Cover
Quantum Field Theory and Critical Phenomena - Fifth Edition
Copyright
Dedication
Preface
Acknowledgements
Some general references for the whole work
Contents
1 Gaussian integrals. Algebraic preliminaries
1.1 Gaussian integrals: Wick’s theorem
1.2 Perturbative expansion. Connected contributions
1.2.1 Perturbation theory
1.2.2 Connected contributions or cumulants
1.3 The steepest descent method
1.4 Complex structures and Gaussian integrals
1.5 Grassmann algebras. Differential forms
1.5.1 Differential forms
1.6 Differentiation and integration in Grassmann algebras
1.6.1 Differentiation in Grassmann algebras
1.6.2 A basis of differential operators
1.6.3 Integration in Grassmann algebras
1.6.4 Change of variables in a Grassmann integral
1.6.5 Mixed change of variables
1.7 Gaussian integrals with Grassmann variables
1.7.1 General Gaussian integrals
1.7.2 Pfaffian and determinant
1.8 Legendre transformation
2 Euclidean path integrals and quantum mechanics (QM)
2.1 Markovian evolution and locality
2.2 Statistical operator: Path integral representation
2.2.1 Short-time evolution
2.2.2 The path integral
2.3 Explicit evaluation of a path integral: The harmonic oscillator
2.4 Partition function: Classical and quantum statistical physics
2.4.1 The quantum partition function
2.4.2 Classical and quantum statistical physics
2.5 Correlation functions. Generating functional
2.5.1 Thermodynamic limit
2.5.2 Generating functional of correlation functions
2.5.3 Functional differentiation and correlation functions
2.6 Harmonic oscillator. Correlation functions and Wick’s theorem
2.6.1 Correlation functions, Wick’s theorem
2.6.2 Harmonic oscillator: Paths and square integrable functions
2.7 Perturbed harmonic oscillator
2.8 Semi-classical expansion
2.8.1 Quantum partition function
2.8.2 WKB spectrum
A2 Additional remarks
A2.1 A useful relation between determinant and trace
A2.2 The two-point function: An integral representation
A2.3 Time-ordered products of operators
3 Quantum mechanics (QM): Path integrals in phase space
3.1 General Hamiltonians: Phase-space path integral
3.1.1 Hamiltonian and Lagrangian
3.1.2 QM: Path integral for time evolution
3.1.3 Separable Hamiltonians: Equivalence
3.2 The harmonic oscillator. Perturbative expansion
3.2.1 The quantum harmonic oscillator
3.2.2 Phase space path integral: Perturbative definition
3.3 Hamiltonians quadratic in momentum variables
3.3.1 Quantization in a static magnetic field
3.3.2 General quadratic Hamiltonians
3.3.3 Phase-space formalism: The δ(0) problem
3.4 The spectrum of the O(2)-symmetric rigid rotator
3.5 The spectrum of the O(N)-symmetric rigid rotator
A3 Quantization. Topological actions: Quantum spins,magnetic monopoles
A3.1 Symplectic form and quantization: General remarks
A3.2 Classical equations of motion and quantization
A3.3 Topological actions
A3.3.1 Spin dynamics and quantization
A3.3.2 Quantization of spin degrees of freedom
A3.3.3 The magnetic monopole
4 Quantum statistical physics: Functional integration formalism
4.1 One-dimensional QM: Holomorphic representation
4.1.1 Hilbert space of analytic functions
4.1.2 Operator kernels
4.2 Holomorphic path integral
4.2.1 The harmonic oscil
4.2.2 Linear coupling to an external source: Generating functional
4.2.3 General one-dimensional Hamiltonian
4.3 Several degrees of freedom. Boson interpretation
4.4 The Bose gas. Field integral representation
4.4.1 Matrix density at thermal equilibrium: Fixed number of particles
4.4.2 Second quantization. Field integral representation
4.4.3 Fock space
4.4.4 Hamiltonian in Fock space
4.4.5 Kernels of operators and field integral representation
4.4.6 The Gaussian model
4.4.7 Pair potentials: The example of the δ(x)-function potential
4.5 Fermion representation and complex Grassmann algebras
4.5.1 Analytic Grassmann functions, scalar product
4.5.2 Operator algebra and kernels
4.5.3 An example: One state system
4.6 Path integrals with fermions
4.6.1 Generalization
4.7 The Fermi gas. Field integral representation
4.7.1 Simple examples
4.7.2 Non-relativistic Fermi gas at low temperatures, in one dimension
5 Quantum evolution: From particles to non-relativistic fields
5.1 Time evolution and scattering matrix in quantum mechanics (QM)
5.1.1 Evolution operator and S-matrix
5.1.2 One-particle system
5.1.3 Path integrals
5.2 Path integral and S-matrix: Perturbation theory
5.3 Path integral and S-matrix: Semi-classical expansions
5.3.1 Path integral and S-matrix
5.3.2 One dimension: Semi-classical limit
5.3.3 Eikonal approximation and path integral
5.4 S-matrix and holomorphic formalism
5.4.1 Path integrals
5.4.2 Time-dependent force
5.5 The Bose gas: Evolution operator
5.6 Fermi gas: Evolution operator
A5 Perturbation theory in the operator formalism
6 The neutral relativistic scalar field
6.1 The relativistic scalar field
6.1.1 Free-field theory and particle-field relation
6.1.2 Field integral and Fock’s space
6.1.3 Hamiltonian and particle number operators
6.1.4 Free-field two-point function
6.2 Quantum evolution and the S-matrix
6.2.1 The S-matrix: Scattering by an external source
6.2.2 General interacting theory
6.3 S-matrix and field asymptotic conditions
6.3.1 The Gaussian integral in an external source and the S-matrix
6.3.2 S-matrix elements from correlation functions
6.3.3 The φ3 example: Tree approximation
