Quantum Field Theory: Feynman Path Integrals and Diagrammatic Techniques in Condensed Matter

✍ Scribed by Lukong Cornelius Fai

• Publisher: CRC Press
• Year: 2019
• Tongue: English
• Leaves: 536
• Edition: 1
• Category: Library

✦ Synopsis

Choice Recommended Title, February 2020

This book explores quantum field theory using the Feynman functional and diagrammatic techniques as foundations to apply Quantum Field Theory to a broad range of topics in physics. This book will be of interest not only to condensed matter physicists but physicists in a range of disciplines as the techniques explored apply to high-energy as well as soft matter physics.

Features:

Comprehensive and rigorous, yet presents an easy to understand approach

Applicable to a wide range of disciplines

Accessible to those with little, or basic, mathematical understanding

✦ Table of Contents

Cover
Half Title
Title Page
Copyright Page
Contents
Preface
About the Author
1. Symmetry Requirements in QFT
1.1.Second Quantization
1.1.1.Fock Space
1.1.2.Creation and Annihilation Operators
1.1.3.(Anti)Commutation Relations
1.1.4.Change of Basis in Second Quantization
1.1.5.Quantum Field Operators
1.1.6.Operators in Second-Quantized Form
1.1.6.1.One-Body Operator
1.1.6.2.Two-Body Operator
2. Coherent States
2.1.Coherent States for Bosons
2.2.Coherent States and Overcompleteness
2.2.1.Overcompleteness of Coherent States
2.2.2.Overlap of Two Coherent States
2.2.3.Overcompleteness Condition
2.2.4.Closure Relation via Schur’s Lemma
2.2.5.Normal-Ordered Operators
2.2.6.The Trace of an Operator
2.3.Grassmann Algebra and Fermions
2.3.1.Grassmann Algebra
2.3.1.1.Differentiation over Grassmann Variables
2.3.1.2.Exponential Function of Grassmann Numbers
2.3.1.3.Involution of Grassmann Numbers
2.3.1.4.Bilinear Form of Operators
2.3.1.5.Berezin Integration
2.3.1.6.Grassmann Delta Function
2.3.1.7.Scalar Product of Grassmann Algebra
2.3.2.Fermions
2.4.Fermions and Coherent States
2.4.1.Coherent State Overcompleteness Relation Proof
2.4.2.Trace of a Physical Quantity
2.4.3.Functional Integral Time-Ordered Property
2.5.Gaussian Integrals
2.5.1.Multidimensional Gaussian Integral
2.5.2.Multidimensional Complex Gaussian Integral
2.5.3.Multidimensional Grassmann Gaussian Integral
2.6.Wick Theorem for Multidimensional Grassmann Integrals
2.6.1.Wick Theorem
3.Fermionic and Bosonic Path Integrals
3.1.Coherent State Path Integrals
3.2.Noninteracting Particles
3.2.1.Bare Partition Function
3.2.2. Inverse Matrix of S(α)
3.3.Bare Green’s Function via Generating Functional
3.3.1.Generating Functional
3.4.Single-Particle Green’s Function
3.4.1.Matsubara Green’s Function
3.5.Noninteracting Green’s Function
3.6 Average Value of a Functional
4.Perturbation Theory and Feynman Diagrams
4.1.Representation as Diagrams
4.2.Generating Functionals
4.3.Wick Theorem
4.4.Perturbation Theory
4.4.1.Linked Cluster Theorem
4.4.2.Green’s Function Generating Functional
4.4.3.Green’s Functions
4.4.3.1.Zeroth Order
4.4.3.2.First Order
4.4.3.3.Second Order
5. (Anti)Symmetrized Vertices
5.1.Fully (Anti)Symmetrized Vertices
6.Generating Functionals
6.1.Connected Green’s Functions
6.2.General Case
6.3.Dyson-Schwinger Equations
6.4.Effective Action For 1PI Green’s Functions
6.4.1.Normal Systems
