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

Cover
From Random Walks to Random Matrices: Selected Topics in Modern Theoretical Physics
Copyright
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