e
�i
e�
i
ab
ov
e
ge
ts
re
pl
ac
ed
by
�i
e�
µ
w
hi
ch
co
rr
ec
tl
y
ac
co
un
ts
fo
r
th
e
(e
�0
)2
pi
ec
e
in
th
e
in
st
an
ta
ne
ou
s
in
te
ra
ct
io
n.
–
14
1
–
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesDtrij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�i.Theindexon�icontractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenbyxy
contributesafactorof
i(e�0)2�(x0�y0)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x0�y0)
4�|�x��y|=
Z
d4p
(2�)4
eip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
�����
�����
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�iabovegetsreplacedby�ie�µwhichcorrectly
accountsforthe(e�0)2pieceintheinstantaneousinteraction.
–141–
6.4.1 Naive Feynman Rules
We want to determine the Feynman rules for this theory. For fermions, the rules are
the same as those given in Section 5. The new pieces are:
• We denote the photon by a wavy line. Each end of the line comes with an i, j =
1, 2, 3 index telling us the component of �A. We calculated the transverse photon
propagator in (6.33): it is and contributes Dtrij =
i
p2 + i�
✓
�ij �
pipj
|�p|2
◆
• The vertex contributes �ie�i. The index on �i contracts with the
index on the photon line.
• The non-local interaction which, in position space, is given by x y
contributes a factor of
i(e�0)2�(x0 � y0)
4�|�x � �y|
These Feynman rules are rather messy. This is the price we’ve paid for working in
Coulomb gauge. We’ll now show that we can massage these expressions into something
much more simple and Lorentz invariant. Let’s start with the o�ending instantaneous
interaction. Since it comes from the A0 component of the gauge field, we could try to
redefine the propagator to include a D00 piece which will capture this term. In fact, it
fits quite nicely in this form: if we look in momentum space, we have
�(x0 � y0)
4�|�x � �y| =
Z
d4p
(2�)4
eip·(x�y)
|�p|2 (6.83)
so we can combine the non-local interaction with the transverse photon propagator by
defining a new photon propagator
Dµ⌫(p) =
�
�����
�����
+
i
|�p|2 µ, � = 0
i
p2 + i�
✓
�ij �
pipj
|�p|2
◆
µ = i 6= 0, � = j 6= 0
0 otherwise
(6.84)
With this propagator, the wavy photon line now carries a µ, � = 0, 1, 2, 3 index, with
the extra µ = 0 component taking care of the instantaneous interaction. We now need
to change our vertex slightly: the �ie�i above gets replaced by �ie�µ which correctly
accounts for the (e�0)2 piece in the instantaneous interaction.
– 141 –
6
.4
.1
N
a
iv
e
F
ey
n
m
a
n
R
u
le
s
W
e
w
an
t
to
d
et
er
m
in
e
th
e
F
ey
n
m
an
ru
le
s
fo
r
th
is
th
eo
ry
.
F
or
fe
rm
io
n
s,
th
e
ru
le
s
ar
e
th
e
sa
m
e
as
th
os
e
gi
ve
n
in
S
ec
ti
on
5.
T
h
e
n
ew
p
ie
ce
s
ar
e:
•
W
e
d
en
ot
e
th
e
p
h
ot
on
by
a
w
av
y
lin
e.
E
ac
h
en
d
of
th
e
lin
e
co
m
es
w
it
h
an
i,
j
=
1,
2,
3
in
d
ex
te
lli
n
g
u
s
th
e
co
m
p
on
en
t
of
� A
.
W
e
ca
lc
u
la
te
d
th
e
tr
an
sv
er
se
p
h
ot
on
p
ro
p
ag
at
or
in
(6
.3
3)
:
it
is
an
d
co
nt
ri
b
u
te
s
D
tr ij
=
i
p2
+
i�
✓
� i
j
�
p i
p j |�p
|2
◆
•
T
h
e
ve
rt
ex
co
nt
ri
b
u
te
s
�
ie
�i .
T
h
e
in
d
ex
on
�
i
co
nt
ra
ct
s
w
it
h
th
e
in
d
ex
on
th
e
p
h
ot
on
lin
e.
•
T
h
e
n
on
-l
oc
al
in
te
ra
ct
io
n
w
h
ic
h
,
in
p
os
it
io
n
sp
ac
e,
is
gi
ve
n
by
x
y
co
nt
ri
b
u
te
s
a
fa
ct
or
of
i(
e�
0
)2
�(
x
0
�
y0
)
4�
|�x
�
�y|
T
h
es
e
F
ey
n
m
an
ru
le
s
ar
e
ra
th
er
m
es
sy
.
T
h
is
is
th
e
p
ri
ce
w
e’
ve
p
ai
d
fo
r
w
or
ki
n
g
in
C
ou
lo
m
b
ga
u
ge
.
W
e’
ll
n
ow
sh
ow
th
at
w
e
ca
n
m
as
sa
ge
th
es
e
ex
p
re
ss
io
n
s
in
to
so
m
et
h
in
g
m
u
ch
m
or
e
si
m
p
le
an
d
L
or
en
tz
in
va
ri
an
t.
L
et
’s
st
ar
t
w
it
h
th
e
o�
en
d
in
g
in
st
an
ta
n
eo
u
s
in
te
ra
ct
io
n
.
S
in
ce
it
co
m
es
fr
om
th
e
A
0
co
m
p
on
en
t
of
th
e
ga
u
ge
fi
el
d
,
w
e
co
u
ld
tr
y
to
re
d
efi
n
e
th
e
p
ro
p
ag
at
or
to
in
cl
u
d
e
a
D
00
p
ie
ce
w
h
ic
h
w
ill
ca
p
tu
re
th
is
te
rm
.