6.4.2.Self-Energy and Dyson Equation
6.4.2.1.Self-Energy and Dyson Equation
6.4.3.Higher-Order Vertices
6.4.4.General Case
6.4.5.Luttinger-Ward Functional and 2PI Vertices
6.4.5.1.Normal Systems
6.4.5.2.The Self-Consistent Dyson Equation
6.4.5.3.Diagrammatic Interpretation of LWF
6.4.5.4.2PI Vertices and Bethe-Salpeter Equation
6.4.5.5.Bethe-Salpeter Equation
7.Random Phase Approximation (RPA)
7.1.Path Integral Formalism
7.1.1.Quantum Three-Dimensional Coulomb Gas
7.1.2.Translationally Invariant System
7.2 RPA Functional Integral
7.2.1.Gaussian Fluctuations
7.2.1.1.Integration over Grassmann Variables
7.2.1.2.Fermionic Determinant Gaussian Expansion
7.2.1.3.Diagrammatic Interpretation of the RPA
7.2.1.4.Saddle-Point Approximation
7.2.1.5.Lindhard Function and Plasmon Oscillations
7.2.1.6.Particle-Hole Pair Excitation
7.2.1.7.Lindhard Formula
7.2.1.8.Spectral Function
7.2.1.9.Plasma Oscillations And Landau Damping
7.2.1.10.Thomas-Fermi Screening
7.2.1.11.Friedel Oscillations
7.2.1.12.Dynamic Polarization Function
7.2.1.13.Ground-State Energy in the RPA
7.2.1.14.Compressibility
7.2.1.15.One-Particle Property: Hartree-Fock Theory
8.Phase Transitions and Critical Phenomena
8.1.Landau Theory of Phase Transition
8.2.Entropy and Specific Heat
8.3.External Field Effect on a Phase Transition
8.4.Ginzburg-Landau Theory
8.5.The Scaling Hypothesis
8.6.Identities from the d-Dimensional Space
8.7.Energy Fluctuation
9.Weakly Interacting Bose Gas
9.1.Bose-Einstein Condensation
9.2.Bogoliubov Transformation
9.3.Nonideal Bose Gas Path Integral Formalism
9.3.1.Beliaev-Dyson Equations
10.Superconductivity Theory
10.1.BCS Superconductivity Theory
10.1.1.Electron-Phonon Interaction in a Solid State
10.1.2.Effective Four-Fermion BCS Theory
10.1.3.Effective Action Functional
10.1.4.Critical Temperature
10.2.Mean-Field Theory
10.3.Green’s Function via Bogoliubov Coefficients
10.4.The BCS Ground State
10.5.Gauge Invariance
10.6.Diagrammatic Approach to Superconductivity
10.6.1.Ladder Approximation
10.6.2.Bethe-Salpeter Equation
10.6.3.Cooper Instability
10.6.3.1.Finite Temperature Calculation
10.6.4.Small Momentum Transfer Vertex Function
10.6.5.Ward Identities: Gauge Invariance
10.6.6.Galilean Invariance
10.6.7.Response on Vector Potential
11.Path Integral Approach to the BCS Theory
11.1.Two-Component Fermi Gas Action Functional
11.2.Hubbard-Stratonovich Fields
11.2.1.Nambu-Gorkov Representation
11.2.2.Pairing-Order Parameter Effective Action
11.2.3.Reciprocal Space
11.3.Saddle-Point Approximation
11.4.Generalized Correlation Functions
11.5.Condensate Fraction
11.6.Pair Correlation Length
11.7.Improvement of the Saddle-Point Solution
11.8.Fluctuation Partition Function
11.9.Fluctuation Bosonic Partition Function
11.10.Number Equation Fluctuation Contributions
11.11.Collective Mode Excitations
12.Green’s Function Averages over Impurities
12.1.Scattering Potential and Disordered System
12.2.Disorder Diagrams
12.3.Perturbation Series T-Matrix Expansion
12.4.T-Matrix Expansion
12.5.Disorder Averaging
12.6.Green’s Function Perturbation Series
12.7.Quenched Average and White Noise Potential
12.8.Average over Impurities’ Locations
12.9.Disorder Average Green’s Function
12.10.Disorder Diagrams
12.11.Gorkov Equation with Impurities
