Introduction to quantum fields on a lattice

Jan Smit0521890519, 9780521890519, 9780511020780

This book provides a concrete introduction to quantum fields on a lattice: a precise and non-perturbative definition of quantum field theory obtained by replacing the space-time continuum by a lattice. Topics covered include quark confinement, chiral symmetry breaking in QCD, quantized non-abelian gauge fields, scaling and universality. The author also discusses the results of simulations on computers.

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Table of contents :
Contents……Page 8
Preface……Page 12
1.1 QED, QCD, and confinement……Page 14
1.2 Scalar field……Page 18
2.1 Path integral in quantum mechanics……Page 21
2.2 Regularization by discretization……Page 23
2.3 Analytic continuation to imaginary time……Page 25
2.4 Spectrum of the transfer operator……Page 26
2.5 Latticization of the scalar field……Page 28
2.6 Transfer operator for the scalar field……Page 31
2.7 Fourier transformation on the lattice……Page 33
2.8 Free scalar field……Page 35
2.9 Particle interpretation……Page 38
2.10 Back to real time……Page 39
2.11 Problems……Page 41
3.1 Goldstone bosons……Page 45
3.2 O(n) models as spin models……Page 47
3.3 Phase diagram and critical line……Page 49
3.4 Weak-coupling expansion……Page 52
3.5 Renormalization……Page 59
3.6 Renormalization-group beta functions……Page 61
3.7 Hopping expansion……Page 64
3.8 Lüscher–Weisz solution……Page 68
3.9 Numerical simulation……Page 73
3.10 Real-space renormalization group and universality……Page 80
3.11 Universality at weak coupling……Page 84
3.12 Triviality and the Standard Model……Page 87
3.13 Problems……Page 92
4.1 QED action……Page 96
4.2 QCD action……Page 98
4.3 Lattice gauge field……Page 103
4.4 Gauge-invariant lattice path integral……Page 108
4.5 Compact and non-compact Abelian gauge theory……Page 110
4.6 Hilbert space and transfer operator……Page 112
4.7 The kinetic-energy operator……Page 115
4.8 Hamiltonian for continuous time……Page 118
4.9 Wilson loop and Polyakov line……Page 120
4.10 Problems……Page 125
5.1 Potential at weak coupling……Page 128
5.2 Asymptotic freedom……Page 134
5.3 Strong-coupling expansion……Page 138
5.4 Potential at strong coupling……Page 142
5.5 Confinement versus screening……Page 145
5.6 Glueballs……Page 148
5.7 Coulomb phase, confinement phase……Page 149
5.8 Mechanisms of confinement……Page 151
5.9 Scaling and asymptotic scaling, numerical results……Page 153
5.10 Problems……Page 157
6.1 Naive discretization of the Dirac action……Page 162
6.2 Species doubling……Page 164
6.3 Wilson’s fermion method……Page 169
6.4 Staggered fermions……Page 173
6.5 Transfer operator for Wilson fermions……Page 174
6.6 Problems……Page 178
7.1 Integrating over the fermion fields……Page 183
7.2 Hopping expansion for the fermion propagator……Page 184
7.3 Meson and baryon propagators……Page 186
7.4 Hadron masses at strong coupling……Page 190
7.5 Numerical results……Page 192
7.6 The parameters of QCD……Page 201
7.8 Problems……Page 203
8.1 Chiral symmetry and effective action in QCD……Page 206
8.2 Pseudoscalar masses and the U(1) problem……Page 212
8.3 Chiral anomalies……Page 215
8.4 Chiral symmetry and the lattice……Page 217
8.5 Spontaneous breaking of chiral symmetry……Page 225
8.6 Chiral gauge theory……Page 230
8.8 Problems……Page 236
A.1 Fundamental representation of SU(n)……Page 242
A.2 Adjoint representation of SU(n)……Page 244
A.3 Left and right translations in SU(n)……Page 247
A.4 Tensor method for SU(n)……Page 249
Appendix B Quantization in the temporal gauge……Page 252
Appendix C Fermionic coherent states……Page 255
Appendix D Spinor fields……Page 266
Notes……Page 271
References……Page 274
C……Page 280
G……Page 281
P……Page 282
S……Page 283
Z……Page 284