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Mathematics of Quantization and Quantum Fields

Book Description

Unifying a range of topics that are currently scattered throughout the literature, this book offers a unique and definitive review of mathematical aspects of quantization and quantum field theory. The authors present both basic and more advanced topics of quantum field theory in a mathematically consistent way, focusing on canonical commutation and anti-commutation relations. They begin with a discussion of the mathematical structures underlying free bosonic or fermionic fields, like tensors, algebras, Fock spaces, and CCR and CAR representations (including their symplectic and orthogonal invariance). Applications of these topics to physical problems are discussed in later chapters. Although most of the book is devoted to free quantum fields, it also contains an exposition of two important aspects of interacting fields: diagrammatics and the Euclidean approach to constructive quantum field theory. With its in-depth coverage, this text is essential reading for graduate students and researchers in departments of mathematics and physics.

Table of Contents

  1. Cover Page
  2. Half Title
  3. Series
  4. Title Page
  5. Copyright
  6. Dedication
  7. Table of Contents
  8. Introduction
    1. Acknowledgement
  9. 1 Vector spaces
    1. 1.1 Elementary linear algebra
      1. 1.1.1 Vector spaces and linear operators
      2. 1.1.2 2 × 2 block matrices
      3. 1.1.3 Duality
      4. 1.1.4 Annihilator
      5. 1.1.5 Transpose of an operator
      6. 1.1.6 Dual pairs
      7. 1.1.7 Bilinear forms
      8. 1.1.8 Symmetric forms
      9. 1.1.9 (Pseudo-)Euclidean spaces
      10. 1.1.10 Inertia of a symmetric form
      11. 1.1.11 Group O(Y) and Lie algebra o(Y)
      12. 1.1.12 Anti-symmetric forms
      13. 1.1.13 Symplectic spaces
      14. 1.1.14 Group Sp(Y) and Lie algebra sp(Y)
      15. 1.1.15 Involutions and super-spaces
      16. 1.1.16 Conjugations on a symplectic space
    2. 1.2 Complex vector spaces
      1. 1.2.1 Anti-linear operators
      2. 1.2.2 Internal conjugations
      3. 1.2.3 Complex conjugate spaces
      4. 1.2.4 Anti-linear functionals
      5. 1.2.5 Adjoint of an operator
      6. 1.2.6 Anti-dual pairs
      7. 1.2.7 Sesquilinear forms
      8. 1.2.8 Hermitian forms
      9. 1.2.9 (Pseudo-)unitary spaces
      10. 1.2.10 Group U(Ƶ) and Lie algebra u(Ƶ)
      11. 1.2.11 Charged symplectic spaces
    3. 1.3 Complex structures
      1. 1.3.1 Anti-involutions
      2. 1.3.2 Conjugations on a space with an anti-involution
      3. 1.3.3 Complexification
      4. 1.3.4 Complexification of a Euclidean space
      5. 1.3.5 Complexification of a symplectic space
      6. 1.3.6 Holomorphic and anti-holomorphic subspaces
      7. 1.3.7 Operators on a space with an anti-involution
      8. 1.3.8 (Pseudo-)Kähler spaces
      9. 1.3.9 Complexification of a (pseudo-)Kähler space
      10. 1.3.10 Conjugations on a (pseudo-)Kähler space
      11. 1.3.11 Real representations of the group U(1)
    4. 1.4 Groups and Lie algebras
      1. 1.4.1 General linear group and Lie algebra
      2. 1.4.2 Homogeneous linear differential equations
      3. 1.4.3 Affine transformations
      4. 1.4.4 Inhomogeneous linear differential equations
      5. 1.4.5 Exact sequences
      6. 1.4.6 Cayley transform
    5. 1.5 Notes
  10. 2 Operators in Hilbert spaces
    1. 2.1 Convergence and completeness
      1. 2.1.1 Nets
      2. 2.1.2 Functions
      3. 2.1.3 Topological vector spaces
      4. 2.1.4 Infinite sums
      5. 2.1.5 Infinite products
    2. 2.2 Bounded and unbounded operators
      1. 2.2.1 Normed vector spaces
      2. 2.2.2 Scalar product spaces
      3. 2.2.3 Operators on Hilbert spaces
      4. 2.2.4 Product of a closed and a bounded operator
      5. 2.2.5 Compact operators
      6. 2.2.6 Hilbert–Schmidt and trace-class operators
      7. 2.2.7 Fredholm determinant
      8. 2.2.8 Derivatives
    3. 2.3 Functional calculus
      1. 2.3.1 Holomorphic functional calculus
      2. 2.3.2 Functional calculus for normal operators
      3. 2.3.3 Spectrum of the product of operators
      4. 2.3.4 Scale of Hilbert spaces associated with a positive operator
      5. 2.3.5 C0-semi-groups
      6. 2.3.6 Local Hermitian semi-groups
      7. 2.3.7 Essential self-adjointness
      8. 2.3.8 Commuting self-adjoint operators
      9. 2.3.9 Conjugations adapted to a self-adjoint operator
