Statistical Approach to Quantum Field Theory: An Introduction

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Preface to the Second Edition Preface to the First Edition Acknowledgments Contents About the Author Acronyms 1 Introduction References 2 Path Integrals in Quantum and Statistical Mechanics 2.1 Summing Over All Paths 2.2 Recalling Quantum Mechanics 2.3 Feynman–Kac Formula 2.4 Euclidean Path Integral 2.4.1 Quantum Mechanics in Imaginary Time 2.4.2 Imaginary Time Path Integral 2.5 Path Integral in Quantum Statistics 2.5.1 Thermal Correlation Functions 2.6 The Harmonic Oscillator 2.7 Problems References 3 High-Dimensional Integrals 3.1 Numerical Algorithms 3.1.1 Newton–Cotes Integration Method 3.2 Monte Carlo Integration 3.2.1 Hit-or-Miss Monte Carlo Method and Binomial Distribution 3.2.2 Sum of Random Numbers and Gaussian Distribution 3.3 Importance Sampling 3.4 Some Basic Facts in Probability Theory 3.5 Programs for This Chapter 3.6 Problems References 4 Monte Carlo Simulations in Quantum Mechanics 4.1 Markov Chains 4.1.1 Fixed Points of Markov Chains 4.2 Detailed Balance 4.2.1 Acceptance Rate 4.2.2 Metropolis–Hastings Algorithm 4.2.3 Heat Bath Algorithm 4.3 The Anharmonic Oscillator 4.3.1 Simulating the Anharmonic Oscillator 4.4 Hybrid Monte Carlo Algorithm 4.4.1 Implementing the HMC Algorithm 4.4.2 HMC Algorithm for Harmonic Oscillator 4.5 Programs for Chap.4 4.6 Problems References 5 Scalar Fields at Zero and Finite Temperature 5.1 Quantization 5.2 Scalar Field Theory at Finite Temperature 5.2.1 Free Scalar Field 5.3 Schwinger Function and Effective Potential 5.3.1 The Legendre–Fenchel Transformation 5.4 Scalar Field on a Spacetime Lattice 5.5 Random Walk Representation of Green Function 5.6 There Is No Leibniz Rule on the Lattice 5.7 Problems References 6 Classical Spin Models: An Introduction 6.1 Simple Spin Models for (Anti)Ferromagnets 6.1.1 Ising Model 6.2 Ising-Type Spin Systems 6.2.1 Standard Potts Models 6.2.2 The Zq Model (Planar Potts Model and Clock Model) 6.2.3 The U(1) Model 6.2.4 Non-linear O(N) Models 6.2.5 Interacting Continuous Spins 6.3 Spin Systems in Thermal Equilibrium 6.4 Variational Principles 6.4.1 Gibbs State and Free Energy 6.4.2 Fixed Average Field 6.5 Programs for Chap.6 6.6 Problems References 7 Mean Field Approximation 7.1 Approximation for General Lattice Models 7.2 The Ising Model 7.2.1 An Alternative Derivation 7.3 Critical Exponents α,β,γ,δ 7.3.1 Susceptibility 7.3.2 Magnetization as a Function of Temperature 7.3.3 Specific Heat 7.3.4 Magnetization as a Function of the Magnetic Field 7.3.5 Comparison with Exact and Numerical Results 7.4 Mean Field Approximation for Standard Potts Models 7.5 Mean Field Approximation for Zq Models 7.6 Landau Theory and Ornstein–Zernike Extension 7.6.1 Critical Exponents in Landau Theory 7.6.2 Two-Point Correlation Function 7.7 Anti-ferromagnetic Systems 7.8 Mean Field Approximation for Lattice Field Theories 7.8.1 ϕ4 and ϕ6 Scalar Theories 7.8.2 Non-linear O(N) Models 7.9 Program for Chap.7 7.10 Problems References 8 Transfer Matrices, Correlation Inequalities, and Roots of Partition Functions 8.1 Transfer-Matrix Method for the Ising Chain 8.1.1 Transfer Matrix 8.1.2 The ``Hamiltonian'' 8.1.3 The Anti-Ferromagnetic Chain 8.2 