In
fa
ct
,
it
fi
ts
qu
it
e
n
ic
el
y
in
th
is
fo
rm
:
if
w
e
lo
ok
in
m
om
en
tu
m
sp
ac
e,
w
e
h
av
e
�(
x
0
�
y0
)
4�
|�x
�
�y|
=
Z
d4
p
(2
�
)4
ei
p
·(
x
�
y
)
|�p
|2
(6
.8
3)
so
w
e
ca
n
co
m
b
in
e
th
e
n
on
-l
oc
al
in
te
ra
ct
io
n
w
it
h
th
e
tr
an
sv
er
se
p
ho
to
n
p
ro
p
ag
at
or
by
d
efi
n
in
g
a
n
ew
p
h
ot
on
p
ro
p
ag
at
or
D
µ
⌫
(p
)
=
� � � � � � � � � � �
+
i |�p
|2
µ
,�
=
0
i
p2
+
i�
✓
� i
j
�
p i
p j |�p
|2
◆
µ
=
i
6=
0,
�
=
j
6=
0
0
ot
h
er
w
is
e
(6
.8
4)
W
it
h
th
is
p
ro
p
ag
at
or
,
th
e
w
av
y
p
h
ot
on
lin
e
n
ow
ca
rr
ie
s
a
µ
,�
=
0,
1,
2,
3
in
d
ex
,
w
it
h
th
e
ex
tr
a
µ
=
0
co
m
p
on
en
t
ta
ki
n
g
ca
re
of
th
e
in
st
an
ta
n
eo
u
s
in
te
ra
ct
io
n
.
W
e
n
ow
n
ee
d
to
ch
an
ge
ou
r
ve
rt
ex
sl
ig
ht
ly
:
th
e
�
ie
�
i
ab
ov
e
ge
ts
re
p
la
ce
d
by
�
ie
�
µ
w
h
ic
h
co
rr
ec
tl
y
ac
co
u
nt
s
fo
r
th
e
(e
�
0
)2
p
ie
ce
in
th
e
in
st
an
ta
n
eo
u
s
in
te
ra
ct
io
n
.
–
14
1
–
6
.4
.1
N
a
iv
e
F
ey
n
m
a
n
R
u
les
W
e
w
ant
to
d
eterm
in
e
th
e
F
eyn
m
an
ru
les
for
th
is
th
eory.
F
or
ferm
ion
s,
th
e
ru
les
are
th
e
sam
e
as
th
ose
given
in
S
ection
5.
T
h
e
n
ew
p
ieces
are:
•
W
e
d
en
ote
th
e
p
h
oton
by
a
w
avy
lin
e.
E
ach
en
d
of
th
e
lin
e
com
es
w
ith
an
i,j
=
1,2,3
in
d
ex
tellin
g
u
s
th
e
com
p
on
ent
of
� A
.
W
e
calcu
lated
th
e
tran
sverse
p
h
oton
p
rop
agator
in
(6.33):
it
is
an
d
contrib
u
tes
D
tr ij
=
i
p
2
+
i�✓
�
ij�
p
ip
j
|�p|2◆
•
T
h
e
vertex
contrib
u
tes
�
ie�
i .
T
h
e
in
d
ex
on
�
i
contracts
w
ith
th
e
in
d
ex
on
th
e
p
h
oton
lin
e.
•
T
h
e
n
on
-local
interaction
w
h
ich
,
in
p
osition
sp
ace,
is
given
by
x
y
contrib
u
tes
a
factor
of
i(e�
0 )
2 �(x
0 �
y
0 )
4�|�x
�
�y|
T
h
ese
F
eyn
m
an
ru
les
are
rath
er
m
essy.
T
h
is
is
th
e
p
rice
w
e’ve
p
aid
for
w
orkin
g
in
C
ou
lom
b
gau
ge.
W
e’ll
n
ow
sh
ow
th
at
w
e
can
m
assage
th
ese
exp
ression
s
into
som
eth
in
g
m
u
ch
m
ore
sim
p
le
an
d
L
orentz
invariant.
L
et’s
start
w
ith
th
e
o�
en
d
ing
in
stantan
eou
s
interaction
.
S
in
ce
it
com
es
from
th
e
A
0
com
p
on
ent
of
th
e
gau
ge
fi
eld
,
w
e
cou
ld
try
to
red
efi
n
e
th
e
p
rop
agator
to
in
clu
d
e
a
D
00
p
iece
w
h
ich
w
ill
cap
tu
re
th
is
term
.
In
fact,
it
fi
ts
qu
ite
n
icely
in
th
is
form
:
if
w
e
look
in
m
om
entu
m
sp
ace,
w
e
h
ave
�(x
0 �
y
0 )
4�|�x
�
�y|
=
Z
d
4 p
(2�
)
4
e
ip·(x�
y
)
|�p|2
(6.83)
so
w
e
can
com
b
in
e
th
e
n
on
-local
interaction
w
ith
th
e
tran
sverse
p
hoton
p
rop
agator
by
d
efi
n
in
g
a
n
ew
p
h
oton
p
rop
agator
D
µ
⌫(p)
=
� � � � � � � � � � �
+
i
|�p|2
µ
,�
=
0
i
p
2
+
i�✓
�
ij�
p
ip
j
|�p|2◆
µ
=
i6=
0,�
=
j
6=
0
0
oth
erw
ise
(6.84)
W
ith
th
is
p
rop
agator,
th
e
w
avy
p
h
oton
lin
e
n
ow
carries
a
µ
,�
=
0,1,2,3
in
d
ex,
w
ith
th
e
extra
µ
=
0
com
p
on
ent
takin
g
care
of
th
e
in
stantan
eou
s
interaction
.
W
e
n
ow
n
eed
to
ch
an
ge
ou
r
vertex
slightly:
th
e
�
ie�
i
ab
ove
gets
rep
laced
by
�
ie�
µ
w
h
ich
correctly
accou
nts
for
th
e
(e�
0 )
2
p
iece
in
th
e
in
stantan
eou
s
interaction
.
–
141
–
6.4.1
N
aive
Feynm
an
R
ules
W
e
want
to
determ
ine
the
Feynm
an
rulesforthis
theory.
Forferm
ions,the
rules
are
the
sam
e
asthose
given
in
Section
5.The
new
piecesare:
•
W
e
denote
the
photon
by
a
wavy
line.Each
end
ofthe
line
com
eswith
an
i,j=
1,2,3
index
telling
usthecom
ponentof�A.W
e
calculated
thetransverse
photon
propagatorin
(6.33):itis
and
contributesDtr ij=
i p2
+
i�
✓
�
ij�p
ip
j |�p|2
◆
•
The
vertex
contributes�
ie�i.