6.4 The non-relativistic limit: The ϕ4 QFT
6.5 Quantum statistical physics
6.5.1 Partition function and correlation functions
6.5.2 The problem of infinities or ultraviolet divergences
6.5.3 Connected and vertex functions
6.5.4 Change of field variables
6.5.5 S-matrix and field representation
6.6 Källen–Lehmann representation and field renormalization
7 Perturbative quantum field theory (QFT): Algebraic methods
7.1 Generating functionals of correlation functions
7.2 Perturbative expansion. Wick’s theorem and Feynman diagrams
7.2.1 Gaussian integral and free field theory
7.2.2 Perturbative expansion: A compact expression
7.2.3 Wick’s theorem
7.2.4 Feynman diagrams
7.3 Connected correlation functions: Generating functional
7.3.1 An alternative proof of connectivity
7.3.2 Inversion
7.4 The example of the ϕ4 QFT
7.5 Algebraic properties of field integrals. Quantum field equations
7.5.1 Integration by parts and quantum field equations
7.5.2 Direct algebraic proof of the quantum field equations
7.5.3 The infinitesimal change of variables
7.5.4 The choice of the Gaussian measure
7.5.5 The functional Dirac δ-function
7.6 Connected correlation functions. Cluster properties
7.7 Legendre transformation. Vertex functions
7.8 Momentum representation
7.9 Loop or semi-classical expansion
7.9.1 Loop expansion at leading order
7.9.2 Order ~ or one-loop contributions
7.9.3 Loop expansion at higher orders
7.10 Vertex functions: One-line irreducibility
7.11 Statistical and quantum interpretation of the vertex functional
7.11.1 Interpretation and variational principle
7.11.2 Vertex functional and free energy at fixed field time average
A7 Additional results and methods
A7.1 Generating functional at two loops
A7.2 The background field method
A7.3 Connected Feynman diagrams: Cluster properties
A7.3.1 Decay of connected Feynman diagrams in Euclidean space
A7.3.2 Threshold effects
8 Ultraviolet divergences: Effective field theory (EFT)
8.1 Gaussian expectation values and divergences: The scalar field
8.2 Divergences of Feynman diagrams: Power counting
8.2.1 UV dimension of fields and interaction vertices
8.2.2 Vertex functions: Power counting, superficial degree of divergence
8.3 Classification of interactions in scalar quamtum field theories
8.3.1 Classification of vertices
8.3.2 Classification of field theories
8.4 Momentum regularization
8.4.1 Effective field theory: Regularization
8.4.2 Terms quadratic in the fields with higher derivatives
8.4.3 Regulator fields
8.5 Example: The φ3d=6 field theory at one-loop order
8.5.1 Perturbation theory at one-loop order
8.5.2 Analysis of the divergences at one-loop order
8.5.3 Universal properties of Yang–Lee’s edge singularity
8.6 Operator insertions: Generating functionals, power counting
8.7 Lattice regularization. Classical statistical physics
8.8 Effective QFT. The fine-tuning problem
8.8.1 Effective action and perturbative assumption
8.8.2 Gaussian renormalization, dimensional analysis
8.8.3 The quadratic action and the fine-tuning problem
8.9 The emergence of renormalizable field theories
8.9.1 Non-renormalizable interactions: The example of four dimensions
A8 Technical details
A8.1 Schwinger’s proper-time representation
A8.2 Regularization and one-loop divergences
A8.3 More general momentum regularizations
9 Introduction to renormalization theory and renormalization group (RG)
9.1 Power counting. Dimensional analysis
9.2 Regularization. Bare and renormalized QFT
9.2.1 Bare and renormalized action: Counter-terms
9.2.2 Bare and renormalized correlation functions
9.3 One-loop divergences
9.4 Divergences beyond one-loop: Skeleton diagrams
9.5 Callan–Symanzik equations
9.6 Inductive proof of renormalizability
9.6.1 Coefficients of the CS equation
9.6.2 The 〈φφφφ〉 vertex function (l = 0, n = 4)
9.6.3 The 〈φ2φφ〉 vertex function (l = 1, n = 2)
9.6.4 The 〈φφ〉 vertex function (l = 0, n = 2)
9.6.5 The large momentum behaviour of superficially divergent vertex functions
9.7 The 〈φ2φ2〉 vertex function
9.8 The renormalized action: General construction
9.9 The massless theory
9.9.1 Large momentum behaviour and massless theory
9.9.2 RG equations in a massless QFT
9.10 Homogeneous RG equations: Massive QFT
9.10.1 Covariance of RG functions
9.11 EFT and RG
9.11.1 Bare (or microscopic) and renormalized vertex functions
9.11.2 Bare or asymptotic microscopic RG equations
9.12 Solution of bare RG equations: The triviality issue
9.12.1 RG functions at leading order
9.12.2 The triviality issue
A9 Functional RG equations. Super-renormalizable QFTs. Normal order
A9.1 Large-momentum mode integration and functional RG equations
A9.1.1 A basic equivalence
A9.1.2 Large-momentum mode partial integration and RG equations
A9.2 The φ4 QFT in three dimensions: Divergences
A9.3 Super-renormalizable scalar QFTs in two dimensions: Normal order
10 Dimensional continuation, regularization, minimal subtraction (MS). Renormalization group (RG) functions
10.1 Dimensional continuation and dimensional regularization
10.1.1 Dimensional continuation
10.1.2 Dimensional regularization: Defining properties of d-dimensional integrals
10.1.3 Dimensional regularization and UV divergences
10.2 RG functions
10.2.1 The massive φ4 field theory
10.2.2 The massless theory
10.3 The structure of renormalization constants
10.4 MS scheme
10.4.1 RG functions in the MS scheme
