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

Format:

Book

Author/Creator:

Wipf, Andreas, author.

Series:

Lecture notes in physics ; Volume 992.

Lecture Notes in Physics ; Volume 992

Language:

English

Subjects (All):

Quantum field theory.

Physical Description:

1 online resource (568 pages)

Edition:

Second edition.

Place of Publication:

Cham, Switzerland : Springer, [2021]

Summary:

This new expanded second edition has been totally revised and corrected.The reader finds two complete new chapters.One covers the exact solution of the finite temperature Schwinger model with periodic boundary conditions.

Contents:

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

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

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

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

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

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

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

9.3 Low-Temperature Expansion of Ising Models

9.3.1 Free Energy and Magnetization of Two-Dimensional Model

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

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

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

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

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

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

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

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

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.

Notes:

Description based on print version record.

Description based on publisher supplied metadata and other sources.

ISBN:

3-030-83263-5

OCLC:

1281239385