Potts Chain 8.3 Perron–Frobenius Theorem 8.4 The General Transfer-Matrix Method 8.5 Continuous Target Spaces 8.5.1 Euclidean Quantum Mechanics 8.5.2 Real Scalar Field 8.6 Correlation Inequalities 8.7 Roots of the Partition Function 8.7.1 Lee–Yang Zeroes of Ising Chain 8.7.2 General Ferromagnetic Systems 8.8 Problems References 9 High-Temperature and Low-Temperature Expansions 9.1 Ising Chain 9.1.1 Low Temperature 9.1.2 High Temperature 9.2 High-Temperature Expansions for Ising Models 9.2.1 General Results and Two-Dimensional Model Correlation Functions Susceptibility Extrapolation to the Critical Point 9.2.2 Three-Dimensional Model Free Energy Density and Specific Heat Susceptibility 9.3 Low-Temperature Expansion of Ising Models 9.3.1 Free Energy and Magnetization of Two-Dimensional Model Extrapolation to the Critical Point 9.3.2 Three-Dimensional Model 9.3.3 Improved Series Studies for Ising-Type Models 9.4 High-Temperature Expansions of Nonlinear O(N) Models 9.4.1 Expansions of Partition Function and Free Energy 9.5 Polymers and Self-Avoiding Walks 9.6 Problems References 10 Peierls Argument and Duality Transformations 10.1 Peierls Argument 10.1.1 Extension to Higher Dimensions 10.2 Duality Transformation of Two-Dimensional Ising Model 10.2.1 An Algebraic Derivation 10.2.2 Two-Point Function 10.2.3 Potts Models 10.2.4 Curl and Divergence on a Lattice 10.3 Duality Transformation of Three-Dimensional Ising Model 10.3.1 Local Gauge Transformations 10.4 Duality Transformation of Three-Dimensional Zn Gauge Model 10.4.1 Wilson Loops 10.4.2 Duality Transformation of U(1) Gauge Model 10.5 Duality Transformation of Four-Dimensional Zn Gauge Model 10.6 Problems References 11 Renormalization Group on the Lattice 11.1 Decimation of Spins 11.1.1 Ising Chain 11.1.2 The Two-Dimensional Ising Model 11.2 Fixed Points 11.2.1 The Vicinity of a Fixed Point 11.2.2 Derivation of Scaling Laws 11.3 Block-Spin Transformation 11.4 Continuum Limit of Noninteracting Scalar Fields 11.4.1 Correlation Length for Interacting Systems 11.5 Continuum Limit of Spin Models 11.6 Programs for Chap.11 11.7 Problems References 12 Functional Renormalization Group 12.1 Scale-Dependent Functionals 12.2 Derivation of the Flow Equation 12.3 Functional Renormalization Applied to Quantum Mechanics 12.3.1 Projection onto Polynomials of Order 12 12.3.2 Changing the Regulator Function 12.3.3 Solving the Flow Equation for Non-convex Potentials 12.4 Scalar Field Theory 12.4.1 Fixed Points 12.4.2 Critical Exponents 12.5 Linear O(N) Models 12.5.1 Large N Limit 12.5.2 Exact Solution of the Flow Equation 12.6 Wave Function Renormalization 12.6.1 RG Equation for Wave Function Renormalization 12.7 Outlook 12.8 Programs for Chap.12 12.9 Problems Appendix: A Momentum Integral References 13 Lattice Gauge Theories 13.1 Continuum Gauge Theories 13.1.1 Parallel Transport 13.2 Gauge-Invariant Formulation of Lattice Higgs Models 13.2.1 Wilson Action of Pure Gauge Theories 13.2.2 Strong- and Weak-Coupling Limits of Higgs Models 13.3 Mean Field Approximation 13.3.1 Z2 Gauge Model 13.3.2 U(1) Gauge Theory 13.3.3 SU(n) Gauge Theories 13.3.4 Higgs Model 13.4 Expected Phase Diagrams at Zero Temperature 13.5 Elitzur's Theorem 13.5.1 Proof for Pure Z2 Gauge Theory 13.5.2 General