Analysis on Fock Spaces and Mathematical Theory of Quantum Fields: An Introduction to Mathematical Analysis of Quantum Fields (Second Edition)

Contents
Preface to the Second Edition
Preface to the First Edition
About the Author
List of Symbols
Part 1: Analysis on Fock Spaces
	1. Theory of Linear Operators
		1.1 Linear Operators on Vector Space
			1.1.1 Definitions
			1.1.2 Eigenvalues
			1.1.3 Sum and product of linear operators
			1.1.4 A partial ordering in linear operators
			1.1.5 Commutator
		1.2 Linear Operators on Banach Space
			1.2.1 Continuous mappings
			1.2.2 Bounded linear operators
			1.2.3 Closed operators
			1.2.4 Resolvent set and spectra of linear operator
			1.2.5 Compact operators
		1.3 Linear Operators on Hilbert Space
			1.3.1 Dual spaces
			1.3.2 Adjoint operators
			1.3.3 Topologies of convergence of bounded linear operators
			1.3.4 Graph and core
			1.3.5 Unitary equivalence
			1.3.6 A decomposition theorem
			1.3.7 Relative boundedness
			1.3.8 Hermitian, symmetric and self-adjoint operators
			1.3.9 Trace class operator
			1.3.10 Orthogonal projections
			1.3.11 Weak commutator
		1.4 Direct Sum Operators
		1.5 Self-Adjoint Operators and Related Subjects
			1.5.1 Criteria on (essential) self-adjointness for symmetric operators
			1.5.2 The Kato–Rellich theorem
			1.5.3 Multiplication operators
			1.5.4 Spectral measures and functional calculus
			1.5.5 Spectral measure induced by the pair of a spectral measure and a measurable mapping
			1.5.6 Spectral theorem
			1.5.7 Spectral measure and spectra
			1.5.8 Essential spectrum
			1.5.9 Functional calculus
			1.5.10 The spectral measure of a multiplication operator
			1.5.11 Strongly continuous one-parameter unitary groups
			1.5.12 Unitary covariance of functional calculus
			1.5.13 Some inequalities
		1.6 Operators of Exponential Type
		1.7 Analytic Vectors
		1.8 Analysis of Strongly Commuting Self-Adjoint Operators
			1.8.1 Strong commutativity
			1.8.2 Joint spectral measure and joint spectrum
			1.8.3 An example in quantum mechanics: A free relativistic quantum particle
			1.8.4 Functional calculus of strongly commuting self-adjoint operators
			1.8.5 Strongly continuous multi-parameter unitary groups
		1.9 Reduction of Linear Operators
		1.10 Convergence of Self-Adjoint Operators
		1.11 Problems
	2. Tensor Product Hilbert Spaces
		2.1 Forms
			2.1.1 Bilinear forms
			2.1.2 Sesquilinear forms and a representation theorem
			2.1.3 Conjugate bilinear forms
		2.2 Tensor Product of Two Hilbert Spaces
		2.3 Contraction of Tensor Products
		2.4 Tensor Product of N Hilbert Spaces with N ≥ 3
		2.5 Hilbert Space Isomorphisms
		2.6 Tensor Product of L2-Spaces
		2.7 Tensor Product of an L2-Space and a Hilbert Space
		2.8 Constant Fiber Direct Integrals
		2.9 Permutation Operators, Symmetric Tensor Products and Anti-symmetric Tensor Products
			2.9.1 Permutation operators
			2.9.2 Basic elements in group theory
			2.9.3 Unitary representation of SN
			2.9.4 Symmetric and anti-symmetric tensor product Hilbert spaces
			2.9.5 CONSs of ⊗NsH
			2.9.6 CONSs of ∧N(H)
		2.10 Occupation Number Representations
			2.10.1 The case of bosonic systems
			2.10.2 The case of fermionic systems
		2.11 Symmetric and Anti-Symmetric State Functions in Quantum Mechanics
			2.11.1 The Hilbert space of symmetric state functions
			2.11.2 The Hilbert space of anti-symmetric state functions
		2.12 Tensor Product of Direct Sum Hilbert Spaces