    4. 2.4 Polar decomposition
      1. 2.4.1 Polar decomposition
      2. 2.4.2 Polar decomposition of self-adjoint and anti-self-adjoint operators
      3. 2.4.3 Polar decomposition of symmetric and anti-symmetric operators
    5. 2.5 Notes
  11. 3 Tensor algebras
    1. 3.1 Direct sums and tensor products
      1. 3.1.1 Direct sums
      2. 3.1.2 Direct sums of operators
      3. 3.1.3 Algebraic tensor product
      4. 3.1.4 Tensor product in the sense of Hilbert spaces
      5. 3.1.5 Bases of tensor products
      6. 3.1.6 Operators in tensor products
      7. 3.1.7 Permutations
      8. 3.1.8 Identifications
      9. 3.1.9 Infinite tensor product of grounded Hilbert spaces
      10. 3.1.10 Infinite tensor product of vectors and operators
    2. 3.2 Tensor algebra
      1. 3.2.1 Full Fock space
      2. 3.2.2 Operators dΓ and Γ in full Fock spaces
    3. 3.3 Symmetric and anti-symmetric tensors
      1. 3.3.1 Fock spaces
      2. 3.3.2 Symmetric and anti-symmetric tensor products
      3. 3.3.3 dΓ and Γ operators
      4. 3.3.4 Identifications
      5. 3.3.5 Bases in bosonic Fock spaces
      6. 3.3.6 Bases in fermionic Fock spaces
      7. 3.3.7 Exponential law for Fock spaces
      8. 3.3.8 Dimension of Fock spaces
      9. 3.3.9 Super-Fock spaces
    4. 3.4 Creation and annihilation operators
      1. 3.4.1 Creation and annihilation operators: abstract approach
      2. 3.4.2 Creation and annihilation operators on Fock spaces
      3. 3.4.3 Exponential law for creation and annihilation operators
      4. 3.4.4 Multiple creation and annihilation operators
    5. 3.5 Multi-linear symmetric and anti-symmetric forms
      1. 3.5.1 Polynomials
      2. 3.5.2 Multiplication and differentiation operators
      3. 3.5.3 Right derivative
      4. 3.5.4 Exponential law in the polynomial notation
      5. 3.5.5 Holomorphic continuation of polynomials
      6. 3.5.6 Polynomials on complex spaces
      7. 3.5.7 Modified Fock spaces
    6. 3.6 Volume forms, determinant and Pfaffian
      1. 3.6.1 Volume forms
      2. 3.6.2 Hodge star operator
      3. 3.6.3 Liouville volume forms
      4. 3.6.4 Liouville volume forms on X# ⊕X
      5. 3.6.5 Densities and Lebesgue measures
      6. 3.6.6 Determinants
      7. 3.6.7 Determinant of a bilinear form
      8. 3.6.8 Orientations of vector spaces
      9. 3.6.9 Volume forms on complex spaces
      10. 3.6.10 Pfaffians
    7. 3.7 Notes
  12. 4 Analysis in L2(Rd)
    1. 4.1 Distributions and the Fourier transformation
      1. 4.1.1 Distributions
      2. 4.1.2 Pullback of distributions
      3. 4.1.3 Schwartz functions and distributions
      4. 4.1.4 Derivatives
      5. 4.1.5 Complex derivatives
      6. 4.1.6 Position and momentum operators
      7. 4.1.7 Fourier transformation
      8. 4.1.8 Gaussian integrals
      9. 4.1.9 Gaussian integrals for complex variables
      10. 4.1.10 Convolution operators
      11. 4.1.11 Sesquilinear forms on S(X)
      12. 4.1.12 Hilbert–Schmidt and trace-class operators on L2(X)
    2. 4.2 Weyl operators
      1. 4.2.1 Definition of Weyl operators
      2. 4.2.2 Quantum Fourier transform
      3. 4.2.3 Stone–von Neumann theorem
    3. 4.3 x, D-quantization
      1. 4.3.1 Quantization of polynomial symbols
      2. 4.3.2 Quantization of distributional symbols
    4. 4.4 Notes
  13. 5 Measures
    1. 5.1 General measure theory
      1. 5.1.1 σ-algebras
      2. 5.1.2 Measures
      3. 5.1.3 Pre-measures
      4. 5.1.4 Borel measures and pre-measures
      5. 5.1.5 Integral
      6. 5.1.6 Lp spaces
      7. 5.1.7 Operators on Lp spaces
      8. 5.1.8 Conditional expectations
      9. 5.1.9 Convergence in measure
      10. 5.1.10 Measure preserving transformations
      11. 5.1.11 Relative continuity
      12. 5.1.12 Moments of a measure
    2. 5.2 Finite measures on real Hilbert spaces
      1. 5.2.1 Cylinder sets and cylinder functions
      2. 5.2.2 Finite-dimensional distributions of a measure
      3. 5.2.3 Characteristic functional of a measure
      4. 5.2.4 Moment functions
      5. 5.2.5 Density of exponentials
      6. 5.2.6 Density of continuous polynomials
    3. 5.3 Weak distributions and the Minlos–Sazonov theorem
      1. 5.3.1 Weak distributions
      2. 5.3.2 Weak distributions generated by a measure
      3. 5.3.3 Characteristic functionals of weak distributions
      4. 5.3.4 Minlos–Sazonov theorem
      5. 5.3.5 Measures on enlarged spaces
      6. 5.3.6 Comparison of enlarged spaces
    4. 5.4 Gaussian measures on real Hilbert spaces