10.5 RG functions at two-loop order: The φ4 QFT
10.5.1 The perturbative expansion
10.5.2 Diagrams: Divergences
10.5.3 The renormalization constants
10.5.4 The φ2 insertion
10.5.5 RG functions
10.5.6 The massless theory at fixed dimension d < 4
10.6 Generalization to N - component fields
10.6.1 Renormalization constants
10.6.2 RG equations
A10 Feynman parametrization
11 Renormalization of local polynomials. Short-distance expansion (SDE)
11.1 Renormalization of operator insertions
11.1.1 The φ2(x) insertion
11.1.2 Operators of dimensions 3 and 4
11.1.3 Operator insertion: General case
11.1.4 Operator insertion and effective field theory
11.2 Quantum field equations
11.3 Short-distance expansion of operator products
11.3.1 SDE at leading order
11.3.2 One-loop calculation of the leading coefficient of the SDE
11.4 Large-momentum expansion of the SDE coefficients: CS equations
11.5 SDE beyond leading order. General operator product
11.6 Light-cone expansion of operator products
12 Relativistic fermions: Introduction
12.1 Massive Dirac fermions
12.1.1 The free massive Dirac fermion
12.1.2 Hamiltonian, spectrum, and particle content
12.1.3 Interacting theory and the S-matrix
12.2 Self-interacting massive fermions: Non-relativistic limit
12.3 Free Euclidean relativistic fermions
12.3.1 Hermitian conjugation
12.3.2 Spin group and reflections
12.4 Partition function. Correlations
12.5 Generating functionals
12.5.1 Boson–fermion action: An example
12.5.2 One-loop calculation
12.6 Connection between spin and statistics
12.7 Divergences and momentum cut-off
12.7.1 Momentum cut-off regularization
12.7.2 Regulator fields
12.8 Dimensional regularization
12.9 Lattice fermions and the doubling problem
12.9.1 Ginsparg–Wilson relation and overlap fermion
A12 Euclidean fermions, spin group and γ matrices
A12.1 Spin group. Dirac γ matrices
A12.1.1 Clifford algebra. Orthogonal groups
A12.1.2 Clifford algebra and group structure
A12.1.3 Spin group and Lie algebra
A12.1.4 The γ matrices: A Hermitian representation
A12.1.5 Spin group: A unitary representation
A12.1.6 Reflections and chiral components
A12.1.7 Charge conjugation
A12.2 The example of dimension 4
A12.3 The Fierz transformation
A12.4 Traces of products of γ matrices
13 Symmetries, chiral symmetry breaking, and renormalization
13.1 Lie groups and algebras: Preliminaries
13.1.1 Orthogonal and unitary representations: Conventions and notation
13.2 Linear global symmetries and WT identities
13.2.1 WT identities
13.3 Linear symmetry breaking
13.3.1 The O(N) symmetric example with linear breaking
13.3.2 Renormalized action and symmetry to all orders
13.4 Spontaneous symmetry breaking
13.4.1 Classical analysis: The O(N) example
13.4.2 General continuous symmetry group
13.4.3 WT identities and SSB
13.5 Chiral symmetry breaking in strong interactions: Effective theory
13.5.1 The chiral symmetry: General structure
13.6 The linear ˙-model
13.6.1 The scalar boson sector
13.6.2 The boson-fermion model
13.6.3 Beyond the tree approximation
13.6.4 Renormalization group: β-functions and triviality issue
13.7 WT identities
13.7.1 Scalar model
13.7.2 Full boson–fermion model
13.8 Quadratic symmetry breaking
13.8.1 Discrete symmetries
A13 Currents and Noether’s theorem
A13.1 Currents in classical-field theory
A13.2 The energy–momentum tensor
A13.3 Euclidean theory: Dilatation and conformal invariance
A13.3.1 The conformal group
A13.4 QFT: Currents and correlation functions
A13.5 Energy-momentum tensor and QFT
14 Critical phenomena: General considerations. Mean-field theory (MFT)
14.1 The transfer matrix
14.1.1 A few properties of the transfer matrix on a finite transverse lattice
14.2 The infinite transverse size limit: Ising-like systems
14.2.1 Order parameter and cluster properties
14.3 Continuous symmetries
14.4 Mean-field approximation
14.4.1 Ising-like ferromagnetic systems
14.4.2 Mean-field approximation and steepest descent method
14.4.3 Thermodynamic potential and phase transition
14.5 Universality within mean-field approximation
14.5.1 Homogeneous observables
14.5.2 The two-point correlation function
14.5.3 Continuous symmetries
14.5.4 Landau’s theory of critical phenomena
14.6 Beyond the mean-field approximation
14.6.1 Perturbative expansion: The two-point function at one-loop order
14.6.2 The role of dimension 4
14.7 Power counting and the role of dimension 4
14.8 Tricritical points
A14 Additional considerations
A14.1 High-temperature expansion
A14.2 Mean-field approximation: General formalism
A14.2.1 Mean-field approximation
A14.3 Mean-field expansion
A14.4 High-, low-temperature, and mean-field expansions
A14.4.1 High temperature
A14.4.2 Low temperature expansion
A14.5 Quenched averages
15 The renormalization group (RG) approach: The critical theory near four dimensions
15.1 RG: The general idea
15.1.1 The RG idea: Fixed points
15.1.2 Hamiltonian flows. Scaling operators
15.1.3 Classification of eigenvectors or scaling fields
15.1.4 Explicit RG equations for correlation functions
15.2 The Gaussian fixed point
15.2.1 Eigenoperators
15.2.2 Beyond the Gaussian fixed point
15.3 Critical behaviour: The effective φ4 field theory
15.4 RG equations near four dimensions
15.5 Solution of the RG equations: The ε-expansion
15.6 Critical correlation functions with φ2(x) insertions
15.6.1 RG equations