12.11.1.Properties of Homogeneous Superconductors
13.Classical and Quantum Theory of Magnetism
13.1 Classical Theory of Magnetism
13.1.1.Molecular Field (Weiss Field)
13.2.Quantum Theory of Magnetism
13.2.1.Spin Wave: Model of Localized Magnetism
13.2.2.Heisenberg Hamiltonian
13.2.3.X-Y Model
13.2.4.Spin Waves in Ferromagnets
13.2.5.Bosonization of Operators
13.2.6.Magnetization
13.2.7.Experiments Revealing Magnons
13.2.8.Spin Waves in Antiferromagnets
13.2.9.Bogoliubov Transformation
13.2.10.Stability
13.2.11.Spin Dynamics, Dynamical Response Function
13.2.11.1.Spin Dynamics
13.2.12.Response Function and Relaxation Time
13.2.12.1.Linear Response Function
13.2.12.2.The Fluctuation-Dissipation Theorem
13.2.12.3.Onsager Relation
13.2.13.Itinerant Ferromagnetism
13.2.13.1.Quantum Impurities and the Kondo Effect
13.2.13.2.Localized and Itinerant Spins Interaction
13.2.13.3.Ruderman-Kittel-Kasuya-Yosida (RKKY) Interaction
13.2.13.4.Abrikosov Technique
13.2.13.5.Self-Energy of the Pseudo-Fermion
13.2.13.6.Effective Spin Screening, Spin Susceptibility
13.2.13.7.Second-Order Self-Energy Diagrams
13.2.13.8.Scattering Amplitudes
13.2.13.9.Scaling and Parquet Equation
13.2.13.10.Kondo Effect and Numerical Renormalization Group
13.2.13.11.Anisotropic Kondo Model
13.2.14.Schwinger-Wigner Representation
13.2.15.Jordan-Wigner
13.2.16.Semi-Fermionic Representation: Hubbard Model
13.2.16.1.Semi-Fermionic Representation
13.2.16.2.Kondo Lattice: Effective Action
14.Nonequilibrium Quantum Field Theory
14.1.Keldysh-Schwinger Technique: Time Contour
14.1.1.Basic Features of the S-Matrix (Operator)
14.1.2.Closed Time Path (CTP) Formalism
14.2.Contour Green’s Functions
14.3.Real-Time Formalism
14.3.1.Real-Time Matrix Representation
14.4.Two-Point Correlation Function Decomposition
14.5.Equilibrium Green’s Function
14.5.1.Spectral Function
14.5.1.1.Kubo-Martin-Schwinger (KMS) Condition
14.5.2.Sum Rule and Physical Interpretation
14.6 Keldysh Rotation
14.7 Path Integral Representation
14.7.1.Gross-Pitaevskii Equation
14.8.Dyson Equation and Self-Energy
14.9.Nonequilibrium Generating Functional
14.10.Gaussian Initial States
14.11.Nonequilibrium 2PI Effective Action
14.11.1.Luttinger-Ward Functional
14.12 Kinetic Equation and the 2PI Effective Action
14.12.1.The Self-Consistent Schwinger–Dyson Equation
14.13.Closed Time Path (CTP) and Extended Keldysh Contours
14.14.Kadanoff-Baym Contour
14.14.1.Green’s Function on the Extended Contour
14.14.2.Kadanoff-Baym Contour
14.15.Kubo-Martin-Schwinger (KMS) Boundary Conditions
14.15.1.Remark on KMS Boundary Conditions
14.15.2.Generalization of an Average Value
14.16.Neglect of Initial Correlations and Schwinger-Keldysh Limit
14.16.1.Equation of Motion for the Nonequilibrium Green’s Function
14.16.1.1.Nonequilibrium Green’s Function Equation of Motion: Auxiliary Fields and Functional Derivatives Technique
14.16.1.2.Keldysh Initial Condition
14.16.1.3.Perturbation Expansion and Feynman Diagrams
14.16.1.4.Right- and Left-Hand Dyson Equations
14.16.1.5.Self-Energy Self-Consistent Equations
14.17.Kadanoff-Baym (KB) Formalism for Bose Superfluids
14.17.1.Kadanoff-Baym Equations
14.17.1.1.Fluctuation-Dissipation Theorem
14.17.1.2.Wigner or Mixed Representation
14.18.Green’s Function Wigner Transformation
References
Index