      1. 5.4.1 Gaussian measures
      2. 5.4.2 Gaussian measures on enlarged spaces
      3. 5.4.3 Exponential law for Gaussian spaces
      4. 5.4.4 Polynomials in Gaussian spaces
      5. 5.4.5 Relative continuity of Gaussian measures
    5. 5.5 Gaussian measures on complex Hilbert spaces
      1. 5.5.1 Holomorphic and anti-holomorphic functions
      2. 5.5.2 Measures on complex Hilbert spaces
      3. 5.5.3 Gaussian measures on complex spaces
      4. 5.5.4 Generalized Gaussian measures on complex spaces
      5. 5.5.5 Isomorphism with modified Fock spaces
    6. 5.6 Notes
  14. 6 Algebras
    1. 6.1 Algebras
      1. 6.1.1 Associative algebras
      2. 6.1.2 *-algebras
      3. 6.1.3 Algebras generated by symbols and relations
      4. 6.1.4 Super-algebras
    2. 6.2 C*-and W*-algebras
      1. 6.2.1 Banach algebras
      2. 6.2.2 C *-algebras
      3. 6.2.3 Representations of C *-algebras
      4. 6.2.4 Intertwiners and unitary equivalence
      5. 6.2.5 States
      6. 6.2.6 GNS representations
      7. 6.2.7 W*-algebras
      8. 6.2.8 Von Neumann algebras
      9. 6.2.9 UHF algebras
      10. 6.2.10 Hyper-finite type II1 factor
      11. 6.2.11 Conditional expectations
    3. 6.3 Tensor products of algebras
      1. 6.3.1 Tensor product of C *-algebras
      2. 6.3.2 Tensor product of W*-algebras
    4. 6.4 Modular theory
      1. 6.4.1 Standard representations
      2. 6.4.2 Tomita-Takesaki theory
      3. 6.4.3 KMS states
      4. 6.4.4 Type I factors: irreducible representation
      5. 6.4.5 Type I factors: representation on Hilbert-Schmidt
    5. 6.5 Non-commutative probability spaces
      1. 6.5.1 Measurable operators
      2. 6.5.2 Non-commutative Lp spaces
      3. 6.5.3 Operators between non-commutative Lp spaces
      4. 6.5.4 Conditional expectations on non-commutative spaces
    6. 6.6 Notes
  15. 7 Anti-symmetric calculus
    1. 7.1 Basic anti-symmetric calculus
      1. 7.1.1 Functional notation
      2. 7.1.2 Change of variables in anti-symmetric polynomials
      3. 7.1.3 Multiplication and differentiation
      4. 7.1.4 Berezin integrals
      5. 7.1.5 Berezin calculus in coordinates
      6. 7.1.6 Differential operators and convolutions
      7. 7.1.7 Anti-symmetric exponential
      8. 7.1.8 Anti-symmetric Gaussians
    2. 7.2 Operators and anti-symmetric calculus
      1. 7.2.1 Berezin integral on X ⊕ X#
      2. 7.2.2 Operators on the space of anti-symmetric polynomials
      3. 7.2.3 Integral kernel of an operator
      4. 7.2.4 x, ∇x-quantization
    3. 7.3 Notes
  16. 8 Canonical commutation relations
    1. 8.1 CCR representations
      1. 8.1.1 Definition of a CCR representation
      2. 8.1.2 CCR representations over a direct sum
      3. 8.1.3 Cyclicity and irreducibility
      4. 8.1.4 Characteristic functions of CCR representations
      5. 8.1.5 Intertwining operators
      6. 8.1.6 Schrödinger representation
      7. 8.1.7 Weighted Schrödinger representations
      8. 8.1.8 Examples of non-regular CCR representations
      9. 8.1.9 Bogoliubov transformations
    2. 8.2 Field operators
      1. 8.2.1 Definition of field operators
      2. 8.2.2 Common domain of field operators
      3. 8.2.3 Non-self-adjoint fields
      4. 8.2.4 CCR over a Kähler space
      5. 8.2.5 Charged CCR representations
      6. 8.2.6 CCR over a symplectic space with conjugation
      7. 8.2.7 CCR over a Kähler space with conjugation
    3. 8.3 CCR algebras
      1. 8.3.1 Polynomial CCR *-algebras
      2. 8.3.2 Stone-von Neumann CCR algebras
      3. 8.3.3 S- and S'-type operators
      4. 8.3.4 Regular CCR algebras
      5. 8.3.5 Weyl CCR algebra
    4. 8.4 Weyl-Wigner quantization
      1. 8.4.1 Quantization of polynomial symbols
      2. 8.4.2 Quantization of distributional symbols
      3. 8.4.3 Weyl-Wigner quantization in the Schrödinger representation
      4. 8.4.4 Parity operator
    5. 8.5 General coherent vectors
      1. 8.5.1 Coherent states transformation
      2. 8.5.2 Contravariant quantization
      3. 8.5.3 Covariant quantization
      4. 8.5.4 Connections between various quantizations
      5. 8.5.5 Gaussian coherent vectors
    6. 8.6 Notes
  17. 9 CCR on Fock space
    1. 9.1 Fock CCR representation
      1. 9.1.1 Field operators on Fock spaces
      2. 9.1.2 Weyl operators on Fock spaces
      3. 9.1.3 Exponentials of creation and annihilation operators
      4. 9.1.4 Gaussian coherent vectors on Fock spaces
    2. 9.2 CCR on anti-holomorphic Gaussian L2 spaces