Argument 13.6 Observables in Pure Gauge Theories 13.6.1 String Tension 13.6.2 Strong-Coupling Expansion for Pure Gauge Theories 13.6.3 Glueballs 13.7 Gauge Theories at Finite Temperature 13.7.1 Center Symmetry 13.7.2 G2 Gauge Theory 13.8 Problems References 14 Two-Dimensional Lattice Gauge Theories and Group Integrals 14.1 Abelian Gauge Theories on the Torus 14.1.1 Z2 Gauge Theory 14.1.2 U(1) Gauge Theory 14.2 Non-Abelian Lattice Gauge Theories on the Torus 14.2.1 Partition Function 14.2.2 Casimir Scaling of Polyakov Loops 14.3 Invariant Measure and Irreducible Representations 14.3.1 The Peter-Weyl Theorem 14.4 Problems References 15 Fermions on a Lattice 15.1 Dirac Equation 15.1.1 Coupling to Gauge Fields 15.2 Grassmann Variables 15.2.1 Gaussian Integrals 15.2.2 Path Integral for Dirac Theory 15.3 Fermion Fields on a Lattice 15.3.1 Lattice Derivative 15.3.2 Naive Fermions on the Lattice 15.3.3 Wilson Fermions 15.3.4 Staggered Fermions 15.3.5 Nielsen–Ninomiya Theorem 15.4 Ginsparg-Wilson Relation and Overlap Fermions 15.4.1 Overlap Fermions 15.4.2 Locality 15.5 Yukawa Models on the Lattice 15.5.1 Higgs Sector of Standard Model 15.5.2 Supersymmetric Yukawa Models 15.6 Coupling to Lattice Gauge Fields 15.7 Finite Temperature and Density 15.8 Problems Appendix: The SLAC Derivative References 16 Finite Temperature Schwinger Model 16.1 The Massless Schwinger Model 16.2 Effective Action: Anomaly-Induced Local Part 16.3 Effective Actions: Global Part 16.3.1 Topologically Trivial Sector 16.3.2 Topologically Non-trivial Sectors 16.4 Computing the Zero Modes 16.5 Chiral Condensate at Finite T and L 16.6 Wilson Loops, Field Strength, and 2-Point Function 16.6.1 Correlation Functions of the Field Strength 16.6.2 Wilson Loops and Charge Screening 16.6.3 Polyakov Loops (Thermal Wilson Loops) 16.6.4 Gauge-Invariant Fermionic Two-Point Functions 16.7 Massive Multi-Flavor Schwinger Model on the Lattice 16.7.1 Lattice Simulations 16.8 Problems References 17 Interacting Fermions 17.1 Symmetries of Fermi Systems 17.2 Four-Fermi Theories 17.2.1 Thirring Model 17.2.2 (Chiral) Gross–Neveu Model 17.2.3 Nambu-Jona-Lasinio Model 17.2.4 Relations Between Different Four-Fermi Theories 17.2.5 Hubbard–Stratonovich Transformation 17.3 Dirac Operator 17.3.1 Spectral Properties and Furry Theorem 17.4 Large Nf Limit of Gross–Neveu Models 17.5 Effective Potential of GN Models at Finite T and μ 17.5.1 Effective Potentials for Homogeneous Background Fields Zero-Temperature Contribution Finite-Temperature Contribution 17.5.2 The Effective Potentials in Two and Three Dimensions 17.5.3 Gap Equations and Phase Diagrams 17.6 Effective Potential of cGN Model with Momentum Cutoff 17.6.1 Non-covariant Momentum Cutoff 17.6.2 Renormalization 17.7 Magnetic Catalysis in the GN Model 17.7.1 The Effective Potential in Three Dimensions 17.8 Inhomogeneous Condensates in the 2d GN Model 17.8.1 The Fermion Determinant for σ=σ(x) 17.9 Beyond Large Nf 17.9.1 Critical Behavior of Three-Dimensional GN Models 17.9.2 Lattice Simulations of Three-Dimensional GN Models 17.9.3 Lattice Simulations of Three-Dimensional Thirring Models 17.9.4 Inhomogeneous Phases 17.9.5 Functional Renormalization Group 17.10 Problems Appendix: Covariant ζ-Function Regularization References Index