The
index
on
�i
contracts
with
the
index
on
the
photon
line.
•
The
non-localinteraction
which,in
position
space,is
given
by
x
y
contributesa
factorofi(e�0)2
�(x0�
y0)
4�|�x�
�y|
These
Feynm
an
rules
are
rather
m
essy.
This
is
the
price
we’ve
paid
for
working
in
Coulom
b
gauge.W
e’llnow
show
thatwecan
m
assagetheseexpressionsinto
som
ething
m
uch
m
ore
sim
ple
and
Lorentz
invariant.Let’sstartwith
the
o�ending
instantaneous
interaction.Since
itcom
esfrom
the
A
0com
ponentofthe
gauge
field,we
could
try
to
redefine
the
propagatorto
include
a
D
00piece
which
willcapture
thisterm
.In
fact,it
fitsquite
nicely
in
thisform
:ifwe
look
in
m
om
entum
space,we
have
�(x0�
y0)
4�|�x�
�y|=Z
d4
p (2�)4eip·(x�
y) |�p|2
(6.83)
so
wecan
com
binethenon-localinteraction
with
thetransverse
photon
propagatorby
defining
a
new
photon
propagator
D
µ⌫(p)=
������
�����
+i |�p|2
µ,�
=
0
i p2
+
i�
✓
�
ij�p
ip
j |�p|2
◆
µ
=
i6=
0,�
=
j6=
0
0
otherwise
(6.84)
W
ith
thispropagator,the
wavy
photon
line
now
carriesa
µ,�
=
0,1,2,3
index,with
theextra
µ
=
0
com
ponenttaking
careoftheinstantaneousinteraction.W
enow
need
to
change
ourvertex
slightly:the�
ie�i
above
getsreplaced
by�
ie�µ
which
correctly
accountsforthe
(e�0
)2
piece
in
the
instantaneousinteraction.
–
141
–
6.
4.
1
N
ai
ve
Fe
yn
m
an
R
ul
es
W
e
wa
nt
to
de
te
rm
in
e
th
e
Fe
yn
m
an
ru
les
fo
r t
hi
s
th
eo
ry
.
Fo
r f
er
m
ion
s,
th
e
ru
les
ar
e
th
e
sa
m
e
as
th
os
e
gi
ve
n
in
Se
ct
io
n
5.
Th
e
ne
w
pi
ec
es
ar
e:
•
W
e
de
no
te
th
e
ph
ot
on
by
a
wa
vy
lin
e.
Ea
ch
en
d
of
th
e
lin
e
co
m
es
wi
th
an
i,
j =
1,
2,
3
in
de
x
te
lli
ng
us
th
e c
om
po
ne
nt
of
�A.
W
e
ca
lcu
la
te
d
th
e t
ra
ns
ve
rs
e
ph
ot
on
pr
op
ag
at
or
in
(6
.3
3)
: i
t i
s
an
d
co
nt
rib
ut
es
D
tr
ij
=
i
p
2 +
i�
✓ � ij
�
p i
p j
|�p|
2
◆
•
Th
e
ve
rt
ex
co
nt
rib
ut
es
�i
e�
i .
Th
e
in
de
x
on
�
i c
on
tr
ac
ts
wi
th
th
e
in
de
x
on
th
e
ph
ot
on
lin
e.
•
Th
e
no
n-
lo
ca
l i
nt
er
ac
tio
n
wh
ich
, i
n
po
sit
io
n
sp
ac
e,
is
gi
ve
n
by
x
y
co
nt
rib
ut
es
a
fa
ct
or
of
i(e
�
0 )
2 �(
x
0 �
y
0 )
4�
|�x
�
�y|
Th
es
e
Fe
yn
m
an
ru
les
ar
e
ra
th
er
m
es
sy
.
Th
is
is
th
e
pr
ice
we
’ve
pa
id
fo
r
wo
rk
in
g
in
Co
ul
om
b
ga
ug
e.
W
e’l
l n
ow
sh
ow
th
at
we
ca
n
m
as
sa
ge
th
es
e e
xp
re
ss
ion
s i
nt
o
so
m
et
hi
ng
m
uc
h
m
or
e
sim
pl
e
an
d
Lo
re
nt
z
in
va
ria
nt
. L
et
’s
st
ar
t w
ith
th
e
o�
en
di
ng
in
st
an
ta
ne
ou
s
in
te
ra
ct
io
n.
Si
nc
e
it
co
m
es
fro
m
th
e
A 0
co
m
po
ne
nt
of
th
e
ga
ug
e
fie
ld
, w
e
co
ul
d
tr
y
to
re
de
fin
e
th
e
pr
op
ag
at
or
to
in
clu
de
a
D 0
0
pi
ec
e
wh
ich
wi
ll
ca
pt
ur
e
th
is
te
rm
. I
n
fa
ct
, i
t
fit
s q
ui
te
ni
ce
ly
in
th
is
fo
rm
: i
f w
e
lo
ok
in
m
om
en
tu
m
sp
ac
e,
we
ha
ve
�(
x
0 �
y
0 )
4�
|�x
�
�y|
=
Z
d
4 p
(2
�)
4
e
ip
·(x
�y
)
|�p|
2
(6
.8
3)
so
we
ca
n
co
m
bi
ne
th
e n
on
-lo
ca
l i
nt
er
ac
tio
n
wi
th
th
e t
ra
ns
ve
rs
e
ph
ot
on
pr
op
ag
at
or
by
de
fin
in
g
a
ne
w
ph
ot
on
pr
op
ag
at
or
D µ
⌫
(p
) =
������
�����
+
i
|�p|
2
µ,
�
=
0
i
p
2 +
i�
✓ � ij
�
p i
p j
|�p|
2
◆
µ
=
i 6=
0,
�
=
j 6=
0
0
ot
he
rw
ise
(6
.8
4)
W
ith
th
is
pr
op
ag
at
or, t
he
wa
vy
ph
ot
on
lin
e
no
w
ca
rr
ies
a
µ,
�
=
0,
1,
2,
3
in
de
x,
wi
th
th
e e
xt
ra
µ
=
0
co
m
po
ne
nt
ta
ki
ng
ca
re
of
th
e i
ns
ta
nt
an
eo
us
in
te
ra
ct
io
n.