      1. 9.2.1 Bosonic complex-wave representation
      2. 9.2.2 Coherent vectors in the complex-wave representation
    3. 9.3 CCR on real Gaussian L2 spaces
      1. 9.3.1 Real-wave CCR representation
      2. 9.3.2 Real-wave CCR representation in finite dimension
      3. 9.3.3 Wick transformation
      4. 9.3.4 Integrals of polynomials with a Gaussian weight
      5. 9.3.5 Operators in the real-wave representation
    4. 9.4 Wick and anti-Wick bosonic quantization
      1. 9.4.1 Wick and anti-Wick ordering
      2. 9.4.2 Relation between Wick, anti-Wick and Weyl–Wigner quantizations
      3. 9.4.3 Wick and anti-Wick quantization as covariant and contravariant quantization
      4. 9.4.4 Wick symbols on Fock spaces
      5. 9.4.5 Wick quantization: the operator formalism
      6. 9.4.6 Estimates on Wick polynomials
      7. 9.4.7 Bargmann kernel of an operator
      8. 9.4.8 Link between the two Wick operations
    5. 9.5 Notes
  18. 10 Symplectic invariance of CCR in finite-dimensions
    1. 10.1 Classical quadratic Hamiltonians
      1. 10.1.1 Symplectic transformations
      2. 10.1.2 Poisson bracket
      3. 10.1.3 Spectrum of symplectic transformations
      4. 10.1.4 Poisson bracket on charged symplectic spaces
    2. 10.2 Quantum quadratic Hamiltonians
      1. 10.2.1 Commutation properties of quadratic Hamiltonians
      2. 10.2.2 Infimum of positive quadratic Hamiltonians
      3. 10.2.3 Scale of oscillator spaces
      4. 10.2.4 Quadratic Hamiltonians as closed operators
      5. 10.2.5 One-parameter groups of Bogoliubov *-automorphisms
    3. 10.3 Metaplectic group
      1. 10.3.1 Implementation of Bogoliubov transformations
      2. 10.3.2 Semi-groups generated by quadratic Hamiltonians
      3. 10.3.3 M p(Y) as the two-fold covering of Sp(Y)
    4. 10.4 Symplectic group on a space with conjugation
      1. 10.4.1 Symplectic transformations on a space with conjugation
      2. 10.4.2 Generating function of a symplectic transformation
      3. 10.4.3 Point transformations
      4. 10.4.4 Transformations fixing X#
      5. 10.4.5 Transformations fixing X
      6. 10.4.6 Harmonic oscillator
      7. 10.4.7 Transformations swapping X and X#
    5. 10.5 Metaplectic group in the Schrödinger representation
      1. 10.5.1 Metaplectic group in L2(R)
      2. 10.5.2 Harmonic oscillator
      3. 10.5.3 Quadratic Hamiltonians in the Schrödinger representation
      4. 10.5.4 Integral kernel of elements of the metaplectic group
    6. 10.6 Notes
  19. 11 Symplectic invariance of the CCR on Fock spaces
    1. 11.1 Symplectic group on a Kähler space
      1. 11.1.1 Basic properties
      2. 11.1.2 Unitary group on a Kähler space
      3. 11.1.3 Symplectic transformations on Kähler spaces
      4. 11.1.4 Positive symplectic transformations
      5. 11.1.5 Polar decomposition of symplectic maps
      6. 11.1.6 Restricted symplectic group
      7. 11.1.7 Anomaly-free symplectic group
      8. 11.1.8 Pairs of Kähler structures on symplectic spaces
      9. 11.1.9 Conjugation adapted to a pair of Kähler involutions
    2. 11.2 Bosonic quadratic Hamiltonians on Fock spaces
      1. 11.2.1 Wick and anti-Wick quantizations of quadratic polynomials
      2. 11.2.2 Bosonic Schwinger term
      3. 11.2.3 Infimum of bosonic quadratic Hamiltonians
      4. 11.2.4 Gaussian vectors
      5. 11.2.5 Gaussian vectors in the real-wave representation
      6. 11.2.6 Two-particle creation and annihilation operators
    3. 11.3 Bosonic Bogoliubov transformations on Fock spaces
      1. 11.3.1 Symplectic transformations in the Fock representation
      2. 11.3.2 One-parameter groups of Bogoliubov transformations
      3. 11.3.3 Implementation of positive symplectic transformations
      4. 11.3.4 Metaplectic group in the Fock representation
    4. 11.4 Fock sector of a CCR representation
      1. 11.4.1 Vacua of CCR representations
      2. 11.4.2 Fock CCR representations
      3. 11.4.3 Unitary equivalence of Fock CCR representations
      4. 11.4.4 Fock sector of CCR representations
      5. 11.4.5 Number operator of regular CCR representations
      6. 11.4.6 Relative continuity of Gaussian measures
    5. 11.5 Coherent sector of CCR representations
      1. 11.5.1 Coherent vectors in a CCR representation
      2. 11.5.2 Coherent vectors in Fock spaces
      3. 11.5.3 Coherent representations
      4. 11.5.4 Coherent sector
    6. 11.6 van Hove Hamiltonians
      1. 11.6.1 Classical van Hove dynamics