15.6.2 Renormalization theory and critical phenomena
15.7 The O(N)-symmetric (˚2)2 field theory
15.8 Statistical properties of long self-repelling chains
15.8.1 A generating function
15.8.2 Some exact results. Flory’s approximation
15.8.3 Equivalence with the (˚2)2 field theory in the N = 0 limit
15.8.4 RG approach to self-avoiding walk (SAW) and statistical properties of polymers
15.9 Liquid–vapour phase transition and φ4 field theory
15.9.1 The classical gas in the continuum: Field integrals
15.9.2 Phase transition
15.10 Superfluid transition
15.10.1 The Bose partition function
15.10.2 Critical properties: Beyond the Gaussian approximation
15.10.3 Superfluid transition and Bose–Einstein condensation
16 Critical domain: Universality, ε-expansion
16.1 Strong scaling above Tc: The renormalized theory
16.1.1 Microscopic and renormalized coupling constants
16.1.2 Strong scaling
16.1.3 Large momentum behaviour
16.2 Critical domain: Homogeneous RG equations
16.3 Scaling properties above Tc
16.4 Correlation functions with φ2 insertions
16.5 Scaling properties in a magnetic field and below Tc
16.5.1 The equation of state
16.5.2 Properties of the universal function f(x)
16.5.3 Correlation functions for non-vanishing magnetization
16.5.4 Correlation functions in zero field below Tc: Spontaneous symmetry breaking
16.6 The N-vector model
16.6.1 The O(N) symmetric N-vector model: IR fixed point
16.6.2 Correlation functions in a field or below Tc
16.7 The general N-vector model
16.7.1 RG equations
16.7.2 Stability of the O(N) symmetric fixed point
16.7.3 Gradient flow
16.8 Asymptotic expansion of the two-point function
16.8.1 Asymptotic expansion from SDE
16.8.2 Next to leading terms in a field or below Tc
16.9 Some universal quantities as ε expansions
16.9.1 RG functions. Critical exponents
16.9.2 The scaling equation of state
16.9.3 Parametric representation of the equation of state
16.9.4 Amplitude ratios
16.10 Conformal bootstrap
17 Critical phenomena: Corrections to scaling behaviour
17.1 Corrections to scaling: Generic dimensions
17.2 Logarithmic corrections at the upper-critical dimension
17.3 Irrelevant operators and the question of universality
17.4 Corrections coming from irrelevant operators. Improved action
17.4.1 Corrections to scaling
17.4.2 Fixed point in Hamiltonian space and improved action
17.5 Application: Uniaxial systems with strong dipolar forces
18 O(N) - symmetric vector models for N large
18.1 The large N action
18.2 Large N limit: Saddle point equations, critical domain
18.2.1 Saddle point equations in zero field
18.2.2 The (φ2)2 field theory: Phases and exponents
18.2.3 Singular free energy and scaling equation of state
18.2.4 The φ2 two-point function
18.3 Renormalization group (RG) functions and leading corrections to scaling
18.4 Small-coupling constant, large-momentum expansions for d < 4
18.5 Dimension 4: Triviality issue for N large
18.6 The (ϕ2)2 field theory and the non-linear σ-model for N large
18.6.1 The non-linear σ-model
18.6.2 The large N limit
18.6.3 Renormalization group
18.6.4 Two dimensions: Critical domain. Borel summability.
18.6.5 Corrections to scaling and the dimension 4
18.6.5 Corrections to scaling and the dimension 4
18.6.6 Higher orders in the large N expansion: Power counting
18.7 The 1/N-expansion: An alternative field theory
18.8 Explicit calculations
18.8.1 Critical exponents at order 1/N
18.8.2 Higher order results
19 The non-linear σ-model near two dimensions: Phase structure
19.1 The non-linear σ-model: Definition
19.2 Perturbation theory. Power counting
19.3 IR divergences
19.4 UV regularization
19.4.1 Perturbative regularizations
19.4.2 Lattice regularization and statistical physics
19.5 WT identities and master equation
19.5.1 WT identities
19.5.2 Master equation
19.6 Renormalization
19.7 The renormalized action: Solution to the master equation
19.7.1 The solution of the master equation for d = 2
19.7.2 Linearized WT identities
19.8 Renormalization of local functionals
19.9 A linear representation
19.10 (ϕ2)2 field theory in the ordered phase and non-linear σ-model
19.10.1 Lattice spin models
19.10.2 The (ϕ2)2 field theory in the ordered phase
19.10.3 Spontaneous symmetry breaking: The role of dimension 2
19.11 Renormalization, RG equations
19.12 RG equations: Solutions (magnetic terminology)
19.12.1 Critical temperature and exponents, for d = 2 + ε, N > 2
19.12.2 Integration of the RG equations: d > 2, t < tc
19.12.3 Scaling of correlation functions, the critical domain
19.13 Results beyond one-loop order
19.14 The dimension 2: Asymptotic freedom
20 Gross–Neveu–Yukawa and Gross–Neveu models
20.1 The GNY model: Spontaneous mass generation
20.1.1 Renormalization and RG near four dimensions
20.1.2 One-loop calculations
20.1.3 The two-point functions
20.1.4 Three and four-point functions
20.2 RG equations near four dimensions
20.2.1 Renormalization group functions
20.2.2 Bare RG equations: Triviality and mass ratio
20.2.3 The general RG flow at one-loop order, in four dimensions
20.2.4 Dimension d = 4 − ε
20.2.5 RG functions beyond one-loop order
20.3 The GNY model in the large N limit
20.3.1 GN and GNY models
20.4 The large N expansion
20.5 The GN model
20.5.1 RG equations near and in two dimensions
20.5.2 Discussion
21 Abelian gauge theories: The framework of quantumel ectrodynamics (QED)