W
e n
ow
ne
ed
to
ch
an
ge
ou
r v
er
te
x
sli
gh
tly
: t
he
�i
e�
i ab
ov
e
ge
ts
re
pl
ac
ed
by
�i
e�
µ w
hi
ch
co
rr
ec
tly
ac
co
un
ts
fo
r t
he
(e
�
0 )
2 p
iec
e
in
th
e
in
st
an
ta
ne
ou
s i
nt
er
ac
tio
n.
–
14
1
–
6.
4.
1
N
ai
ve
Fe
yn
m
an
R
ul
es
W
e
wa
nt
to
de
te
rm
in
e
th
e
Fe
yn
m
an
ru
les
fo
rt
hi
s
th
eo
ry
.
Fo
rf
er
m
ion
s,
th
e
ru
les
ar
e
th
e
sa
m
e
as
th
os
e
gi
ve
n
in
Se
ct
io
n
5.
Th
e
ne
w
pi
ec
es
ar
e:
•
W
e
de
no
te
th
e
ph
ot
on
by
a
wa
vy
lin
e.
Ea
ch
en
d
of
th
e
lin
e
co
m
es
wi
th
an
i,
j=
1,
2,
3
in
de
x
te
lli
ng
us
th
ec
om
po
ne
nt
of
�A
.W
e
ca
lcu
la
te
d
th
et
ra
ns
ve
rs
e
ph
ot
on
pr
op
ag
at
or
in
(6
.3
3)
:i
ti
s
an
d
co
nt
rib
ut
es
Dtr
ij
=
i p2
+
i�
✓
�i
j�p
ip
j |�p|2◆
•
Th
e
ve
rt
ex
co
nt
rib
ut
es
�i
e�i
.
Th
e
in
de
x
on
�i
co
nt
ra
ct
s
wi
th
th
e
in
de
x
on
th
e
ph
ot
on
lin
e.
•
Th
e
no
n-
lo
ca
li
nt
er
ac
tio
n
wh
ich
,i
n
po
sit
io
n
sp
ac
e,
is
gi
ve
n
by
x
y
co
nt
rib
ut
es
a
fa
ct
or
ofi
(e
�0
)2
�(
x0
�
y0
)
4�
|�x
�
�y|
Th
es
e
Fe
yn
m
an
ru
les
ar
e
ra
th
er
m
es
sy
.
Th
is
is
th
e
pr
ice
we
’ve
pa
id
fo
r
wo
rk
in
g
in
Co
ul
om
b
ga
ug
e.
W
e’l
ln
ow
sh
ow
th
at
we
ca
n
m
as
sa
ge
th
es
ee
xp
re
ss
ion
si
nt
o
so
m
et
hi
ng
m
uc
h
m
or
e
sim
pl
e
an
d
Lo
re
nt
z
in
va
ria
nt
.L
et
’s
st
ar
tw
ith
th
e
o�
en
di
ng
in
st
an
ta
ne
ou
s
in
te
ra
ct
io
n.
Si
nc
e
it
co
m
es
fro
m
th
e
A
0
co
m
po
ne
nt
of
th
e
ga
ug
e
fie
ld
,w
e
co
ul
d
tr
y
to
re
de
fin
e
th
e
pr
op
ag
at
or
to
in
clu
de
a
D
00
pi
ec
e
wh
ich
wi
ll
ca
pt
ur
e
th
is
te
rm
.I
n
fa
ct
,i
t
fit
sq
ui
te
ni
ce
ly
in
th
is
fo
rm
:i
fw
e
lo
ok
in
m
om
en
tu
m
sp
ac
e,
we
ha
ve
�(
x0
�
y0
)
4�
|�x
�
�y|=Z
d4
p (2
�)4eip
·(x
�y
) |�p|2
(6
.8
3)
so
we
ca
n
co
m
bi
ne
th
en
on
-lo
ca
li
nt
er
ac
tio
n
wi
th
th
et
ra
ns
ve
rs
e
ph
ot
on
pr
op
ag
at
or
by
de
fin
in
g
a
ne
w
ph
ot
on
pr
op
ag
at
or
D
µ⌫
(p
)=
��
����
����
�
+i
|�p|2
µ,
�
=
0
i p2
+
i�
✓
�i
j�p
ip
j |�p|2◆
µ
=
i6=
0,
�
=
j6=
0
0
ot
he
rw
ise
(6
.8
4)
W
ith
th
is
pr
op
ag
at
or
,t
he
wa
vy
ph
ot
on
lin
e
no
w
ca
rr
ies
a
µ,
�
=
0,
1,
2,
3
in
de
x,
wi
th
th
ee
xt
ra
µ
=
0
co
m
po
ne
nt
ta
ki
ng
ca
re
of
th
ei
ns
ta
nt
an
eo
us
in
te
ra
ct
io
n.
W
en
ow
ne
ed
to
ch
an
ge
ou
rv
er
te
x
sli
gh
tly
:t
he
�i
e�i
ab
ov
e
ge
ts
re
pl
ac
ed
by
�i
e�µ
wh
ich
co
rr
ec
tly
ac
co
un
ts
fo
rt
he
(e
�0
)2
pi
ec
e
in
th
e
in
st
an
ta
ne
ou
si
nt
er
ac
tio
n.
–
14
1
–
6.4.1
N
aive
Feynm
an
R
ules
W
e
want
to
determ
ine
the
Feynm
an
rules for this
theory.
For ferm
ions, the
rules
are
the
sam
e
as those
given
in
Section
5. The
new
pieces are:
•
W
e
denote
the
photon
by
a
wavy
line. Each
end
of the
line
com
es with
an
i, j =
1, 2, 3
index
telling
us the com
ponent of
�A. W
e
calculated
the transverse
photon
propagator in
(6.33): it is
and
contributes D
tr
ij
=
i
p
2 +
i�
✓ � ij�
p ip j
|�p|
2
◆
•
The
vertex
contributes �
ie�
i .
The
index
on
�
i contracts
with
the
index
on
the
photon
line.