      2. 11.6.2 Classical van Hove Hamiltonians
      3. 11.6.3 Quantum van Hove dynamics
      4. 11.6.4 Quantum van Hove Hamiltonians
      5. 11.6.5 Nine classes of van Hove Hamiltonians
    7. 11.7 Notes
  20. 12 Canonical anti-commutation relations
    1. 12.1 CAR representations
      1. 12.1.1 Definition of a CAR representation
      2. 12.1.2 CAR representations over a direct sum
      3. 12.1.3 Cyclicity and irreducibility
      4. 12.1.4 Intertwining operators
      5. 12.1.5 Volume element
      6. 12.1.6 CAR over Kähler spaces
      7. 12.1.7 Charged CAR representations
      8. 12.1.8 Bogoliubov rotations
    2. 12.2 CAR representations in finite dimensions
      1. 12.2.1 Volume element
      2. 12.2.2 Pauli matrices
      3. 12.2.3 Jordan-Wigner representation
      4. 12.2.4 Unitary equivalence of the CAR in finite dimensions
    3. 12.3 CAR algebras: finite dimensions
      1. 12.3.1 CAR algebra
      2. 12.3.2 Parity
      3. 12.3.3 Complex conjugation and transposition
      4. 12.3.4 Bogoliubov automorphisms
    4. 12.4 Anti-symmetric quantization and real-wave CAR representation
      1. 12.4.1 Anti-symmetric quantization
      2. 12.4.2 Real-wave CAR representation
      3. 12.4.3 Real-wave CAR representation in coordinates
    5. 12.5 CAR algebras: infinite dimensions
      1. 12.5.1 Algebraic CAR algebra
      2. 12.5.2 C*-CAR algebra
      3. 12.5.3 W*-CAR algebra
      4. 12.5.4 Conditional expectations between CAR algebras
      5. 12.5.5 Irreducibility of infinite-dimensional CAR algebras
    6. 12.6 Notes
  21. 13 CAR on Fock spaces
    1. 13.1 Fock CAR representation
      1. 13.1.1 Field operators
      2. 13.1.2 Extended Fock representation
      3. 13.1.3 Slater determinants
    2. 13.2 Real-wave and complex-wave CAR representation on Fock spaces
      1. 13.2.1 Real-wave CAR representation on Fock spaces
      2. 13.2.2 Operators in the real-wave CAR representation
      3. 13.2.3 Complex-wave CAR representation in finite dimensions
      4. 13.2.4 Complex-wave CAR representation: the general case
    3. 13.3 Wick and anti-Wick fermionic quantization
      1. 13.3.1 Wick and anti-Wick ordering
      2. 13.3.2 Relation between Wick, anti-Wick and anti-symmetric quantizations
      3. 13.3.3 Wick quantization: the operator formalism
      4. 13.3.4 Estimates on Wick polynomials
    4. 13.4 Notes
  22. 14 Orthogonal invariance of CAR algebras
    1. 14.1 Orthogonal groups
      1. 14.1.1 Group O1(Y)
      2. 14.1.2 Group Op(Y)
    2. 14.2 Quadratic fermionic Hamiltonians
      1. 14.2.1 Fermionic harmonic oscillator
      2. 14.2.2 Commutation properties of quadratic fermionic Hamiltonians
      3. 14.2.3 Quadratic Hamiltonians in C*-CAR algebras
      4. 14.2.4 Quadratic Hamiltonians in W*-CAR algebras
    3. 14.3 PinC and Pin groups
      1. 14.3.1 Pin C and Pin groups in finite dimensions
      2. 14.3.2 Pin1C and Pin1 groups
      3. 14.3.3 Pin2C and Pin2 groups
      4. 14.3.4 Symbol of elements of Spin(Y)
    4. 14.4 Notes
  23. 15 Clifford relations
    1. 15.1 Clifford algebras
      1. 15.1.1 Representations of Clifford relations
      2. 15.1.2 Clifford algebras
      3. 15.1.3 Complex Clifford algebras
    2. 15.2 Quaternions
      1. 15.2.1 Basic definitions
      2. 15.2.2 Quaternionic vector spaces
      3. 15.2.3 Embedding complex numbers in quaternions
      4. 15.2.4 Matrix representation of quaternions
      5. 15.2.5 Real simple algebras
    3. 15.3 Clifford relations over Rq,p
      1. 15.3.1 Basic facts
      2. 15.3.2 Charge reversal
      3. 15.3.3 Real spinors
      4. 15.3.4 Quaternionic spinors
      5. 15.3.5 Representations of Clifford relations on pseudo-unitary
    4. 15.4 Clifford algebras over Rq,p
      1. 15.4.1 Form of Clifford algebras for a general signature
      2. 15.4.2 Pseudo-Euclidean group
      3. 15.4.3 Pin group for a general signature
    5. 15.5 Notes
  24. 16 Orthogonal invariance of the CAR on Fock spaces
    1. 16.1 Orthogonal group on a Kähler space
      1. 16.1.1 Basic properties
      2. 16.1.2 j-non-degenerate orthogonal maps
      3. 16.1.3 j-self-adjoint maps
      4. 16.1.4 j-polar decomposition
      5. 16.1.5 Conjugations on Kähler spaces
      6. 16.1.6 Partial conjugations on Kähler spaces
      7. 16.1.7 Decomposition of orthogonal operators
      8. 16.1.8 Restricted orthogonal group
      9. 16.1.9 Anomaly-free orthogonal group
      10. 16.1.10 Pairs of Kähler structures on real Hilbert
