From Random Walks to Random Matrices - Selected Topics in Modern Theoretical Physics

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From Random Walks to Random Matrices: Selected Topics in Modern Theoretical Physics
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Preface
Contents
1 The random walk: Universality and continuum limit
	1.1 Random walk invariant under space and discrete time translations
		1.1.1 Translation invariant random walk in continuum space
	1.2 Fourier representation
		1.2.1 Generating function of cumulants
	1.3 Random walk: Asymptotic behaviour from a direct calculation
		1.3.1 Continuum time limit
	1.4 Corrections to continuum limit
	1.5 Random walk: Fixed points of transformations and universality
		1.5.1 Time scale transformation and renormalization
		1.5.2 Fixed points: generic situation: w1 6= 0
		1.5.3 Centred distribution
	1.6 Local and global stability of fixed points
		1.6.1 General analysis and RG terminology
		1.6.2 Fixed point stability: w1 6= 0
		1.6.3 Fixed point stability: w1 = 0
		1.6.4 Random walk on a lattice of points with integer coordinates
	1.7 Brownian motion and path integral
2 Functional integration: From path to field integrals
	2.1 Random walk, Brownian motion and path integral
		2.1.1 Continuum limit and path integral
		2.1.2 Positive measure and correlation functions
		2.1.3 Brownian paths
		2.1.4 Explicit calculation
	2.2 The Wiener measure and statistical physics
		2.2.1 Classical statistical physics
		2.2.2 Quantum statistical physics
	2.3 Generalization
		2.3.1 Path integral and local Markov process
		2.3.2 Path integrals and statistical physics
	2.4 Gaussian path integrals: The quantum harmonic oscillator
	2.5 Path integrals: Perturbation theory
		2.5.1 Gaussian expectation values and Wick’s theorem
		2.5.2 Path integral: Perturbative calculation
	2.6 Path integral: Quantum time evolution
	2.7 Barrier penetration in the semi-classical limit
	2.8 Path integrals: A few generalizations
	2.9 Path integrals for bosons and fermions
		2.9.1 Holomorphic formalism and bosons
		2.9.2 Grassmann path integrals and fermions
	2.10 Field integrals: New issues
		2.10.1 More general quantum field theories
		2.10.2 Regularization and effective field theories
		2.10.3 Renormalization and renormalization group
3 The essential role of functional integrals in modern physics
	3.1 Classical physics: The mystery of the variational principle
		3.1.1 Euler–Lagrange equations
		3.1.2 The particle in a static magnetic field
		3.1.3 Electromagnetism and Maxwell’s equations
		3.1.4 General Relativity
		3.1.5 Quantum mechanics and the variational principle
	3.2 Quantum evolution: From Hamiltonian to Lagrangian formalism
		3.2.1 Quantum evolution
		3.2.2 Relativistic quantum field theory
	3.3 From quantum evolution to statistical physics
		3.3.1 The single particle on an axis
		3.3.2 Quantum field theory: Quantum and classical statistical physics
	3.4 Statistical models at criticality and quantum field theory
	3.5 Barrier penetration, vacuum instability: Instanton calculus
	3.6 Large order behaviour and Borel summability: Critical exponents
	3.7 Quantization of gauge theories
		3.7.1 QED
		3.7.2 Quantization of non-Abelian gauge theories
		3.7.3 Covariant quantization: Faddeev–Popov’s method
	3.8 Numerical simulations in quantum field theory
	3.9 Quantization of the non-linear σ-model
	3.10 N-component fields: Large N techniques
4 From infinities in quantum electrodynamics to the general renormalization group
	4.1 QFT, RG: Some major steps
	4.2 QED and the problem of infinities
		4.2.1 First calculations: The problem of infinities
		4.2.2 Infinities and charged scalar bosons
	4.3 The renormalization strategy
	4.4 The nature of divergences and the meaning of renormalization
	4.5 QFT and RG
		4.5.1 The triumph of renormalizable QFT: The Standard Model
