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IFT Instituto de F´ısica Torica
Notes for a Course on
CLASSICAL FIELDS
R. Aldrovandi and J. G. Pereira
March - June / 2008

Contents
1 Sp ecial Relativity: A Recall 1
1.1 Intro duction ................................ 1
1.2 Classical Mechanics ............................ 2
1.3 Hints Toward Relativity ......................... 9
1.4 Relativistic Spacetime .......................... 11
1.5 Lorentz Vectors and Tensors ....................... 23
1.6 Particle Dynamics ............................. 26
2 Transformations 33
2.1 Transformation Groups .......................... 33
2.2 Orthogonal Transformations ....................... 37
2.3 The Group of Rotations ......................... 42
2.4 The Poincar´e Group ........................... 49
2.5 The Lorentz Group ............................ 50
3 Intro ducing Fields 59
3.1 The Standard Prototyp e ......................... 60
3.2 Non-Material Fields ............................ 66
3.2.1 Optional reading: the Quantum Line .............. 67
3.3 Waveﬁelds ................................. 69
3.4 Internal Transformations ......................... 70
4 General Formalism 74
4.1 Lagrangian Approach ........................... 77
4.1.1 Relativistic Lagrangians ..................... 77
4.1.2 Simpliﬁed Treatment ....................... 79
4.1.3 Rules of Functional Calculus ................... 81
4.1.4 Variations ............................. 83
4.2 The First No ether Theorem ....................... 86
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4.2.1 Symmetries and Conserved Charges ............... 88
4.2.2 The Basic Spacetime Symmetries ................ 90
4.2.3 Internal Symmetries ....................... 94
4.3 The Second No ether Theorem ...................... 96
4.4 Top ological Conservation Laws ..................... 98
5 Bosonic Relativistic Fields 102
5.1 Scalar Fields ................................102
5.1.1 Real Scalar Fields .........................102
5.1.2 Complex Scalar Fields ......................104
5.2 Vector Fields ...............................109
5.2.1 Real Vector Fields ........................110
5.2.2 Complex Vector Fields ......................112
6 Electromagnetic Field 115
6.1 Maxwell’s Equations ...........................115
6.2 Transformations of ~
Eand ~
H.......................117
6.3 Covariant Form of Maxwell’s Equations .................122
6.4 Lagrangian, Spin, Energy .........................125
6.5 Motion of a Charged Particle ......................128
6.6 Electrostatics and Magnetostatics ....................132
6.7 Electromagnetic Waves ..........................136
7 Dirac Fields 142
7.1 Dirac Equation ..............................142
7.2 Non-Relativistic Limit: Pauli Equation .................147
7.3 Covariance .................................148
7.4 Lagrangian Formalism ..........................155
7.5 Parity ...................................156
7.6 Charge Conjugation ............................158
7.7 Time Reversal and CPT.........................159
8 Gauge Fields 162
8.1 Intro duction ................................162
8.2 The Notion of Gauge Symmetry .....................164
8.3 Global Transformations ..........................166
8.4 Lo cal Transformations ..........................167
8.5 Lo cal No ether Theorem .........................168
8.6 Field Strength and Bianchi Identity ...................171
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