•
The
non-local interaction
which, in
position
space, is
given
by
x
y
contributes a
factor of
i(e�
0 )
2 �(x
0 �
y
0 )
4�|�x�
�y|
These
Feynm
an
rules
are
rather
m
essy.
This
is
the
price
we’ve
paid
for
working
in
Coulom
b
gauge. W
e’ll now
show
that we can
m
assage these expressions into
som
ething
m
uch
m
ore
sim
ple
and
Lorentz
invariant. Let’s start with
the
o�ending
instantaneous
interaction. Since
it com
es from
the
A
0
com
ponent of the
gauge
field, we
could
try
to
redefine
the
propagator to
include
a
D
00
piece
which
will capture
this term
. In
fact, it
fits quite
nicely
in
this form
: if we
look
in
m
om
entum
space, we
have
�(x
0 �
y
0 )
4�|�x�
�y|
=
Z
d
4 p
(2�)
4
e
ip·(x�
y)
|�p|
2
(6.83)
so
we can
com
bine the non-local interaction
with
the transverse
photon
propagator by
defining
a
new
photon
propagator
D
µ⌫(p) =
��
����
����
�
+
i
|�p|
2
µ, �
=
0
i
p
2 +
i�
✓ � ij�
p ip j
|�p|
2
◆
µ
=
i 6=
0, �
=
j 6=
0
0
otherwise
(6.84)
W
ith
this propagator, the
wavy
photon
line
now
carries a
µ, �
=
0, 1, 2, 3
index, with
the extra
µ
=
0
com
ponent taking
care of the instantaneous interaction. W
e now
need
to
change
our vertex
slightly: the �
ie�
i above
gets replaced
by �
ie�
µ which
correctly
accounts for the
(e�
0 )
2 piece
in
the
instantaneous interaction.
–
141
–
in which the electron runs around a loop, radiating external photons as it goes. We
recognise the 1-loop vacuum polarization graph as a member of this infinite series, with
two external photons.
56Here, we’ve rescaled the photon to have canonically normalized kinetic terms, so /r = �µ(@µ� ieAµ) 
and e is dimensionless as we’re in exactly four Euclidean dimensions.
57In the problem sheets, you’ll show that the terms involving an odd number of external photons neces-
sarily vanish, as a consequence of Furry’s theorem.
– 109 –
Quite remarkably, in the case of a constant electromagnetic field, Euler and Heisenberg
were able to find a closed form expression for ln det( /r + m) that sums up the e↵ects of all
of these diagrams. They did this in 1936, well before QFT was on a firm footing and over a
decade before Feynman first presented his diagrams! To obtain their result, we first simplify
the e↵ective action slightly using the following observation: ln det( /r + m) = tr ln( /r + m)
includes both a trace over the Dirac spinor indices as well as a functional trace. Since the
trace of any odd number of � matrices vanishes, we have
tr ln( /r + m) = tr ln(� /r + m) (5.94)
and therefore
tr ln( /r + m) = 1
2
⇥
ln det( /r + m) + ln det(� /r + m)
⇤
=
1
2
ln det(� /r2 + m2) .
(5.95)
We also have
/r2 = �µ�⌫rµr⌫ =
✓
1
2
{�µ, �⌫} + 1
2
[�µ, �⌫ ]
◆
rµr⌫
= rµrµ � eSµ⌫Fµ⌫ ,
(5.96)
where Sµ⌫ = i4 [�
µ, �⌫ ] are the (Hermitian) generators of SO(4) ⇠= SU(2) ⇥ SU(2)/Z2 in
the (12 , 0) � (0,
1
2) representation appropriate for a Dirac spinor. This final term involving
Sµ⌫ is the origin of the electron’s magnetic moment in the Dirac equation.
Our task is to understand the functional58
1
2
tr ln(�r2 + eSµ⌫Fµ⌫ + m2)
=
1
2
tr ln
"
(�(@ � ieA)2 + m2)
 
1 0
0 1
!
+ e
 
(B + iE) · � 0
0 (B � iE) · �
!#
, (5.97)
where � are the usual Pauli matrices and the trace is a sum over the eigenvalues of the
operator, acting on spinor–valued functions on R4, as well as a Dirac trace. Since the trace
is basis independent, we’re free to evaluate this in any basis, and we’ll choose to work
with the position basis. We’ll do this by making use of the connection between QFT and
worldline quantum mechanics that we saw in section 3.4, so
�e↵ [A] =
Z 
1
4
Fµ⌫Fµ⌫ +
1
2
⌦
x
�� ln(�r2 + eSµ⌫Fµ⌫ + m2)
��x
↵ �
d4x (5.98)
where {|xi} is a basis of position eigenstates (we temporarily suppress the spinor indices).
Next, we use the asymptotic relation
Z 1
s0
e�sX
ds
s
⇠ � ln X � ln s0 + finite (5.99)
58Our treatment here follows closely chapter 33 of M. Schwartz, Quantum Field Theory and the Standard
Model.
– 110 –
as s0 ! 0+ to rewrite the logarithm of the operator as
1
2
⌦
x
�� ln(�r2 + eSµ⌫Fµ⌫ + m2)
��x
↵
= lim
s0!0+
1
2
Z 1
s0
e�sm
2/2
D
x
���e�s(�r
2+eSµ⌫Fµ⌫)
���x
E ds
s
,
(5.100)
up to a divergent term that is independent of Aµ. Noting that, in the position space
representation of Euclidean quantum mechanics, r2 = (@ � ieA(x))2 = (p̂ � ieA(x̂))2 in
terms of the momentum operator p̂ = �@/@x, our problem has reducedto evaluating the
quantum mechanical expectation value hx|e�sĤ |xi with Hamiltonian Ĥ = (p̂ + eA(x̂))2 +
e
2S
µ⌫Fµ⌫(x̂). Indeed, if {| ni} form a basis of eigenstates of Ĥ, then the new term in the
e↵ective action involves
Z
d4xhx|e�sĤ |xi =
Z
d4x
X
n
hx| ni h n|e�sĤ |xi
=
Z
d4x
X
n
| n(x)|2e�sEn .
(5.101)
We’ll be able to do this in the case of constant electromagnetic fields, so that @µF↵� = 0.