21.1 The free massive vector field: Quantization
21.1.1 The classical action: Degrees of freedom
21.1.2 Field integral quantization
21.2 The Euclidean free action. The two-point function
21.2.1 More general propagators, interpretation
21.3 Coupling to matter
21.4 The massless limit: Gauge invariance
21.4.1 Massless vector field and matter
21.4.2 The massless vector field as a limit
21.5 Massless vector field, gauge invariance, and quantization
21.5.1 Coulomb’s gauge
21.5.2 The temporal (or Weyl’s) gauge
21.6 Equivalence with covariant quantization
21.6.1 Interpretation: The Faddeev–Popov quantization
21.7 Gauge symmetry and parallel transport
21.8 Perturbation theory: Regularization
21.8.1 Momentum cut-off regularization
21.8.2 Lattice regularization
21.9 WT identities and renormalization
21.10 Gauge dependence: The fermion two-point function
21.11 Renormalization and RG equations
21.12 One-loop β function and the triviality issue
21.12.1 Charged Dirac fermions
21.12.2 The sign of the β-function
21.12.3 Charged scalar boson fields
21.13 The Abelian Landau–Ginzburg–Higgs model
21.13.1 Classical approximation
21.13.2 Quantization: Unitary gauge
21.13.3 Renormalizable gauge
21.14 The Landau–Ginzburg–Higgs model: WT identities
21.15 Spontaneous symmetry breaking: Decoupling gauge
21.16 Physical observables. Unitarity of the S-matrix
21.17 Stochastic quantization: The example of gauge theories
A21 Additional remarks
A21.1 Vacuum energy and Casimir effect
A21.1.1 The free electromagnetic field: Vacuum energy
A21.1.2 Casimir effect
A21.2 Gauge dependence
A21.3 Divergences at one loop from Schwinger’s representation
22 Non-Abelian gauge theories: Introduction
22.1 Geometric construction: Parallel transport
22.2 Gauge-invariant actions
22.3 Hamiltonian formalism. Quantization in the temporal gauge
22.3.1 Classical field equations
22.3.2 Temporal or Weyl’s gauge
22.4 Covariant gauges
22.4.1 BRST symmetry
22.5 Perturbation theory, regularization
22.5.1 Feynman rules in the Fourier representation
22.5.2 Regularization
22.6 The non-Abelian Higgs mechanism
22.6.1 Simple compact Lie symmetry groups
22.6.2 The G × G symmetry with simple compact Lie groups
22.6.3 The SU(2) × SU(2) example
22.6.4 Gauge fixing of the Higgs model in a covariant gauge
22.6.5 Renormalization
A22 Massive Yang–Mills fields
23 The Standard Model (SM) of fundamental interactions
23.1 Weak and electromagnetic interactions: Gauge and scalar fields
23.1.1 Gauge and scalar fields: The action
23.1.2 The Higgs mechanism: Classical approximation
23.2 Leptons: Minimal SM extension with Dirac neutrinos
23.2.1 Lepton gauge action
23.2.2 Lepton masses
23.2.3 The Fermi constant
23.3 Quarks and weak–electromagnetic interactions
23.3.1 Elementary scalar fields: Parameter proliferation and fine tuning
23.4 QCD. RG equations and β function
23.4.1 QCD
23.4.2 Semi-classical vacuum and fine-tuning problem
23.4.3 RG equations in a covariant gauge
23.5 General RG β-function at one-loop order: Asymptotic freedom
23.5.1 The RG β-function at one-loop order
23.5.2 AF
23.6 Axial current, chiral gauge theories, and anomalies
23.6.1 Abelian axial current and Abelian vector gauge field
23.6.2 Regulator fields and explicit anomaly calculation
23.6.3 Non-Abelian vector gauge theories and Abelian axial current
23.6.4 Anomaly and index of the Dirac operator
23.6.5 Non-Abelian anomaly: General axial current
23.6.6 Obstruction to gauge invariance
23.6.7 Wess–Zumino consistency conditions
23.7 Anomalies: Applications in particle physics
23.7.1 Weak–electromagnetic interactions: Anomaly cancellation
23.7.2 Electromagnetic π0 decay
23.7.3 The solution of the U(1) problem
24 Large-momentum behaviour in quantum field theory (QFT)
24.1 The (ϕ2)2 Euclidean field theory: Large-momentum behaviour
24.1.1 Dimensions d < 4
24.1.2 The renormalized (ϕ2)2 field theory for d = 4: The triviality issue
24.1.3 The φ44field theory for negative renormalized coupling
24.2 General ˚4-like field theories: d=4
24.3 Theories with scalar bosons and Dirac fermions
24.4 Gauge theories
24.5 Applications: The theory of strong interactions
25 Lattice gauge theories: Introduction
25.1 Gauge invariance on the lattice: Parallel transport
25.1.1 Parallel transport on the lattice
25.2 The matterless gauge theory
25.2.2 Low-temperature analysis
25.3 Wilson’s loop and confinement
25.3.1 Wilson’s loop in continuum Abelian gauge theories
25.3.2 Non-Abelian gauge theories
25.4 Mean-field approximation
25.4.1 Fermions
A25 Gauge theory and confinement in two dimensions
26 Becchi–Rouet–Stora–Tyutin (BRST) symmetry. Gauge theories: Zinn-Justin equation (ZJ) and renormalization
26.1 ST identities: The origin
26.1.1 A simple identity
26.1.2 An invariant measure
26.2 From ST symmetry to BRST symmetry
26.2.1 BRST symmetry
26.3 BRST symmetry: More general coordinates. Group structure
26.3.1 Group manifolds and gauge invariance
26.3.2 Direct construction
26.3.3 Generalization and compatibility conditions
26.4 Stochastic equations
26.4.1 A simple example: Stochastic equations linear in the noise
26.5 BRST symmetry, Grassmann coordinates, and gradient equations
26.5.1 BRST symmetry and Grassmann coordinates
26.5.2 Gradient equations and Grassmann coordinates
26.6 Gauge theories: Notation and algebraic structure
26.7 Gauge theories: Quantization