	4.6 Critical phenomena: Other infinities
	4.7 The failure of scale decoupling: The RG idea
		4.7.1 Scale decoupling in physics
		4.7.2 The RG idea
	4.8 Phase transitions: Exact RG in the continuum
		4.8.1 The exact RG
		4.8.2 Asymptotic or perturbative RG equations
	4.9 Effective field theory: From critical phenomena to particle physics
5 Renormalization group: From a general concept to numbers
	5.1 Scale decoupling in physics: A basic paradigm
	5.2 Fundamental microscopic interactions
	5.3 Macroscopic phase transitions
		5.3.1 The RG idea: Simple ferromagnetic systems
		5.3.2 Fixed points
		5.3.3 Scale non-decoupling and fixed points, a geometric analogue: Fractals
	5.4 Fixed points: The QFT framework
		5.4.1 The Gaussian fixed point
		5.4.2 QFT perturbative RG
	5.5 RG, correlation functions and scaling relations
	5.6 Exponents: Practical QFT calculations
	5.7 Results for three-dimensional critical exponents
6 Critical phenomena: The field theory approach
	6.1 Universality and RG
		6.1.1 Quantum field theory: Renormalization and universality
		6.1.2 Macroscopic continuous phase transitions: Universality
		6.1.3 From Wilson’s momentum-shell integration to functional RG equations
	6.2 RG in the continuum: Abstract formulation
	6.3 Effective field theory
	6.4 The Gaussian field theory
		6.4.1 The Gaussian critical theory
		6.4.2 The non-critical Gaussian theory
		6.4.3 Short distance singularities
	6.5 Gaussian fixed point and Gaussian renormalization
		6.5.1 Perturbing the Gaussian fixed point (d > 2)
		6.5.2 Gaussian renormalization
	6.6 Statistical scalar field theory: Perturbation theory
		6.6.1 The perturbed Gaussian or quasi-Gaussian model
	6.7 Dimensional continuation and regularization
		6.7.1 Dimensional continuation
		6.7.2 Dimensional regularization and ε-expansion
	6.8 Perturbative RG
		6.8.1 Critical theory: The renormalization theorem
		6.8.2 RG equations for the critical theory
		6.8.3 RG equations in the critical domain above Tc
		6.8.4 Renormalized RG equations
	6.9 RG equations: Solutions
	6.10 Wilson–Fisher’s fixed point: ε-Expansion
		6.10.1 The Ising class fixed point from the φ4 field theory
		6.10.2 ε-Expansion: A few general results
	6.11 Critical exponents as ε-expansions
	6.12 Three-dimensional exponents: Summation of the ε-expansion
7 Stability of renormalization group fixed points and decay of correlations
	7.1 Models with only one correlation length
	7.2 Cubic anisotropy, a model with two couplings
		7.2.1 RG and fixed points
		7.2.2 Linearized flow and eigenvalues
		7.2.3 Corresponding values of the exponent η
	7.3 General quartic Hamiltonian: RG functions
	7.4 Running coupling constants and gradient flows
		7.4.1 The gradient property of the RG β-functions
		7.4.2 A few consequences
	7.5 Fixed point stability and value of the potential
		7.5.1 First derivative
		7.5.2 Second derivative
	7.6 Fixed point stability and field dimension
8 Quantum field theory: An effective theory
	8.1 Effective local field theory: The scalar field
	8.2 Perturbative assumption and Gaussian renormalization
		8.2.1 Gaussian renormalization and dimensional analysis
		8.2.2 Classification of interactions and the fine tuning problem
		8.2.3 Renormalizable field theory
		8.2.4 Non-renormalizable interactions
		8.2.5 Renormalizable field theories and RG: The example of the φ44 field theory
	8.3 Fundamental interactions at the microscopic scale
	8.4 Field theory with a large mass: An explicit toy model
		8.4.1 Local expansion
	8.5 An effective field theory: The Gross–Neveu model
		8.5.1 The GNY model
		8.5.2 Symmetric phase: The effective GN model
		8.5.3 The GN model: Four dimensions
		8.5.4 The GN model in two dimensions
		8.5.5 Beyond perturbation theory: d > 2
	8.6 Non-linear σ-model: Another effective field theory
		8.6.1 The O(N) symmetric (˚2)2 field theory in the ordered phase
