Foreword by Dennis Sullivan ................................. ix
Preface ..................................................... xi
1 Classical mechanics .......................................... 1
1.1 Newtonian mechanics ..................................... 1
1.2 Lagrangian mechanics .................................... 4
1.3 Hamiltonian mechanics ................................... 7
1.4 Poisson brackets and Lie algebra structure of
observables ............................................ 10
1.5 Symmetry and conservation laws: Noether's theorem ...... 11
2 Quantum mechanics ........................................... 14
2.1 The birth of quantum theory ............................ 14
2.2 The basic principles of quantum mechanics .............. 16
2.3 Canonical quantization ................................. 21
2.4 From classical to quantum mechanics: the C* algebra
approach ............................................... 24
2.5 The Weyl C* algebra .................................... 26
2.6 The quantum harmonic oscillator ........................ 29
2.7 Angular momentum quantization and spin ................. 35
2.8 Path integral quantization ............................. 40
2.9 Deformation quantization ............................... 49
3 Relativity, the Lorentz group, and Dirac's equation ......... 51
3.1 Relativity and the Lorentz group ....................... 51
3.2 Relativistic kinematics ................................ 56
3.3 Relativistic dynamics .................................. 57
3.4 The relativistic Lagrangian ............................ 58
3.5 Dirac's equation ....................................... 60
4 Fiber bundles, connections, and representations ............. 65
4.1 Fiber bundles and cocycles ............................. 65
4.2 Principal bundles ...................................... 68
4.3 Connections ............................................ 71
4.4 The gauge group ........................................ 73
4.5 The Hodge operator ..................................... 75
4.6 Clifford algebras and spinor bundles ................... 77
4.7 Representations ........................................ 82
5 Classical field theory ...................................... 93
5.1 Introduction ........................................... 93
5.2 Electromagnetic field .................................. 94
5.3 Conservation laws in field theory ...................... 99
5.4 The Dirac field ....................................... 103
5.5 Scalar fields ......................................... 108
5.6 Yang-Mills fields ..................................... 110
5.7 Gravitational fields .................................. 111
6 Quantization of classical fields ........................... 117
6.1 Quantization of free fields: general scheme ........... 117
6.2 Axiomatic field theory ................................ 118
6.3 Quantization of bosonic free fields ................... 122
6.4 Quantization of fermionic fields ...................... 128
6.5 Quantization of the free electromagnetic field ........ 140
6.6 Wick rotations and axioms for Euclidean QFT ........... 141
6.7 The CPT theorem ....................................... 142
6.8 Scattering processes and LSZ reduction ................ 144
7 Perturbative quantum field theory .......................... 153
7.1 Discretization of functional integrals ................ 153
7.2 Gaussian measures and Wick's theorem .................. 154
7.3 Discretization of Euclidean scalar fields ............. 159
7.4 Perturbative quantum field theory ..................... 164
7.5 Perturbative Yang-Mills theory ........................ 182
8 Renormalization ............................................ 192
8.1 Renormalization in perturbative QFT ................... 192
8.2 Constructive field theory ............................. 201
9 The Standard Model ......................................... 204
9.1 Particles and fields .................................. 205
9.2 Particles and their quantum numbers ................... 206
9.3 The quark model ....................................... 207
9.4 Non-abelian gauge theories ............................ 209
9.5 Lagrangian formulation of the standard model .......... 213
9.6 The intrinsic formulation of the Lagrangian ........... 225
Appendix A: Hilbert spaces and operators ...................... 232
A.1 Hilbert spaces ........................................ 232
A.2 Linear operators ...................................... 233
A.3 Spectral theorem for compact operators ................ 235
A.4 Spectral theorem for normal operators ................. 237
A.5 Spectral theorem for unbounded operators .............. 238
A.6 Functional calculus ................................... 245
A.7 Essential self-adjointness ............................ 247
A.8 A note on the spectrum ................................ 249
A.9 Stone's theorem ....................................... 250
A.10 The Kato-Rellich theorem .............................. 253
Appendix В: C* algebras and spectral theory ................... 258
B.l Banach algebras ....................................... 258
B.2 C* algebras ........................................... 262
B.3 The spectral theorem .................................. 268
B.4 States and GNS representation ......................... 271
B.5 Representations and spectral resolutions .............. 276
B.6 Algebraic quantum field theory ........................ 282
Bibliography ............................................... 289
Index ...................................................... 293
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