    2. 16.2 Fermionic quadratic Hamiltonians on Fock spaces
      1. 16.2.1 Quadratic anti-commuting polynomials and their quantization
      2. 16.2.2 Fermionic Schwinger term
      3. 16.2.3 Infimum of quadratic fermionic Hamiltonians
      4. 16.2.4. Two-particle creation and annihilation operators
      5. 16.2.5 Fermionic Gaussian vectors
    3. 16.3 Fermionic Bogoliubov transformations on Fock spaces
      1. 16.3.1 Extending parity and complex conjugation
      2. 16.3.2 Group Pinj C(Y)
      3. 16.3.3 Implementation of partial conjugations
      4. 16.3.4 Implementation of j-non-degenerate transformations
      5. 16.3.5 End of proof of the Shale - Stinespring theorem
      6. 16.3.6 One-parameter groups of Bogoliubov transformations
      7. 16.3.7 Implementation of j-non-degenerate j-positive transformations
      8. 16.3.8 Pin group in the Fock representation
    4. 16.4 Fock sector of a CAR representation
      1. 16.4.1 Vacua of CAR representations
      2. 16.4.2 Fock CAR representations
      3. 16.4.3 Unitary equivalence of Fock CAR representations
      4. 16.4.4 Fock sector of a CAR representation
      5. 16.4.5 Number operator of a CAR representation
    5. 16.5 Notes
  25. 17 Quasi-free states
    1. 17.1 Bosonic quasi-free states
      1. 17.1.1 Definitions of bosonic quasi-free states
      2. 17.1.2 Gauge-invariant bosonic quasi-free states
      3. 17.1.3 Quasi-free charged representations
      4. 17.1.4 Gibbs states of bosonic quadratic Hamiltonians
        1. Density matrix
        2. Standard representations on Hilbert-Schmidt operators
        3. Standard representations on the double Fock spaces
        4. Araki-Woods form of standard representation
      5. 17.1.5 Araki-Woods representations
      6. 17.1.6 Quasi-free CCR representations as Araki-Woods representations
      7. 17.1.7 Free Bose gas at positive temperatures
        1. Thermodynamic limit
        2. W* approach
        3. C* approach
    2. 17.2 Fermionic quasi-free states
      1. 17.2.1 Definition of fermionic quasi-free states
      2. 17.2.2 Gauge-invariant fermionic quasi-free states
      3. 17.2.3 Charged quasi-free CAR representations
      4. 17.2.4 Gibbs states of fermionic quadratic Hamiltonians
        1. Density matrix
        2. Standard representations on Hilbert-Schmidt operators
        3. Standard representations on double Fock spaces
        4. Araki-Wyss form of standard representation
      5. 17.2.5 Araki-Wyss representations
      6. 17.2.6 Quasi-free CAR representations as Araki-Wyss representations
      7. 17.2.7 Free Fermi gas at positive temperatures
        1. W* approach
        2. C* approach
    3. 17.3 Lattices of von Neumann algebras on a Fock space
      1. 17.3.1 Pair of subspaces in a Hilbert space
      2. 17.3.2 Real subspaces in a Hilbert space
      3. 17.3.3 Complete lattices
      4. 17.3.4 Lattice of von Neumann algebras on a bosonic Fock space
      5. 17.3.5 Lattice of von Neumann algebras on a fermionic Fock space
      6. 17.3.6 Even fermionic von Neumann algebras
    4. 17.4 Notes
  26. 18 Dynamics of quantum fields
    1. 18.1 Neutral systems
      1. 18.1.1 Neutral bosonic systems
        1. Algebraic quantization of a symplectic dynamics
        2. Stable symplectic dynamics
        3. Fock quantization of symplectic dynamics
        4. Criterion for a weakly stable symplectic dynamics
      2. 18.1.2 Neutral fermionic systems
        1. Algebraic quantization of an orthogonal dynamics
        2. Kähler structure for a non-degenerate orthogonal dynamics
        3. Fock quantization of orthogonal dynamics
      3. 18.1.3 Time reversal in neutral systems
        1. Time reversal and its algebraic quantization
          1. Fock quantization of time reversal
      4. 18.2 Charged systems
        1. 18.2.1 Charged bosonic systems
          1. Algebraic quantization of a charged symplectic dynamics
          2. Fock quantization of a charged symplectic dynamics
        2. 18.2.2 Charged fermionic systems
          1. Algebraic quantization of a unitary dynamics
          2. Fock quantization of a unitary dynamics
        3. 18.2.3 Charge reversal
        4. Charge reversal and its algebraic quantization
        5. Fock quantization of charge reversal
        6. Neutral subspace
      5. 18.2.4 Time reversal in charged systems
        1. Wigner time reversal and its algebraic quantization
        2. Fock quantization of Wigner time reversal
        3. Racah time reversal