		8.6.2 The non-linear σ-model
9 The non-perturbative renormalization group
	9.1 Intuitive RG formulation
	9.2 Non-perturbative RG equations
		9.2.1 General local statistical field theory
		9.2.2 Functional RG equations
		9.2.3 Perturbative fixed points
	9.3 Partial field integration: Some identities
		9.3.1 A basic identity
		9.3.2 Other form of the identity: Partial integration
	9.4 Partial field integration in differential form
10 O(N) vector model in the ordered phase: Goldstone modes
	10.1 Classical lattice spin model and regularized non-linear σ-model
		10.1.1 Low temperature limit
		10.1.2 Local parametrization
	10.2 Perturbative or low temperature expansion
		10.2.1 The π-integration
		10.2.2 The configuration energy and the measure
		10.2.3 The propagator
		10.2.4 Gaussian fixed point and perturbations
	10.3 Zero momentum or IR divergences
		10.3.1 IR regularization
	10.4 Formal continuum limit: The non-linear σ-model
		10.4.1 Correlation functions with σ insertions
	10.5 The continuum theory: Regularization
		10.5.1 Dimensional regularization
		10.5.2 Derivative or Pauli–Villars’s regularizations
	10.6 Symmetry and renormalization
		10.6.1 WT identities and master equation
		10.6.2 Renormalization constants and renormalized action
	10.7 Correlation functions in dimension d = 2 + ε at one loop
		10.7.1 The field expectation value at one-loop order
		10.7.2 The two-point vertex function at one-loop order
	10.8 RG equations
		10.8.1 RG functions: One-loop calculation
	10.9 Zeros of the RG β-function: Fixed points
	10.10 Correlation functions: Scaling form below Tc
		10.10.1 Critical exponents
		10.10.2 Non-linear σ-model and (σ2)2 field theory
	10.11 Linear formulation
		10.11.1 IR divergences and O(N) symmetric functions
	10.12 Two dimensions
		10.12.1 Non-Abelian group: N > 2
		10.12.2 O(N) invariant functions and IR singularities
		10.12.3 The Abelian SO(2) model
11 Gauge invariance and gauge fixing
	11.1 Gauge invariance: A few historical remarks
	11.2 Variational principle, charged particle and gauge invariance
		11.2.1 Euler–Lagrange equations
		11.2.2 The motion of the charged particle: The principle of gauge invariance
		11.2.3 Enforcing gauge invariance: A dynamic principle
		11.2.4 The classical Hamiltonian in a magnetic and electric field
	11.3 Gauge invariance: A charged quantum particle
		11.3.1 Quantum Hamiltonian in a static magnetic field and gauge invariance
		11.3.2 The Schr¨odinger representation
		11.3.3 Time-dependent gauge transformations
	11.4 Evolution of a charged particle: Path integral representation
	11.5 Classical electromagnetism and Maxwell’s equations
	11.6 Gauge fixing in classical gauge theories
	11.7 QED
		11.7.1 Gauge field coupled to a conserved current
		11.7.2 Charged matter fields
		11.7.3 Parallel transport
	11.8 Non-Abelian gauge theories
		11.8.1 Classical field theory
		11.8.2 Gauge fields and differential geometry
	11.9 Quantization of non-Abelian gauge theories: Gauge fixing
		11.9.1 Gauge fixing in gauge field integrals
	11.10 General Relativity
12 The Higgs boson: A major discovery and a problem
	12.1 Perturbative quantum field theory: The construction
	12.2 Spontaneous symmetry breaking
		12.2.1 Relativistic quantum field theory
	12.3 Non-Abelian gauge theories
		12.3.1 Classical theory
		12.3.2 The problem of quantization
	12.4 The classical Abelian Landau–Ginzburg–Higgs mechanism
	12.5 Abelian and non-Abelian Higgs mechanism
	12.6 Non-Abelian gauge theories: Quantization and renormalization
		12.6.1 Non-Abelian gauge theories: Renormalization
		12.6.2 BRST symmetry
	12.7 The self-coupled Higgs field: A simple RG analysis
		12.7.1 The self-coupling approximation
	12.8 The Gross–Neveu–Yukawa model: A Higgs–top toy model
		12.8.1 The GNY model
		12.8.2 RG equations and mass ratio