26.7.1 Quantization and ST symmetry
26.7.2 Quantization and BRST symmetry
26.7.3 BRST transformations
26.7.4 BRST differential operator and BRST symmetry
26.8 WT identities and ZJ equation
26.9 Renormalization: General considerations
26.9.1 Counter-terms and ZJ equation
26.10 The renormalized gauge action
26.10.1 General renormalizable gauges: Power counting
26.10.2 From the ZJ equation back to BRST symmetry
26.10.3 BRST symmetry and renormalized action
26.10.4 Comments
26.10.5 Linear gauges
26.11 Gauge independence: Physical observables
A26 BRST symmetry and ZJ equation: Additional remarks
A26.1 BRST symmetry and ZJ equation
A26.2 Canonical invariance of the ZJ equation
A26.3 Elements of BRST cohomology
A26.4 From BRST symmetry to supersymmetry
27 Supersymmetric quantum field theory (QFT): Introduction
27.1 Scalar superfields in three dimensions
27.1.1 Supersymmetry and Majorana spinors in d = 3
27.1.2 Superfields and covariant derivatives
27.1.3 Supersymmetry generators and Ward–Takahashi (WT) identities
27.1.4 General O(N)-symmetric action
27.1.5 Perturbative expansion
27.2 The O(N) supersymmetric non-linear σ model
27.3 Supersymmetry in four dimensions
27.3.1 Grassmann coordinates and supersymmetry
27.3.2 Scalar chiral superfields
27.4 Vector superfields and gauge invariance
27.4.1 Abelian gauge invariance and supersymmetry
27.4.2 Supersymmetric curvature tensor
28 Elements of classical and quantum gravity
28.1 Manifolds. Change of coordinates. Tensors
28.1.1 Fields on manifolds: Classification
28.1.2 Infinitesimal change of coordinates
28.1.3 Differential forms
28.2 Parallel transport: Connection, covariant derivative
28.2.1 Parallel transport
28.2.2 Affine connection
28.2.3 The covariant derivative
28.3 Riemannian manifold. The metric tensor
28.3.1 Covariant volume element
28.4 The curvature (Riemann) tensor
28.4.1 Curvature tensor and metric
28.4.2 Holonomy variables, holonomy group
28.5 Fermions, vielbein, spin connection
28.5.1 Local frame: Vielbein
28.5.2 Gauge invariance and spin connection
28.6 Classical GR. Equations of motion
28.6.1 The classical action
28.6.2 Classical equation of motion
28.7 Quantization in the temporal gauge: Pure gravity
28.7.1 The action in the temporal or Weyl gauge
28.7.2 Quantization: A few remarks
28.8 Observational cosmology: A few comments
28.8.1 An emerging model of the Universe and its evolution
29 Generalized non-linear σ-models in two dimensions
29.1 Homogeneous spaces and Goldstone modes
29.1.1 Spontaneous symmetry breaking, Goldstone modes, and homogeneous spaces
29.1.2 Goldstone mode effective action
29.1.3 Metric and action in general coordinates
29.2 WT identities and renormalization in linear coordinates
29.2.1 Linear coordinates
29.2.2 Correlation functions, WT identities
29.2.3 Renormalization
29.2.4 Field renormalizations
29.3 Renormalization in general coordinates: BRST symmetry
29.3.1 Infinitesimal group transformations
29.3.2 WT identities
29.3.3 The renormalized action
29.4 Symmetric spaces: Definition
29.5 Classical field equations. Conservation laws
29.5.1 Non-local conserved currents
29.6 QFT: Perturbative expansion and RG
29.6.1 RG functions at one-loop order
29.6.2 One-loop β-function and background field method
29.7 Generalizations
A29 Homogeneous spaces: A few algebraic properties
A29.1 Pure gauge. Maurer–Cartan equations
A29.2 Metric and curvature in homogeneous spaces
A29.3 Explicit expressions for the metric
A29.4 Symmetric spaces: Classification
A29.4.1 Definition
A29.4.2 A basic property
A29.4.3 The principal chiral models
30 A few solvable two-dimensional quantum field theories (QFT)
30.1 The free massless scalar field
30.1.1 m = 0: IR finite correlation functions
30.1.2 Symmetries and currents
30.1.3 Complex coordinates
30.2 The free massless Dirac fermion
30.2.1 Fermion correlation functions
30.2.2 Bilinear operator correlations and boson–fermion correspondence
30.2.3 The massive free fermion and the sG model
30.3 The gauge-invariant fermion determinant and the anomaly
30.3.1 Current correlation functions
30.4 The sG model
30.4.1 Perturbative expansion
30.4.2 RG equations
30.5 The Schwinger model
30.5.1 The massless model
30.5.2 Confinement and chiral symmetry breaking
30.5.3 The massive Schwinger model
30.6 The massive Thirring model
30.6.1 Bosonization: The sG model
30.6.2 The massless Thirring model
30.6.3 RG properties. Mass spectrum
30.7 A generalized Thirring model with two fermions
30.7.1 The model
30.7.2 Derivation
30.8 The SU(N) Thirring model
30.8.1 Derivation
A30 Two-dimensional models: A few additional results
A30.1 Four-fermion current interactions: RG β-function
A30.2 The Schwinger model: The anomaly
A30.2.1 The general anomaly
A30.2.2 The two-point function: One-loop calculation
A30.3 Solitons in the sG model
31 O(2) spin model and the Kosterlitz–Thouless’s (KT) phase transition
31.1 The spin correlation functions at low temperature
31.2 Correlation functions in a field
31.3 The Coulomb gas in two dimensions
31.3.1 Coulomb gas and sine-Gordon field theory
31.3.2 Renormalization and RG
31.3.3 The correlation length near the phase transition
31.4 O(2) spin model and Coulomb gas
31.5 The critical two-point function in the O(2) model
31.6 The generalized Thirring model
32 Finite-size effects in field theory. Scaling behaviour
32.1 RG in finite geometries