    2. 18.3 Abstract Klein–Gordon equation and its quantization
      1. 18.3.1 Splitting into configuration and momentum space
      2. 18.3.2 Neutral Klein–Gordon equation
      3. 18.3.3 Neutral Klein–Gordon equation in an external potential
      4. 18.3.4 Splitting into complex configuration and momentum spaces
      5. 18.3.5 Charged Klein–Gordon equation
      6. 18.3.6 Charged Klein–Gordon equation in an external potential
      7. 18.3.7 Quantization of the Klein–Gordon equation
      8. 18.3.8 Two-point functions for the Klein–Gordon equation
      9. 18.3.9 Green's functions of the abstract Klein–Gordon equation
      10. 18.3.10 Green's functions of the Klein–Gordon equation as operators
      11. 18.3.11 Euclidean Green's function of the Klein–Gordon equation
      12. 18.3.12 Thermal Green's function of the Klein–Gordon equation
    3. 18.4 Abstract Dirac equation and its quantization
      1. 18.4.1 Abstract Dirac equation
      2. 18.4.2 Neutral Dirac equation
      3. 18.4.3 Charged Dirac equation
      4. 18.4.4 Quantization of the Dirac equation
      5. 18.4.5 Two-point functions for the Dirac equation
      6. 18.4.6 Green's functions of the Dirac equation
      7. 18.4.7 Green's functions of the Dirac equation as operators
      8. 18.4.8 Euclidean Green's function of the Dirac equation
      9. 18.4.9 Thermal Green's function for the abstract Dirac equation
    4. 18.5 Notes
  27. 19 Quantum fields on space-time
    1. 19.1 Minkowski space and the Poincaré group
      1. 19.1.1 Minkowski space
      2. 19.1.2 The Lorentz group
      3. 19.1.3 Pin groups for the Lorentzian signature
      4. 19.1.4 Positive energy representations of Clifford relations
    2. 19.2 Quantization of the Klein–Gordon equation
      1. 19.2.1 Klein–Gordon operator
      2. 19.2.2 Lagrangian of the Klein–Gordon equation
      3. 19.2.3 Green's functions
      4. 19.2.4 Cauchy problem
      5. 19.2.5 Symplectic form on the space of solutions
      6. 19.2.6 Solutions parametrized by test functions
      7. 19.2.7 Algebraic quantization
      8. 19.2.8 Fock quantization
      9. 19.2.9 Charge symmetry and charge reversal
      10. 19.2.10 Covariance under the Poincaré group
      11. 19.2.11 Parity reversal
      12. 19.2.12 Time reversal
      13. 19.2.13 Klein–Gordon equation in the momentum representation
    3. 19.3 Quantization of the Dirac equation
      1. 19.3.1 Dirac operator
      2. 19.3.2 Lagrangian of the Dirac equation
      3. 19.3.3 Green's functions
      4. 19.3.4 Cauchy problem
      5. 19.3.5 Scalar product in the space of solutions
      6. 19.3.6 Solutions parametrized by test functions
      7. 19.3.7 Algebraic quantization
      8. 19.3.8 Fock quantization
      9. 19.3.9 Charge symmetry
      10. 19.3.10 Charge reversal
      11. 19.3.11 Inversion of the sign in front of the mass
      12. 19.3.12 Covariance under the Poincaré group
      13. 19.3.13 Parity reversal
      14. 19.3.14 Time reversal
      15. 19.3.15 Dirac equation in the momentum representation
    4. 19.4 Partial differential equations on manifolds
      1. 19.4.1 Manifolds
      2. 19.4.2 Integration on pseudo-Riemannian manifolds
      3. 19.4.3 Lorentzian manifolds
      4. 19.4.4 First-order partial differential equations
    5. 19.5 Generalized Klein–Gordon equation on curved space-time
      1. 19.5.1 Klein–Gordon operators
      2. 19.5.2 Lagrangian of the Klein–Gordon equation
      3. 19.5.3 Green's functions of hyperbolic Klein–Gordon equations
      4. 19.5.4 Cauchy problem
      5. 19.5.5 Symplectic space of solutions of the Klein–Gordon equation
      6. 19.5.6 Solutions parametrized by test functions
      7. 19.5.7 Algebraic quantization
    6. 19.6 Generalized Dirac equation on curved space-time
      1. 19.6.1 Dirac operators
      2. 19.6.2 Lagrangian of the Dirac equation
      3. 19.6.3 Green's functions of hyperbolic Dirac equations
      4. 19.6.4 Cauchy problem
      5. 19.6.5 Unitary space of solutions of the Dirac equation
      6. 19.6.6 Solutions parametrized by test functions
      7. 19.6.7 Algebraic quantization
    7. 19.7 Notes
  28. 20 Diagrammatics
    1. 20.1 Diagrams and Gaussian integration
      1. 20.1.1 Vertices
      2. 20.1.2 Diagrams
      3. 20.1.3 Connected diagrams
      4. 20.1.4 Particle spaces and Gaussian integration
      5. 20.1.5 Monomials
      6. 20.1.6 Evaluation of diagrams
      7. 20.1.7 Gaussian integration of products of monomials
      8. 20.1.8 Identical diagrams