	12.9 GNY model: The general RG flow at one loop
	12.10 The fine tuning issue
13 Quantum chromodynamics: A non-Abelian gauge theory
	13.1 Geometry of gauge theories: Parallel transport
		13.1.1 Gauge transformations, gauge invariance and parallel transport
		13.1.2 Gauge theories in the continuum
	13.2 Gauge invariant action
		13.2.1 Component form
	13.3 Hamiltonian formalism. Quantization in the temporal gauge
		13.3.1 Classical field equations
		13.3.2 Weyl’s or temporal gauge: Classical theory
		13.3.3 Quantum gauge theory in the temporal gauge
		13.3.4 Covariant generalized Landau’s gauge
		13.3.5 BRST symmetry
	13.4 Perturbation theory, regularization
		13.4.1 Regularization
		13.4.2 WT identities and renormalization
	13.5 QCD: Renormalization group
	13.6 Anomalies: General remarks
	13.7 QCD: The semi-classical vacuum and instantons
		13.7.1 The θ-vacuum and instantons
		13.7.2 Physics application: The solution of the U(1) problem
	13.8 Lattice gauge theories: Generalities
		13.8.1 Gauge invariance and parallel transport on the lattice
	13.9 Pure lattice gauge theory
		13.9.1 Gauge invariant action and partition function
		13.9.2 Low coupling analysis
	13.10 Wilson loop and the confinement property
		13.10.1 Wilson’s loop in continuum: d-Dimensional Abelian gauge theories
		13.10.2 Non-Abelian gauge theories
	13.11 Fermions on the lattice. Chiral symmetry
		13.11.1 Numerical methods: Computer simulations
14 From BRST symmetry to the Zinn-Justin equation
	14.1 Non-Abelian gauge theories: Classical field theory
		14.1.1 Gauge transformations and gauge fields
		14.1.2 Covariant derivatives and curvature
	14.2 Non-Abelian gauge theories: The quantized action
		14.2.1 Quantized gauge action: The field integral viewpoint
	14.3 BRST symmetry of the quantized action
		14.3.1 BRST symmetry
	14.4 The ZJ equation and remormalization
		14.4.1 Regularization
		14.4.2 Renormalization with counter-terms
		14.4.3 ZJ equation
	14.5 The ZJ equation: A few general properties
		14.5.1 Special solutions
		14.5.2 Perturbative solutions
		14.5.3 Canonical invariance
		14.5.4 Infinitesimal canonical transformations
	14.6 BRST symmetry: The algebraic origin
		14.6.1 BRST symmetry
		14.6.2 BRST symmetry and group elements
15 Quantum field theory: Asymptotic safety
	15.1 RG and consistency
		15.1.1 The non-Abelian gauge theory revolution: Asymptotic freedom
		15.1.2 Wilson’s theory of critical phenomena
	15.2 Super-renormalizable effective field theories: The (φ2)2 example
		15.2.1 The renormalized field theory and Callan–Symanzik equations
		15.2.2 The effective critical field theory
	15.3 A renormalizable field theory: The (φ2)2 theory in dimension 4
	15.4 The non-linear σ-model
		15.4.1 Dimension 2
		15.4.2 Higher dimensions
	15.5 The Gross–Neveu model
	15.6 QCD
	15.7 General interactions and summary
16 Symmetries: From classical to quantum field theories
	16.1 Symmetries and regularization
		16.1.1 UV divergences
		16.1.2 Symmetries and regularization
	16.2 Higher derivatives and momentum cut-off regularization
		16.2.1 Scalar fields: Higher derivative regularization
		16.2.2 Schwinger’s proper time regularization
		16.2.3 Regularization: Spin 1/2 fermions
		16.2.4 Regularization and determinants
		16.2.5 Application to global linear symmetries
	16.3 Regulator fields
		16.3.1 Scalar fields
		16.3.2 Fermions
	16.4 Abelian gauge theory, the theoretical framework of QED
		16.4.1 Charged fermions in a gauge field background
		16.4.2 The fermion determinant
		16.4.3 Boson determinant in a gauge background
		16.4.4 The gauge field propagator
	16.5 Non-Abelian gauge theories
	16.6 Dimensional regularization and chiral symmetry
		16.6.1 Dimensional regularization
		16.6.2 The problem with fermions: The example of dimension 4
	16.7 Lattice regularization
		16.7.1 Scalar bosons on the lattice