32.1.1 The (ϕ2)2 field theory for d < 4
32.1.2 Low-temperature expansion and finite-size effects
32.2 Momentum quantization
32.2.1 Periodic boundary conditions and the zero mode
32.2.2 Twisted boundary conditions
32.3 The φ4 field theory in a periodic hypercube
32.3.1 Dimension d > 4
32.3.2 Higher order corrections
32.3.3 Dimension d = 4 − ε
32.4 The φ4 field theory: Cylindrical geometry
32.4.1 Correlation length: Dimensions d > 4
32.4.2 Dimensions d = 4 − ε
32.5 Finite size effects in the non-linear σ-model
32.5.1 The hypercubic geometry
32.5.2 The cylindrical geometry
A32 Additional remarks
A32.1 Perturbation theory in a finite volume
A32.2 Discrete symmetries and finite-size effects
A32.2.1 Finite volume
A32.2.2 Finite size correlation length in Ising-like systems below Tc
33 Quantum field theory (QFT) at finite temperature: Equilibrium properties
33.1 Finite- (and high-) temperature field theory
33.1.1 Finite temperature QFT
33.1.2 The role of the zero mode, dimensional reduction
33.1.3 The EFT
33.2 The example of the φ41,d−1 field theory
33.2.1 RG at finite temperature
33.2.2 One-loop effective action
33.3 High temperature and critical limits
33.3.1 Dimension d = 5
33.3.2 Dimension d ≤ 4
33.4 The non-linear σ-model in the large N limit
33.4.1 The large N limit: Finite temperature saddle point equations
33.4.2 Dimension d = 3
33.4.3 Dimensions d > 3: Critical and high temperatures
33.5 The perturbative non-linear σ-model at finite temperature
33.5.1 RG equations at finite temperature
33.5.2 Dimensional reduction at one-loop order
33.5.3 Matching conditions
33.6 The GN model in the large N expansion
33.6.1 The gap equation
33.6.2 Scalar mass
33.7 Abelian gauge theories: The QED framework
33.7.1 Mode expansion and gauge transformations
33.7.2 Gauge field coupled to fermions: Quantization
33.7.3 Dimensional reduction
33.7.4 The Abelian Higgs model
33.8 Non-Abelian gauge theories
33.8.1 Quantization and mode expansion
33.8.2 Dimensional reduction
A33 Feynman diagrams at finite temperature
A33.1 One-loop calculations
A33.1.1 General remarks
A33.1.2 Γ, ψ, ζ, θ-functions: A few useful identities
A33.2 Group measure
34 Stochastic differential equations: Langevin, Fokker–Planck (FP) equations
34.1 The Langevin equation
34.1.1 The discretized Langevin equation
34.2 Time-dependent probability distribution and FP equation
34.2.1 Markov property and FP Hamiltonian
34.2.2 The FP equation
34.3 Equilibrium distribution. Correlation functions
34.3.1 Equilibrium distribution
34.3.2 Correlation functions
34.3.3 Time evolution of observables
34.4 A special class: Dissipative Langevin equations
34.4.1 The FP Hamiltonian
34.4.2 Detailed balance
34.5 The linear Langevin equation
34.6 Path integral representation
34.7 BRST and supersymmetry
34.7.1 Dynamic action
34.7.2 Grassmann coordinates: Supersymmetry
34.8 Gradient time-dependent force and Jarzynski’s relation
34.9 More general Langevin equations. Motion in Riemannian manifolds
34.9.1 The FP equation
A34 Markov’s stochastic processes: A few remarks
A34.1 Discrete spaces: Markov’s processes, phase transitions
A34.1.1 Evolution or master equation
A34.1.2 The spectrum of the transition matrix
A34.1.3 Detailed balance
A34.2 Stochastic process with prescribed equilibrium distribution
A34.3 Stochastic processes and phase transitions
35 Langevin field equations: Properties and renormalization
35.1 Langevin and Fokker–Planck (FP) equations
35.2 Time-dependent correlation functions and equilibrium
35.2.1 Dynamic action
35.2.2 Slavov–Taylor symmetry and equilibrium correlation functions
35.3 Renormalization and BRST symmetry
35.3.1 Power counting analysis
35.3.2 BRST symmetry
35.3.3 Power counting and renormalization
35.4 Dissipative Langevin equation and supersymmetry
35.4.1 Supersymmetry
35.4.2 WT identities
35.4.3 Renormalization of the dissipative Langevin equation
35.5 Supersymmetry and equilibrium correlation functions
35.6 Stochastic quantization of two-dimensional chiral models
35.7 Langevin equation and Riemannian manifolds
A35 The random field Ising model: Supersymmetry
36 Critical dynamics and renormalization group (RG)
36.1 Dissipative equation: RG equations in dimension d = 4 − ε
36.1.1 Supersymmetry and the fluctuation–dissipation theorem
36.1.2 RG equations at Tc for d < 4
36.1.3 Correlation functions above Tc, in the critical domain
36.2 Dissipative dynamics: RG equations in dimension d = 2 + ε
36.3 Conserved order parameter
36.4 Relaxational model with energy conservation
36.5 A non-relaxational model
36.6 Finite size effects and dynamics
36.6.1 The (ϕ2)2 field theory
36.6.2 The non-linear σ-model: The bare RG
36.6.3 Dynamics in the ordered phase
A36 RG functions at two loops
A36.1 Supersymmetric perturbative calculations at two loops
A36.1.2 The non-linear σ-model
37 Instantons in quantum mechanics (QM)
37.1 The quartic anharmonic oscillator for negative coupling
37.2 A toy model: A simple integral
37.3 QM: Instantons
37.4 Instanton contributions at leading order
37.4.1 Collective coordinates and Gaussian integration
37.4.2 The result at leading order
37.5 General analytic potentials: Instanton contributions
37.6 Evaluation of the determinant: The shifting method
37.6.1 The shifting method
37.6.2 The partition function
37.7 Zero temperature limit: The ground state
A37 Exact Jacobian. WKB method.
