Statistical approach to quantum field theory : an introduction / / Andreas Wipf

• Autore: Wipf Andreas
• Pubblicazione: Cham, Switzerland : , : Springer, , [2021]; ©2021
• Edizione: Second edition.
• Descrizione fisica: 1 online resource (568 pages)
• Disciplina: 530.143
• Soggetto topico: Quantum field theory
• Nota di contenuto: Intro -- 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.
• Titolo autorizzato: Statistical approach to quantum field theory
• ISBN: 3-030-83263-5
• Formato: Materiale a stampa
• Livello bibliografico: Monografia
• Lingua di pubblicazione: Inglese
• Record Nr.: 996466850503316
• Lo trovi qui: Univ. di Salerno
• Serie: Lecture notes in physics ; ; Volume 992.