		16.7.2 Fermions, chiral symmetry and the doubling problem
		16.7.3 Gauge theories: Gauge fields and scalar bosons
17 Quantum anomalies: A few physics applications
	17.1 Electromagnetic decay of the neutral pion and Abelian anomaly
		17.1.1 Abelian axial current and Abelian vector gauge field
		17.1.2 Regulator fields and explicit anomaly calculation
		17.1.3 The electromagnetic decay of the neutral pion
		17.1.4 Chiral gauge theories
	17.2 A two-dimensional illustration: The Schwinger model
		17.2.1 The classical theory
		17.2.2 Quantum theory: Spectrum and anomaly
		17.2.3 Two dimensions: The chiral anomaly
		17.2.4 Currents: Field equations, anomaly and spectrum
	17.3 Abelian axial current and non-Abelian gauge fields
		17.3.1 The axial anomaly
		17.3.2 Anomaly and eigenvalues of the Dirac operator
	17.4 Non-Abelian anomaly and chiral gauge theories
		17.4.1 General axial current
	17.5 Weak and electromagnetic interactions: Anomaly cancellation
		17.5.1 Obstruction to gauge invariance
		17.5.2 An application: Weak–electromagnetic interactions
	17.6 Wess–Zumino consistency conditions
	17.7 Lattice fermions: Ginsparg–Wilson relation
		17.7.1 Lattice: New implementation of chiral symmetry
		17.7.2 Eigenvalues and index of the Dirac operator in a gauge background
		17.7.3 Chiral transformations on the lattice: Non-Abelian generalization
		17.7.4 Explicit realization: Overlap fermions
	17.8 Supersymmetric quantum mechanics and domain wall fermions
		17.8.1 Supersymmetric quantum mechanics
		17.8.2 Domain wall fermions: Continuum formulation
18 Periodic semi-classical vacuum, instantons and anomalies
	18.1 The periodic cosine potential
		18.1.1 The structure of the ground state
		18.1.2 Path integral representation and topology
		18.1.3 Instantons
	18.2 Instantons, anomalies and θ-vacua: CPN−1 models
		18.2.1 The CPN−1 models
		18.2.2 The CPN−1 action
		18.2.3 The structure of the semi-classical vacuum
		18.2.4 Instantons and topology
		18.2.5 CP1 and O(3) non-linear σ-models
	18.3 Non-Abelian gauge theories: Instantons and anomalies
		18.3.1 Instantons, chiral anomaly and topology
		18.3.2 The topological charge: Quantization
	18.4 The semi-classical vacuum and the strong CP violation
	18.5 Fermions in an instanton background: The U(1) problem
		18.5.1 Solutions to the strong CP problem
		18.5.2 The solution of the U(1) problem
19 Field theory in a finite geometry: Finite size scaling
	19.1 Periodic boundary conditions and the problem of the zero mode
		19.1.1 Finite size and finite temperature quantum mechanics
		19.1.2 The role of the zero mode: Effective integral
	19.2 Cylindrical geometry: Two-dimensional field theory
		19.2.1 Super-renormalizable perturbative field theory
		19.2.2 Finite size with periodic boundary conditions
		19.2.3 The effective theory of the zero mode
		19.2.4 Leading order calculation
		19.2.5 One-loop corrections
	19.3 Effective (φ2)2 field theory at criticality in finite geometries
		19.3.1 The (φ2)2 field theory for 2 < d ≤ 4
		19.3.2 RG equations
		19.3.3 RG in finite geometries: Finite size scaling
	19.4 Momentum quantization in finite geometries
		19.4.1 Periodic boundary conditions and the problem of the zero mode
	19.5 The (φ2)2 field theory in a periodic hypercube
		19.5.1 Moments: Leading order calculation
		19.5.2 Moments: One-loop corrections at Tc
		19.5.3 Moments: Dimensions d > 4
		19.5.4 Universality at Tc for d > 4
		19.5.5 Moments: Dimension d = 4 − ε
	19.6 The (φ2)2 field theory: Cylindrical geometry
		19.6.1 Finite size correlation length: Leading order calculation
		19.6.2 Finite size correlation length: One-loop corrections
		19.6.3 Finite size correlation length: Dimensions d > 4
		19.6.4 Finite size correlation length: Dimensions d = 4 − ε
	19.7 Continuous symmetries: Finite size effects at low temperature
20 The weakly interacting Bose gas at the critical temperature
