Contents -- Preface -- 1 Relativistic equations -- 1.1 Introduction -- 1.2 Notations. -- 1.3 Klein-Gordon equation -- 1.3.1 Klein paradox -- 1.4 Dirac equation. -- 1.5 References -- 2 Solutions of the Dirac equation -- 2.1 Plane wave solutions -- 2.2 Normalization of the wave function -- 2.3 Spin of the Dirac particle. -- 2.4 Continuity equation. -- 2.5 Dirac's hole theory -- 2.6 Properties of the Dirac matrices -- 2.6.1 Fierz rearrangement -- 2.7 References -- 3 Properties of the Dirac equation -- 3.1 Lorentz transformations -- 3.2 Covariance of the Dirac equation -- 3.3 Transformation of bilinears. -- 3.4 Projection operators, completeness relation -- 3.5 Helicity -- 3.6 Massless Dirac particle -- 3.7 Chirality -- 3.8 Non-relativistic limit of the Dirac equation. -- 3.9 Electron in an external magnetic field -- 3.10 Foldy-Wouthuysen transformation. -- 3.11 Zitterbewegung -- 3.12 References -- 4 Representations of Lorentz and Poincaré groups -- 4.1 Symmetry algebras -- 4.1.1 Rotation -- 4.1.2 Translation -- 4.1.3 Lorentz transformation -- 4.1.4 Poincaré transformation -- 4.2 Representations of the Lorentz group -- 4.2.1 Similarity transformations and representations -- 4.3 Unitary representations of the Poincaré group -- 4.3.1 Massive representation -- 4.3.2 Massless representation -- 4.4 References -- 5 Free Klein-Gordon field theory -- 5.1 Introduction -- 5.2 Lagrangian density -- 5.3 Quantization. -- 5.4 Field decomposition. -- 5.5 Creation and annihilation operators. -- 5.6 Energy eigenstates -- 5.7 Physical meaning of energy eigenstates -- 5.8 Green's functions -- 5.9 Covariant commutation relations -- 5.10 References -- 6 Self-interacting scalar field theory -- 6.1 Nöther's theorem -- 6.1.1 Space-time translation -- 6.2 Self-interacting 4 theory. -- 6.3 Interaction picture and time evolution operator -- 6.4 S-matrix.

6.5 Normal ordered product and Wick's theorem -- 6.6 Time ordered products and Wick's theorem -- 6.7 Spectral representation and dispersion relation -- 6.8 References -- 7 Complex scalar field theory -- 7.1 Quantization. -- 7.2 Field decomposition. -- 7.3 Charge operator -- 7.4 Green's functions -- 7.5 Spontaneous symmetry breaking and the Goldstone theorem -- 7.6 Electromagnetic coupling. -- 7.7 References -- 8 Dirac field theory. -- 8.1 Pauli exclusion principle -- 8.2 Quantization of the Dirac field. -- 8.3 Field decomposition. -- 8.4 Charge operator -- 8.5 Green's functions -- 8.6 Covariant anti-commutation relations -- 8.7 Normal ordered and time ordered products -- 8.8 Massless Dirac fields -- 8.9 Yukawa interaction -- 8.10 Feynman diagrams -- 8.11 References -- 9 Maxwell field theory -- 9.1 Maxwell's equations. -- 9.2 Canonical quantization -- 9.3 Field decomposition. -- 9.4 Photon propagator -- 9.5 Quantum electrodynamics -- 9.6 Physical processes -- 9.7 Ward-Takahashi identity in QED -- 9.8 Covariant quantization of the Maxwell theory -- 9.9 References -- 10 Dirac method for constrained systems -- 10.1 Constrained systems -- 10.2 Dirac method and Dirac bracket. -- 10.3 Particle moving on a sphere -- 10.4 Relativistic particle -- 10.5 Dirac field theory -- 10.6 Maxwell field theory -- 10.7 References -- 11 Discrete symmetries -- 11.1 Parity. -- 11.1.1 Parity in quantum mechanics -- 11.1.2 Spin zero field -- 11.1.3 Photon field -- 11.1.4 Dirac field -- 11.2 Charge conjugation -- 11.2.1 Spin zero field -- 11.2.2 Dirac field -- 11.2.3 Majorana fermions -- 11.2.4 Eigenstates of charge conjugation -- 11.3 Time reversal -- 11.3.1 Spin zero field and Maxwell's theory -- 11.3.2 Dirac fields -- 11.3.3 Consequences of T invariance -- 11.3.4 Electric dipole moment of neutron -- 11.4 CPT theorem -- 11.4.1 Equality of mass for particles and antiparticles.

11.4.2 Electric charge for particles and antiparticles -- 11.4.3 Equality of lifetimes for particles and antiparticles -- 11.5 References -- 12 Yang-Mills theory -- 12.1 Non-Abelian gauge theories -- 12.2 Canonical quantization of Yang-Mills theory -- 12.3 Path integral quantization of gauge theories -- 12.4 Path integral quantization of tensor fields -- 12.5 References -- 13 BRST invariance and its consequences -- 13.1 BRST symmetry -- 13.2 Covariant quantization of Yang-Mills theory -- 13.3 Unitarity -- 13.4 Slavnov-Taylor identity -- 13.5 Feynman rules -- 13.6 Ghost free gauges -- 13.7 References -- 14 Higgs phenomenon and the standard model -- 14.1 Stückelberg formalism -- 14.2 Higgs phenomenon -- 14.3 The standard model. -- 14.3.1 Field content -- 14.3.2 Lagrangian density -- 14.3.3 Spontaneous symmetry breaking -- 14.4 References -- 15 Regularization of Feynman diagrams -- 15.1 Introduction -- 15.2 Loop expansion -- 15.3 Cut-off regularization -- 15.3.1 Calculation in the Yukawa theory -- 15.4 Pauli-Villars regularization -- 15.5 Dimensional regularization -- 15.5.1 Calculations in QED -- 15.6 References -- 16 Renormalization theory -- 16.1 Superficial degree of divergence -- 16.2 A brief history of renormalization -- 16.3 Schwinger-Dyson equation -- 16.4 BPHZ renormalization -- 16.5 Renormalization of gauge theories -- 16.6 Anomalous Ward identity -- 16.7 References -- 17 Renormalization group and equation -- 17.1 Gell-Mann-Low equation -- 17.2 Renormalization group -- 17.3 Renormalization group equation -- 17.4 Solving the renormalization group equation -- 17.5 Callan-Symanzik equation -- 17.6 References -- Index.