Let’s begin with just a constant magnetic field B, and choose to align the z-axis with
the direction of B. Then we can pick the gauge Ay = Bx with Aµ = 0 for µ 6= y. In this
gauge the Hamiltonian becomes
Ĥ = p̂20 + p̂
2
x + p̂
2
z + (p̂y � eBx)2 � eB�z (5.102)
where eB�z arises from the magnetic moment Sµ⌫Fµ⌫ and �z is the usual Pauli matrix. You
should have met this system in undergraduate quantum mechanics: it’s just a harmonic
oscillator in the disguise of Landau levels for an electron in a constant magnetic field.
This should come as no surprise: the Feynman diagrams we’ve drawn show an electron
moving around a circle in the presence of a background magnetic field. The energy of the
nth level is
En(pt, py, pz, ±) = p2t + p2z + 2eB
✓
n +
1
2
◆
⌥ eB (5.103)
where pt, pz are the eigenvalues of the momenta and the choice of sign ± corresponds to
the electron having its spin aligned or anti-aligned with the magnetic field. (Note that the
energy eigenvalues are independent of py, so there is a large degeneracy in this system.)
The corresponding energy eigenstate is
hx| ni = �±n
⇣
x � py
eB
⌘
e�i(ptt+pyy+pzz) (5.104)
where �±n is the n
th excited state of a harmonic oscillator, tensored with a spin-up or
spin-down state.
FILL IN DETAILS OF DENSITY OF STATES CALCULATION
Thus we obtain
Z
d4xhx|e�Ĥs|xi = eBL
4
(2⇡)3
X
±
e±eB
Z
R2
ei(p
2
t
+p2z)sdpt dpz
1X
n=0
e�se(2n+1)B
=
2L4eB
8⇡2
1
s
cosh(esB)
sinh(esB)
.
(5.105)
– 111 –
Had we started with a constant electric field, from (5.97) we would have the same result,
but with B ! iE. The resulting e↵ective Lagrangian for general constant Fµ⌫ can be
written as
1
4
Fµ⌫Fµ⌫ � lim
s0!0+
e2
32⇡2
Z 1
s0
ds
s
e�sm
2 Re cosh(esX)
Im cosh(esX)
F̃µ⌫F⇢� , (5.106)
where
X2 =
1
2
Fµ⌫Fµ⌫ +
1
2
F̃µ⌫Fµ⌫ = (B + iE)
2 (5.107)
and F̃µ⌫ = i2✏
µ⌫⇢�F⇢�.
This expression is singular as s0 ! 0, as we anticipated in (5.99). Physically, s repre-
sents worldline proper time – a Schwinger parameter – along the electron’s S1 worldline,
so the divergence at the lower limit corresponds to the UV region where the electron EX-
PLAIN.
We can regularize the answer by simply imposing a cut–o↵ at s0, but then we need to
renormalize (tune our initial coupling) so as to obtain a finite e↵ective action as s0 ! 0.
Expanding (5.106) as a series in the coupling e we have
Re cosh(esX)
Im cosh(esX)
✏µ⌫⇢�Fµ⌫F⇢� =
4
e2s2
+
2
3
Fµ⌫Fµ⌫ �
e2s2
45

(Fµ⌫Fµ⌫)
2 +
7
4
(F̃µ⌫Fµ⌫)
2
�
+ O(e4)
(5.108)
and the first two terms cause divergences in the s ! 0 region of the integral ds/s. The
leading divergence is independent of the fields and can be removed by tuning the initial
vacuum energy density. In fact, this term comes from the field independent term ln det(/@+
m) corresponding to the 1-loop vacuum Feynman graph. It can also be removed by dividing
by the partition function of a free (uncharged) electron. The subleading term leads to a
logarithmic divergence in the coe�cient of the usual photon kinetic term Fµ⌫Fµ⌫ . It can
thus be removed by including a wavefunction renormalization – a counterterm for this
kinetic term. The simplest choice of renormalization scheme is just minimal subtraction,
where we just remove these divergent pieces by hand. Altogether, we’re left with the
e↵ective action
�[A] =
Z ⇢
1
4
Fµ⌫Fµ⌫ �
e2
32⇡2
Z 1
0
e�sm
2

Re cosh(esX)
Im cosh(esX)
F̃µ⌫Fµ⌫ �
4
e2s2
� 2
3
Fµ⌫Fµ⌫
�
ds
s
�
d4x
(5.109)
as found by Euler & Heisenberg.
This e↵ective action for the photon takes into account the quantum e↵ects of the
electron. In our derivation, we needed to assume the electromagnetic field was constant.
We thus expect that the true e↵ective action contains further terms such as (@F )2 that
involve derivatives of the Maxwell fieldstrength. On dimensional grounds, every derivative
must be suppressed by some mass scale, and the only scale in the problem is the mass
of the electron. Thus we expect the Euler-Heisenberg action (5.109) to be an accurate
description of very high intensity light (where the non-linear F 2k terms are significant) but
of frequency much lower than the electron mass. For example, expanding (5.109) we find
a term
↵2
90
1
m4

(Fµ⌫Fµ⌫)
2 +
7
4
(F̃µ⌫Fµ⌫)
2
�
– 112 –
which describes a four-photon interaction. This is the leading term in light-by-light scat-
tering, leading to an amplitude for this process / ↵2(!/m)4, valid in the regime where the
photon energy transfer ! ⌧ m.
5.3.1 Physical interpretation of vacuum polarization
When light propagates through a region containing an insulating medium with no relevant
degrees of freedom, on general grounds we expect the low–energy e↵ective field theory
to be governed by an action that modifies the coe�cients of the electric and magnetic
fields in the usual Maxwell action by terms that respect the microscopic symmetries of the
medium. In the present case, the medium is simply the vacuum itself ! Since the vacuum
is Lorentz invariant, these modifications must be proportional to the Lorentz invariant
combination Fµ⌫Fµ⌫ = 2(E2 �B2). In (5.83) we see explicitly that this is true. If we place
a medium such as water in the presence of an electric field, it will become polarized due to
the large dipole moment of the H2O molecules. Likewise, at distances & 1/m the vacuum
itself becomes a dielectric medium in which virtual electron–positron pairs form dipoles,
polarizing the vacuum.