	20.1 Bose gas: Field integral formulation
		20.1.1 Euclidean Bose gas action
		20.1.2 Equation of state and two-point function
	20.2 Independent bosons: Bose–Einstein condensation
	20.3 The weakly interacting Bose gas and the Helium phase transition
		20.3.1 Phase transition and dimensional reduction
	20.4 RG and universality
		20.4.1 Solution of the RG equations: The IR fixed point
		20.4.2 RG equation: Another form of the solution and crossover scale
	20.5 The shift of the critical temperature for weak interaction
		20.5.1 The variation of the equation of state
		20.5.2 The N-vector model: The large N expansion at order 1/N
		20.5.3 The saddle point equations
		20.5.4 The two-point function: 1/N correction
21 Quantum field theory at finite temperature
	21.1 Finite temperature QFT: General considerations
		21.1.1 Classical statistical field theory and RG
		21.1.2 Mode expansion and dimensional reduction
	21.2 Scalar field theory: Effective theory for the zero mode
		21.2.1 Effective reduced action: Leading order
		21.2.2 One-loop correction to the effective action
		21.2.3 Reduced action at higher orders
		21.2.4 Renormalization
	21.3 The (φ2)21,d scalar QFT: Phase transitions
		21.3.1 Phase transitions at zero temperature
		21.3.2 Phase transitions at finite temperature
	21.4 Temperature effects: The temperature-dependent mass
	21.5 Phase structure at finite temperature at one loop
		21.5.1 Thermodynamic potential density at one loop
		21.5.2 Critical temperature
		21.5.3 Two-point function: One-loop calculation in the symmetric phase
	21.6 RG at finite temperature
		21.6.1 Dimension d ≤ 3
	21.7 Effective action: Perturbative calculation
		21.7.1 Effective action at leading order
		21.7.2 Effective action: One-loop correction
	21.8 Effective action: φ-Expansion
		21.8.1 The φ2 term
	21.9 The (φ2)2 field theory at finite temperature in the large N limit
		21.9.1 The large N limit
		21.9.2 Zero temperature
		21.9.3 Finite temperature
	21.10 The non-linear σ-model at finite temperature for large N
		21.10.1 The O(N) symmetric non-linear σ-model
		21.10.2 The large N limit at zero temperature
		21.10.3 The σ two-point function
		21.10.4 The large N limit at finite temperature: The gap equations
		21.10.5 The symmetric phase
		21.10.6 Dimension d = 2
		21.10.7 Dimension d ≥ 3
		21.10.8 The spontaneously broken phase
	21.11 The GN model at finite temperature for large N
		21.11.1 The GN model
		21.11.2 The GN model at zero temperature for N large: Gap equation and mass spectrum
		21.11.3 Finite temperature: Gap or saddle point equation
		21.11.4 Phase transition at finite temperature: The critical temperature
		21.11.5 The σ two-point function
		21.11.6 Dimension d = 1
	21.12 Abelian gauge theories: The QED example
		21.12.1 Massive vector field coupled to fermion matter
		21.12.2 Finite temperature
		21.12.3 From physical to temporal (Weyl) gauge: zero temperature
		21.12.4 From physical to temporal (Weyl) gauge: Finite temperature
		21.12.5 Dimensional reduction
		21.12.6 The action density
		21.12.7 Discussion
	A21 Appendix: One-loop contributions
		A21.1 Γ and ζ functions
		A21.2 The one-loop two-point contribution at T = 0
		A21.3 The thermal corrections at one loop
			A21.3.1 The gap equation
			A21.3.2 The two-point function
22 From random walk to critical dynamics
	22.1 Random walk with gradient driving force
		22.1.1 The trajectory probability distribution
		22.1.2 The purely dissipative Langevin equation
	22.2 An elementary example: The linear driving force
		22.2.1 The Brownian motion: ω = 0
		22.2.2 Case ω 6= 0
		22.2.3 Addition of a time-dependent linear potential: Jarzinsky’s relation
	22.3 The Fokker–Planck formalism
		22.3.1 The FP equation
		22.3.2 Dissipative Langevin equation
		22.3.3 Detailed balance