      9. 20.1.9 Gaussian integration of exponentials
    2. 20.2 Perturbations of quantum dynamics
      1. 20.2.1 Time-ordered exponentials
      2. 20.2.2 Perturbation theory
      3. 20.2.3 Standard Møller and scattering operators
      4. 20.2.4 Stone formula
      5. 20.2.5 Stationary formulas for Møller and scattering operators
      6. 20.2.6 Problem with eigenvalues
      7. 20.2.7 Adiabatic dynamics
      8. 20.2.8 Bound state energy
    3. 20.3 Friedrichs diagrams and products of Wick monomials
      1. 20.3.1 Friedrichs vertices
      2. 20.3.2 Friedrichs diagrams
      3. 20.3.3 Connected Friedrichs diagrams
      4. 20.3.4 One-particle spaces
      5. 20.3.5 Incoming and outgoing diagram spaces
      6. 20.3.6 Evaluation of a Friedrichs diagram
      7. 20.3.7 Products of operators
    4. 20.4 Friedrichs diagrams and the scattering operator
      1. 20.4.1 Multiplication of Friedrichs diagrams
      2. 20.4.2 One-particle dynamics
      3. 20.4.3 Time-dependent Wick monomials
      4. 20.4.4 Diagrams for the scattering operator
      5. 20.4.5 Stationary evaluation of a diagram
      6. 20.4.6 Scattering operator for a time-independent Hamiltonian
      7. 20.4.7 Goldstone theorem
    5. 20.5 Feynman diagrams and vacuum expectation value
      1. 20.5.1 Free neutral particles
      2. 20.5.2 Free charged particles
      3. 20.5.3 Full Hilbert space
      4. 20.5.4 Wick's time-ordered product
      5. 20.5.5 Feynman 2-point functions: general remarks
      6. 20.5.6 Feynman's phase space 2-point functions for neutral particles
      7. 20.5.7 Feynman's phase space 2-point functions for charged particles
      8. 20.5.8 Feynman's configuration space 2-point functions for neutral bosons
      9. 20.5.9 Feynman's configuration space 2-point functions for~charged~bosons
      10. 20.5.10 Wick quantization of Feynman polynomials
      11. 20.5.11 Evaluation of Feynman diagrams with no external legs
      12. 20.5.12 Vacuum expectation value of the scattering operator
      13. 20.5.13 Energy shift
      14. 20.5.14 Polynomials on path spaces
      15. 20.5.15 Feynman formalism and Gaussian integration
    6. 20.6 Feynman diagrams and the scattering operator
      1. 20.6.1 Feynman diagrams with external legs
      2. 20.6.2 Feynman diagrams with incoming and outgoing external legs
      3. 20.6.3 Scattering operator and Feynman diagrams
      4. 20.6.4 Gell-Mann-Low scattering operator for time-independent perturbations
      5. 20.6.5 Friedrichs diagrams as Feynman diagrams
    7. 20.7 Notes
  29. 21 Euclidean approach for bosons
    1. 21.1 A simple example: Brownian motion
    2. 21.2 Euclidean approach at zero temperature
      1. 21.2.1 Markov path spaces
      2. 21.2.2 Reconstruction theorem
      3. 21.2.3 Gaussian path spaces I
      4. 21.2.4 Gaussian path spaces II
      5. 21.2.5 From a positivity preserving semi-group to a Markov path space
    3. 21.3 Perturbations of Markov path spaces
      1. 21.3.1 Feynman–Kac–Nelson kernels
      2. 21.3.2 Feynman–Kac–Nelson formula
      3. 21.3.3 Perturbed Hamiltonians
    4. 21.4 Euclidean approach at positive temperatures
      1. 21.4.1 β-Markov path spaces
      2. 21.4.2 Reconstruction theorem
      3. 21.4.3 Proof of the KMS condition
      4. 21.4.4 Identification of the modular conjugation
      5. 21.4.5 Gaussian β-Markov path spaces I
      6. 21.4.6 Gaussian β-Markov path spaces II
    5. 21.5 Perturbations of β-Markov path spaces
      1. 21.5.1 Perturbed path spaces
      2. 21.5.2 Perturbed Liouvilleans
    6. 21.6 Notes
  30. 22 Interacting bosonic fields
    1. 22.1 Free bosonic fields
      1. 22.1.1 Klein-Gordon equation
      2. 22.1.2 Quantization of linear Klein-Gordon equation
      3. 22.1.3 Free dynamics and free local algebras
      4. 22.1.4 Q-space representation
    2. 22.2 P(p) interaction
      1. 22.2.1 Nonlinear Klein-Gordon equation
      2. 22.2.2 Formal quantization of non-linear Klein-Gordon equation
      3. 22.2.3 P(p)2 interaction as a Wick polynomial
      4. 22.2.4 P(p)2 interaction as a multiplication operator
      5. 22.2.5 Space-cutoff P(p)2 Hamiltonian
      6. 22.2.6 Interacting dynamics and local algebras
    3. 22.3 Scattering theory for space-cutoff P (p)2 Hamiltonians
      1. 22.3.1 Domain of the space-cutoff P(p)2 Hamiltonian
      2. 22.3.2 Spectrum of space-cutoff P(p)2 Hamiltonians
      3. 22.3.3 Asymptotic fields
    4. 22.4 Notes
  31. Refereces
  32. Symbols index
  33. Subject index