A37.1 The exact Jacobian
A37.1.1 Example: Time translation
A37.1.2 Time translation and O(N) internal rotations
A37.2 The WKB method
38 Metastable vacua in quantum field theory (QFT)
38.1 The φ4 QFT for negative coupling
38.1.1 Instantons: Classical solutions and classical action
38.1.2 The Gaussian integration for d < 4
38.1.3 UV divergences and renormalization for d < 4
38.2 General potentials: Instanton contributions
38.2.1 Calculation of the instanton contribution
38.3 The ˚4 QFT in dimension 4
38.4 Instanton contributions at leading order
38.4.1 The Jacobian
38.4.2 The determinant
38.5 Coupling constant renormalization
38.6 The imaginary part of the n-point function
38.7 The massive theory
38.8 Cosmology: The decay of the false vacuum
A38 Instantons: Additional remarks
A38.1 Virial theorem
A38.2 Sobolev inequalities
A38.3 Instantons and RG equations
A38.4 Conformal invariance
39 Degenerate classical minima and instantons
39.1 The quartic double-well potential
39.1.1 The structure of the ground state
39.1.2 Instanton contributions
39.2 The periodic cosine potential
39.2.1 The structure of the ground state
39.2.2 The instanton contributions
39.3 Instantons and stochastic dynamics
39.3.1 Random walk
39.3.2 Quantum field theory (QFT)
39.4 Instantons in stable boson field theories: General remarks
39.5 Instantons in CP(N − 1) models
39.5.1 The semi-classical vacuum: Temporal gauge
39.6 Instantons in the SU(2) gauge theory
A39 Trace formula for periodic potentials
40 Large order behaviour of perturbation theory
40.1 QM
40.1.1 Real instantons
40.1.2 Complex instantons
40.1.3 Degenerate classical minima
40.2 Scalar field theories: The example of the ˚4 field theory
40.3 The ( 2)2 field theory in dimension 4 and 4 − ε
40.3.1 Semi-classical contribution
40.3.2 Ultraviolet (UV) and infrared (IR) (renormalons) contributions
40.3.3 Wilson–Fisher’s ε-expansion
40.4 Field theories with fermions
40.4.1 Example of a Yukawa-like QFT
40.4.2 Evaluation of the fermion determinant for large fields
40.4.3 The large-order behaviour
40.4.4 The example of QED
A40 large-order behaviour: Additional remarks
41 Critical exponents and equation of state from series summation
41.1 Divergent series: Borel summability, Borel summation
41.1.1 Borel summability. Borel summation
41.1.2 Large-order behaviour and Borel summability
41.2 Borel transformation: Series summation
41.3 Summing the perturbative expansion of the (φ2)2 field theory
41.3.1 RG functions and exponents
41.3.2 Equation of state
41.4 Summation method: Practical implementation
41.5 Field theory estimates of critical exponents for the O(N) model
41.5.1 Dimension 2
41.5.2 Dimension 3
41.6 Other three-dimensional theoretical estimates
41.7 Critical exponents from experiments
41.8 Amplitude ratios
A41 Some other summation methods
A41.1 Order-dependent mapping method (ODM)
A41.2 Linear differential approximants
42 Multi-instantons in quantum mechanics (QM)
42.1 The quartic double-well potential
42.1.1 The two-instanton configuration
42.1.2 The n-instanton configuration and action
42.1.3 The n-instanton contribution
42.1.4 The calculation
42.2 The periodic cosine potential
42.3 General potentials with degenerate minima
42.3.1 The n-instanton action
42.3.2 The n-instanton contribution
42.3.3 Large-order estimates of perturbation theory
42.4 The O(ν)-symmetric anharmonic oscillator
42.5 Generalized Bohr–Sommerfeld quantization formula
A42 Additional remarks
A42.1 Multi-instantons: The determinant
A42.2 The instanton interaction
A42.2.1 Multi-instantons from constraints
A42.3 A simple example of non-Borel summability
A42.4 Multi-instantons and WKB approximation
A42.4.1 The conjecture
A42.4.2 The WKB expansion
Bibliography
Index