		22.3.4 Gradient time-dependent force and Jarzinsky’s relation
	22.4 Path integral representation
		22.4.1 Dissipative Langevin equation
		22.4.2 Detailed balance and path integral
		22.4.3 Time-dependent force deriving from a potential
		22.4.4 Time-dependent force and Jarzinsky’s relation
	22.5 The dissipative Langevin equation: Supersymmetric formulation
		22.5.1 Grassmann coordinates and algebraic properties
		22.5.2 Superpaths and covariant derivatives
		22.5.3 Supersymmetry
		22.5.4 Supersymmetry and detailed balance
	22.6 Critical dynamics: The Langevin equation in field theory
		22.6.1 The associated FP equation
		22.6.2 The linear Langevin equation
	22.7 Time-dependent correlation functions and dynamic action
		22.7.1 Dynamic action
		22.7.2 The divergent determinant
	22.8 The dissipative Langevin equation and supersymmetry
		22.8.1 Supersymmetry
		22.8.2 WT identities
	22.9 Renormalization of the dissipative Langevin equation
	22.10 Dissipative Langevin equation: RG equations in 4−ε dimensions
		22.10.1 RG equations at and above Tc
		22.10.2 The infrared fixed point
		22.10.3 Correlation functions above Tc in the critical domain
23 Field theory: Perturbative expansion and summation methods
	23.1 Divergent series in quantum field theory
		23.1.1 A special class of divergent series
		23.1.2 Borel summable series. Borel transformation
	23.2 An example: The perturbative (φ2)2 field theory
		23.2.1 The perturbative expansion: Large order behaviour
	23.3 Renormalized perturbation theory: Callan–Symanzik equations
		23.3.1 CS equations
		23.3.2 RG functions in three dimensions in the CS formalism
	23.4 Summation methods and critical exponents
		23.4.1 Pad´e approximants
		23.4.2 Methods based on Borel transformation
		23.4.3 Borel transformation and conformal mapping
	23.5 ODM summation method
		23.5.1 The general method
		23.5.2 Functions analytic in a cut-plane: Heuristic convergence analysis
		23.5.3 Examples
	23.6 Application: The simple integral d = 0
		23.6.1 The optimal mapping
		23.6.2 Numerical application
		23.6.3 Alternative mapping
	23.7 The quartic anharmonic oscillator: d = 1
	23.8 φ4 field theory in d = 3 dimensions
24 Hyper-asymptotic expansions and instantons
	24.1 Divergent series and Borel summability
	24.2 Perturbative expansion and path integral
	24.3 The quartic anharmonic oscillator: A Borel summable example
		24.3.1 Cauchy representation and barrier penetration
		24.3.2 Barrier penetration and instantons: The ground state N = 0
	24.4 The double-well potential: Generalized Bohr–Sommerfeld quantization formulae
		24.4.1 The quartic double-well potential: Perturbative expansion
		24.4.2 Quantum tunnelling and energy splitting
		24.4.3 The hyper-asymptotic expansion
		24.4.4 Generalized Bohr–Sommerfeld quantization formula
		24.4.5 Multi-instanton contributions at leading order
		24.4.6 Large order behaviour of perturbative series
	24.5 Instantons and multi-instantons
		24.5.1 Partition function and symmetries
		24.5.2 Potentials with symmetric degenerate minima
		24.5.3 Multi-instantons
		24.5.4 The general multi-instanton action
		24.5.5 The multi-instanton contribution
		24.5.6 The sum of leading order instanton contributions
	24.6 Perturbative and exact WKB expansions
		24.6.1 Riccati equation and complex Bohr–Sommerfeld quantization formula
		24.6.2 Complex WKB expansion
	24.7 Other analytic potentials: A few examples
		24.7.1 Asymmetric wells
		24.7.2 The periodic cosine potential
25 Renormalization group approach to matrix models
	25.1 One-Hermitian matrix models and random surfaces: A summary
	25.2 Continuum and double scaling limits
		25.2.1 Continuum limit: The cubic example
		25.2.2 The double scaling limit
		25.2.3 Generalizations
	25.3 The RG approach
		25.3.1 The matrix RG flow
		25.3.2 Linear perturbative approximation
		25.3.3 Stability analysis
Bibliography
Index