Statistical Approach to Quantum Field Theory : An Introduction.

Intro -- Statistical Approach to Quantum Field Theory -- Preface -- Acknowledgments -- Contents -- Notations -- Chapter 1: Introduction -- References -- Chapter 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 -- Comment -- References -- Chapter 3: High-Dimensional Integrals -- 3.1 Numerical Algorithms -- 3.1.1 Newton-Cotes Integration Method -- Composite Integration Formulas -- 3.2 Monte Carlo Integration -- 3.2.1 Hit-or-Miss Monte Carlo Method and Binomial Distribution -- Numerical Experiment -- 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 Chap. 3 -- 3.6 Problems -- References -- Chapter 4: Monte Carlo Simulations in Quantum Mechanics -- 4.1 Markov Chains -- Stochastic Matrices and Stochastic Vectors -- 4.1.1 Fixed Points of Markov Chains -- 4.2 Detailed Balance -- 4.2.1 Acceptance Rate -- 4.2.2 Metropolis Algorithm -- 2-State System -- 3-State System -- 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 -- Header-Files -- 4.6 Problems -- References -- Chapter 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 -- Zeta-Function Regularization -- Heat Kernel of a Differential Operator.

Free Energy of Non-interacting Scalars -- High-Temperature Expansion -- 5.3 Schwinger Function and Effective Potential -- Generalizations -- 5.3.1 The Legendre-Frenchel Transformation -- 5.4 Scalar Field on a Space-Time Lattice -- Boundary Conditions -- 5.5 Random Walk Representation of Green's Function -- 5.6 There Is No Leibniz Rule on the Lattice -- 5.7 Programs for Chap. 5 -- 5.8 Problems -- References -- Chapter 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, Clock Model) -- 6.2.3 The U(1) Model -- 6.2.4 O(N) Models -- 6.2.5 Interacting Continuous Spins -- 6.3 Spin Systems in Thermal Equilibrium -- 6.4 Variational Principles -- 6.4.1 Principle for Gibbs State and Free Energy -- 6.4.2 Fixed Average Field -- 6.5 Programs for the Simulation of the Ising Chain -- 6.6 Problems -- References -- Chapter 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 alpha,beta,gamma,delta -- 7.3.1 Susceptibility -- 7.3.2 Magnetization as a Function of Temperature -- 7.3.3 Speci c 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 Antiferromagnetic Systems -- 7.8 Mean-Field Approximation for Lattice Field Theories -- 7.8.1 phi4 and phi6 Scalar Theories -- 7.8.2 O(N) Models -- 7.9 Program for Chap. 7 -- 7.10 Problems -- Hint -- References.

Chapter 8: Transfer Matrices, Correlation Inequalities and Roots of Partition Functions -- 8.1 Transfer-Matrix Method for the Ising Chain -- 8.1.1 Transfer Matrix -- Thermodynamic Potentials -- Correlation Functions -- 8.1.2 The "Hamiltonian -- Two Dimensions -- 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 -- Continuum Limit -- 8.5.2 Real Scalar Field -- 8.6 Correlation Inequalities -- Application of 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 -- Chapter 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 Speci c 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 O(N) Sigma Models -- 9.4.1 Expansions of Partition Function and Free Energy -- Order beta2 -- Order beta4 -- Order beta6 -- 9.5 Polymers and Self-Avoiding Walks -- 9.6 Problems -- References -- Chapter 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 -- Chapter 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 Non-interacting 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 -- Chapter 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 -- Comparison with Weak-Coupling Perturbation Expansion -- 12.4 Scalar Field Theory -- 12.4.1 Fixed Points -- Scalar Fields in Three Dimensions -- Numerical Solution -- 12.4.2 Critical Exponents -- 12.5 Linear O(N) Models -- 12.5.1 Large N Limit -- Fixed-Point Analysis -- 12.5.2 Exact Solution of the Flow Equation -- Symmetry Breaking -- 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 -- The Case |p+q|&lt -- k -- The Case |p+q|&lt -- k -- References -- Chapter 13: Lattice Gauge Theories -- 13.1 Continuum Gauge Theories.

13.1.1 Parallel Transport -- Composition of Paths -- Stokes' Theorem -- Gauge Transformation -- Matter Fields -- 13.2 Gauge-Invariant Formulation of Lattice Higgs Models -- 13.2.1 Wilson Action of Pure Gauge Theories -- 13.2.2 Weak and Strong Coupling Limits of Higgs Models -- Vanishing beta and Unitary Gauge -- In nite beta and Axial Gauge on Periodic Lattices -- Vanishing kappa -- The Limit kappa-&gt -- infty -- 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 -- Cubic Group -- Projecting on Fixed Quantum Numbers -- 13.7 Gauge Theories at Finite Temperature -- 13.7.1 Center Symmetry -- 13.7.2 G2 Gauge Theory -- 13.8 Problems -- References -- Chapter 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 2d Torus -- Gluing Loops and Migdal's Recursion Relation -- 14.2.1 Partition Function -- 14.2.2 Casimir Scaling of Polyakov Loops -- 14.3 Invariant Measure and Irreducible Representations -- Haar Measure of SU(2) -- Haar Measure for a General Lie Group -- 14.3.1 The Peter-Weyl Theorem -- 14.4 Problems -- References -- Chapter 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 -- Forward and Backward Derivative.

Antisymmetric Derivative.

This book opens with a self-contained introduction to path integrals in Euclidean quantum mechanics and statistical mechanics, and moves on to cover lattice field theory, spin systems, gauge theories and more. Each chapter ends with illustrative problems.

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Electronic reproduction. Ann Arbor, Michigan : ProQuest Ebook Central, 2024. Available via World Wide Web. Access may be limited to ProQuest Ebook Central affiliated libraries.