The first e↵ect is that vacuum polarization leads to a measureable change in the
Coulomb potential. Recall that in the non-relativistic limit (in Lorentzian signature),
the Fourier transform of the (Feynman gauge) photon propagator �⇢�/k2 is the Coulomb
potential V (r) = e2/4⇡r, as I hope is familiar from Rutherford scattering. Let’s compute
the 1-loop quantum corrections to this result. We consider a scattering process in which two
spin-12 charged particles interact electromagnetically. Using the exact photon propagator,
we have
S(1, 2 ! 10, 20) = �e1e2
4⇡2
�4(k1 + k2 � k10 � k20) ū10�µu1 �µ⌫(p) ū20�⌫u2
= �e1e2
4⇡2
�4(k1 + k2 � k10 � k20) ū10�µu1
�0µ⌫(p)
1 � ⇡(p2) ū2
0�⌫u2
(5.110)
where p = k1 + k2 = k10 + k20 is the total incoming (or outgoing) momenta, u1,2 and ū10,20
are the momentum space on–shell Dirac wavefunctions for the incoming and outgoing
particles, respectively. Using the fact that these initial and final Dirac wavefunctions obey
(i/k1,2 + m)u1,2 = 0 and (�i/k10,20 + m)ū10,20 = 0 we have
ū10�
µu1
✓
�µ⌫ �
pµp⌫
p2
◆
ū20�µu2 = ū10�
µu1
1
p2
ū20�µu2 . (5.111)
The photon self–energy term ⇡(p2) comes from quantum corrections and so is O(~). Thus,
to leading order in ~, the scattering amplitude is given by
S(1, 2 ! 10, 20) = � e1e2
4⇡2p2
�4(k1 + k2 � k10 � k20) ū10�µu1 [1 + ⇡(p2)] ū20�µu2 . (5.112)
Of course, we can think of these terms as coming from the Feynman diagrams
– 113 –
= �0µ� �
0
µ�⇧
�
��
0�
� + �
0
µ�⇧
�
��
0�
�⇧
�
��
0�
� · · ·� �
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesDtrij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�i.Theindexon�icontractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenbyxy
contributesafactorof
i(e�0)2�(x0�y0)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomethingmuchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x0�y0)
4�|�x��y|=
Z
d4p
(2�)4
eip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
�����
�����
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�iabovegetsreplacedby�ie�µwhichcorrectly
accountsforthe(e�0)2pieceintheinstantaneousinteraction.
–141–
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesDtrij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�i.Theindexon�icontractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenbyxy
contributesafactorof
i(e�0)2�(x0�y0)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x0�y0)
4�|�x��y|=
Z
d4p
(2�)4
eip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
�����
�����
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�iabovegetsreplacedby�ie�µwhichcorrectly
accountsforthe(e�0)2pieceintheinstantaneousinteraction.
–141–
1PI
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesDtrij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�i.Theindexon�icontractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenbyxy
contributesafactorof
i(e�0)2�(x0�y0)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x0�y0)
4�|�x��y|=
Z
d4p
(2�)4
eip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
�����
�����
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�iabovegetsreplacedby�ie�µwhichcorrectly
accountsforthe(e�0)2pieceintheinstantaneousinteraction.
–141–
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesD
tr
ij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�
i
.Theindexon�
i
contractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenby
xy
contributesafactorof
i(e�
0
)
2
�(x
0
�y
0
)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x
0
�y
0
)
4�|�x��y|
=
Zd4p
(2�)4
e
ip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
���
��
���
��
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�
i
abovegetsreplacedby�ie�
µ
whichcorrectly
accountsforthe(e�
0
)
2
pieceintheinstantaneousinteraction.
–141–
1PI1PI+ · · ·=�µ� � �
· · ·S(k) ̄ 1PI1PI ̄ 1PI ̄ 
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesDtrij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�i.Theindexon�icontractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenbyxy
contributesafactorof
i(e�0)2�(x0�y0)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x0�y0)
4�|�x��y|=
Z
d4p
(2�)4
eip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
�����
�����
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�iabovegetsreplacedby�ie�µwhichcorrectly
accountsforthe(e�0)2pieceintheinstantaneousinteraction.
–141–
��Z3
6.4.1 Naive Feynman Rules
We want to determine the Feynman rules for this theory. For fermions, the rules are
the same as those given in Section 5. The new pieces are:
• We denote the photon by a wavy line. Each end of the line comes with an i, j =
1, 2, 3 index telling us the component of �A. We calculated the transverse photon
propagator in (6.33): it is and contributes D
tr
ij =
i
p2 + i�
✓
�ij �
pipj
|�p|2
◆
• The vertex contributes �ie�
i
. The index on �
i
contracts with the
index on the photon line.
• The non-local interaction which, in position space, is given by
x y
contributes a factor of
i(e�
0
)
2
�(x
0
� y
0
)
4�|�x � �y|
These Feynman rules are rather messy. This is the price we’ve paid for working in
Coulomb gauge. We’ll now show that we can massage these expressions into something
much more simple and Lorentz invariant. Let’s start with the o�ending instantaneous
interaction. Since it comes from the A0 component of the gauge field, we could try to
redefine the propagator to include a D00 piece which will capture this term. In fact, it
fits quite nicely in this form:if we look in momentum space, we have
�(x
0
� y
0
)
4�|�x � �y|
=
Z d4p
(2�)4
e
ip·(x�y)
|�p|2 (6.83)
so we can combine the non-local interaction with the transverse photon propagator by
defining a new photon propagator
Dµ⌫(p) =
�
���
��
���
��
+
i
|�p|2 µ, � = 0
i
p2 + i�
✓
�ij �
pipj
|�p|2
◆
µ = i 6= 0, � = j 6= 0
0 otherwise
(6.84)
With this propagator, the wavy photon line now carries a µ, � = 0, 1, 2, 3 index, with
the extra µ = 0 component taking care of the instantaneous interaction. We now need
to change our vertex slightly: the �ie�
i
above gets replaced by �ie�
µ
which correctly
accounts for the (e�
0
)
2
piece in the instantaneous interaction.
