Introduction to quantum fields on a lattice 1st Edition by Jan Smit ISBN 0521890519 978-0521890519
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Product details:
ISBN 10: 0521890519
ISBN 13: 978-0521890519
Author: Jan Smit
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.
Table of contents:
1 Introduction
1.1 QED, QCD, and confinement
1.2 Scalar field
2 Path-integral and lattice regularization
2.1 Path integral in quantum mechanics
2.2 Regularization by discretization
2.3 Analytic continuation to imaginary time
2.4 Spectrum of the transfer operator
2.5 Latticization of the scalar field
2.6 Transfer operator for the scalar field
2.7 Fourier transformation on the lattice
2.8 Free scalar field
2.9 Particle interpretation
2.10 Back to real time
2.11 Problems
3 O(n) models
3.1 Goldstone bosons
3.2 O(n) models as spin models
3.3 Phase diagram and critical line
3.4 Weak-coupling expansion
3.5 Renormalization
3.6 Renormalization-group beta functions
3.7 Hopping expansion
3.8 Lüscher-Weisz solution
3.9 Numerical simulation
3.10 Real-space renormalization group and universality
3.11 Universality at weak coupling
3.12 Triviality and the Standard Model
3.13 Problems
4 Gauge field on the lattice
4.1 QED action
4.2 QCD action
4.3 Lattice gauge field
4.4 Gauge-invariant lattice path integral
4.5 Compact and non-compact Abelian gauge theory
4.6 Hilbert space and transfer operator
4.7 The kinetic-energy operator
4.8 Hamiltonian for continuous time
4.9 Wilson loop and Polyakov line
4.10 Problems
5 U(1) and SU(n) gauge theory
5.1 Potential at weak coupling
5.2 Asymptotic freedom
5.3 Strong-coupling expansion
5.4 Potential at strong coupling
5.5 Confinement versus screening
5.6 Glueballs
5.7 Coulomb phase, confinement phase
5.8 Mechanisms of confinement
5.9 Scaling and asymptotic scaling, numerical results
5.10 Problems
6 Fermions on the lattice
6.1 Naive discretization of the Dirac action
6.2 Species doubling
6.3 Wilson’s fermion method
6.4 Staggered fermions
6.5 Transfer operator for Wilson fermions
6.6 Problems
7 Low-mass hadrons in QCD
7.1 Integrating over the fermion fields
7.2 Hopping expansion for the fermion propagator
7.3 Meson and baryon propagators
7.4 Hadron masses at strong coupling
7.5 Numerical results
7.6 The parameters of QCD
7.7 Computing the gauge coupling from the masses
7.8 Problems
8 Chiral symmetry
8.1 Chiral symmetry and effective action in QCD
8.2 Pseudoscalar masses and the U(1) problem
8.3 Chiral anomalies
8.4 Chiral symmetry and the lattice
8.5 Spontaneous breaking of chiral symmetry
8.6 Chiral gauge theory
8.7 Outlook
8.8 Problems
Appendix A SU(n)
A.1 Fundamental representation of SU(n)
A.2 Adjoint representation of SU(n)
A.3 Left and right translations in SU(n)
A.4 Tensor method for SU(n)
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Jan Smit,quantum fields,a lattice