This book introduces the lattice approach to quantum field theory. The spectacular successes of this technique include compelling evidence that exchange of gauge gluons can confine the quarks within subnuclear matter. The lattice framework enables novel schemes for quantitative calculation and has caused considerable cross-disciplinary activity between elementary particle and solid state physicists. The treatment begins with the lattice definition of a path integral and ends on Monte Carlo simulation methods. Other topics include invariant group integration, duality, mean field theory and renormalization group techniques. The reader is assumed to have a basic background in relativistic quantum mechanics and some exposure to gauge theories.

Preface page vi

Quarks and gluons 1

2 Lattices 5

3 Path integrals and statistical mechanics 8

4 Scalar fields 14

5 Fermions 20

6 Gauge fields 29

7 Lattice gauge theory 34

8 Group integration 39

9 Gauge-invariance and order parameters 51

10 Strong coupling 60

11 Weak coupling 77

12 Renormalization and the continuum limit 81

13 Asymptotic freedom and dimensional transmutation 88

14 Mean field theory 94

15 The Hamiltonian approach 101

16 Discrete groups and duality 108

17 Migdal-Kadanoff recursion relations 117

18 Monte Carlo simulation I: the method 127

19 Monte Carlo simulation II: measuring observables 140

20 Beyond the Wilson action 151

References 162

Index 167

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