– 141 –
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesD
tr
ij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�
i
.Theindexon�
i
contractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenby
xy
contributesafactorof
i(e�
0
)
2
�(x
0
�y
0
)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x
0
�y
0
)
4�|�x��y|
=
Zd4p
(2�)4
e
ip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
���
��
���
��
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�
i
abovegetsreplacedby�ie�
µ
whichcorrectly
accountsforthe(e�
0
)
2
pieceintheinstantaneousinteraction.
–141–
A� A�
k
= + + +
6.4.1 Naive Feynman Rules
We want to determine the Feynman rules for this theory. For fermions, the rules are
the same as those given in Section 5. The new pieces are:
• We denote the photon by a wavy line. Each end of the line comes with an i, j =
1, 2, 3 index telling us the component of �A. We calculated the transverse photon
propagator in (6.33): it is and contributes D
tr
ij =
i
p2 + i�
✓
�ij �
pipj
|�p|2
◆
• The vertex contributes �ie�
i
. The index on �
i
contracts with the
index on the photon line.
• The non-local interaction which, in position space, is given by
x y
contributes a factor of
i(e�
0
)
2
�(x
0
� y
0
)
4�|�x � �y|
These Feynman rules are rather messy. This is the price we’ve paid for working in
Coulomb gauge. We’ll now show that we can massage these expressions into something
much more simple and Lorentz invariant. Let’s start with the o�ending instantaneous
interaction. Since it comes from the A0 component of the gauge field, we could try to
redefine the propagator to include a D00 piece which will capture this term. In fact, it
fits quite nicely in this form: if we look in momentum space, we have
�(x
0
� y
0
)
4�|�x � �y|
=
Z d4p
(2�)4
e
ip·(x�y)
|�p|2 (6.83)
so we can combine the non-local interaction with the transverse photon propagator by
defining a new photon propagator
Dµ⌫(p) =
�
���
��
���
��
+
i
|�p|2 µ, � = 0
i
p2 + i�
✓
�ij �
pipj
|�p|2
◆
µ = i 6= 0, � = j 6= 0
0 otherwise
(6.84)
With this propagator, the wavy photon line now carries a µ, � = 0, 1, 2, 3 index, with
the extra µ = 0 component taking care of the instantaneous interaction. We now need
to change our vertex slightly: the �ie�
i
above gets replaced by �ie�
µ
which correctly
accounts for the (e�
0
)
2
piece in the instantaneous interaction.
– 141 –
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesD
tr
ij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�
i
.Theindexon�
i
contractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenby
xy
contributesafactorof
i(e�
0
)
2
�(x
0
�y
0
)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x
0
�y
0
)
4�|�x��y|
=
Zd4p
(2�)4
e
ip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
���
��
���
��
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�
i
abovegetsreplacedby�ie�
µ
whichcorrectly
accountsforthe(e�
0
)
2
pieceintheinstantaneousinteraction.
–141–
6.4.1NaiveFeynmanRules
WewanttodeterminetheFeynmanrulesforthistheory.Forfermions,therulesare
thesameasthosegiveninSection5.Thenewpiecesare:
•Wedenotethephotonbyawavyline.Eachendofthelinecomeswithani,j=
1,2,3indextellingusthecomponentof�A.Wecalculatedthetransversephoton
propagatorin(6.33):itisandcontributesDtrij=
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
•Thevertexcontributes�ie�i.Theindexon�icontractswiththe
indexonthephotonline.
•Thenon-localinteractionwhich,inpositionspace,isgivenbyxy
contributesafactorof
i(e�0)2�(x0�y0)
4�|�x��y|
TheseFeynmanrulesarerathermessy.Thisisthepricewe’vepaidforworkingin
Coulombgauge.We’llnowshowthatwecanmassagetheseexpressionsintosomething
muchmoresimpleandLorentzinvariant.Let’sstartwiththeo�endinginstantaneous
interaction.SinceitcomesfromtheA0componentofthegaugefield,wecouldtryto
redefinethepropagatortoincludeaD00piecewhichwillcapturethisterm.Infact,it
fitsquitenicelyinthisform:ifwelookinmomentumspace,wehave
�(x0�y0)
4�|�x��y|=
Z
d4p
(2�)4
eip·(x�y)
|�p|2(6.83)
sowecancombinethenon-localinteractionwiththetransversephotonpropagatorby
defininganewphotonpropagator
Dµ⌫(p)=
�
�����
�����
+
i
|�p|2µ,�=0
i
p2+i�
✓
�ij�
pipj
|�p|2
◆
µ=i6=0,�=j6=0
0otherwise
(6.84)
Withthispropagator,thewavyphotonlinenowcarriesaµ,�=0,1,2,3index,with
theextraµ=0componenttakingcareoftheinstantaneousinteraction.Wenowneed
tochangeourvertexslightly:the�ie�iabovegetsreplacedby�ie�µwhichcorrectly
accountsforthe(e�0)2pieceintheinstantaneousinteraction.
–141–
+
so the 1–loop diagram modifies the classical answer by the factor [1 + ⇡(p2)].
Let’s now take the non–relativistic limit, as appropriate for Rutherford scattering. In
this limit, the energy transfer p0 ⌧ |p| and
ū10�
µu1 ⇡
 
�i��1,�10
0
!
,
where �i, is the spin angular momentum quantum number of the ith external particle. The
factor of ���0 enforces that the spins of the two particles should be aligned. Thus, in the
non–relativistic limit we have
S(1, 2 ! 10, 20) ⇡ e1e2
4⇡2p2
�4(k1 + k2 � k10 � k20)
⇥
1 + ⇡(p2)
⇤
��1�10 ��2,�20 . (5.113)
By comparison, in non–relativistic quantum mechanics, the Born approximation for scat-
tering two charged particles o↵ a scalar potential V (r) is given by
SBorn(1, 2 ! 10, 20) =
e1e2
4⇡2
�4(k1 + k2 � k10 � k20) ��1,�10 ��2,�20
Z
d3rV (r) e�ik·r . (5.114)
This shows that the 1-loop corrected amplitude (5.113) looks just like the amplitude we
would find from