		2.13 Notes
		2.14 Problems
	3. Tensor Product of Linear Operators
		3.1 Algebraic Tensor Product of Linear Operators
		3.2 Tensor Product of Linear Operators
		3.3 Tensor Product of Bounded Linear Operators
		3.4 Reduction of Tensor Product Operators
		3.5 Basic Properties of Simple Tensor Product Operators
			3.5.1 Some estimates
			3.5.2 Spectral properties of A ⊗ I
		3.6 Eigenvalues of Tensor Product Operators
		3.7 Tensor Product of N Linear Operators with N ≥ 3
		3.8 Tensor Product of Self-Adjoint Operators
		3.9 Point Spectra of Tensor Product of Self-Adjoint Operators
		3.10 Exponential Type Operators Formed Out of Tensor Products of Self-Adjoint Operators
		3.11 Tensor Product Operators and Direct Integral Operators
		3.12 Problems
	4. Full Fock Spaces and Second Quantization Operators
		4.1 Infinite Direct Sum Hilbert Spaces
		4.2 Infinite Direct Sum Operators
		4.3 Sum and Product of Two Infinite Direct Sum Operators
		4.4 Diagonal and Non-Diagonal Operators on H
			4.4.1 Shift-type operators
			4.4.2 A general class of non-diagonal operators
		4.5 Infinite Direct Sum Operator of Self-Adjoint Operators
		4.6 Full Fock Spaces
		4.7 Second Quantization Operators
		4.8 Ground State
		4.9 Γ-Operators
		4.10 Anti-Unitary Γ-Operators
		4.11 Representations of Unitary Groups and Semi-Groups
		4.12 Relations Between Γ(·) and dΓ(·)
		4.13 Commutation Properties of Second Quantization Operators
		4.14 Creation and Annihilation Operators
		4.15 Problems
	5. Boson Fock Spaces
		5.1 Introduction: Some Physical Backgrounds
		5.2 The Boson Fock Space over a Hilbert Space
		5.3 Boson Second Quantization Operators
			5.3.1 Reduction of a class of tensor product operators
			5.3.2 Boson second quantization operators and their basic properties
			5.3.3 Spectral properties of boson second quantization operators
			5.3.4 eitdΓb(T)-invariant vectors
			5.3.5 Boson number operator
			5.3.6 Commutation properties
		5.4 Boson Γ-Operators
		5.5 Spectral Properties of Γb(T)
		5.6 Trace of Γb-Operator
		5.7 Boson Annihilation and Creation Operators
			5.7.1 Definitions and basic properties
			5.7.2 Commutation relations
			5.7.3 Irreducibility
			5.7.4 Reduction of creation and annihilation operators
		5.8 Occupation Number Representations for a System of Finitely Many Bosons
		5.9 Relations among the Creation and Annihilation Operators and the Second Quantization Operators
			5.9.1 Relative boundedness of the creation and annihilation operators with respect to second quantization operators
			5.9.2 Commutation relations among dΓb(T) and A(f)#
			5.9.3 Relative boundedness of A(f)#A(g)# with respect to dΓb(T)
			5.9.4 Continuity of boson second quantization operators in one-particle operators
			5.9.5 Representation of boson second quantizations in terms of creation and annihilation operators
			5.9.6 Infinite series representation for the square root of a second quantization operator
		5.10 The Segal Field Operator
			5.10.1 Basic properties
			5.10.2 Self-adjointness of the Segal field operator
			5.10.3 Irreducibility of Segal field operators
		5.11 Exponential Operator for a General Linear Operator
			5.11.1 Commutation relations
			5.11.2 Baker–Campbell–Hausdorff formula
		5.12 Exponential Operators for Segal Field Operators
		5.13 Decomposition of the Segal Field Operator
		5.14 Transformations of Annihilation and Creation Operators by Exponential Operators
		5.15 Spectrum of Segal Field Operators
		5.16 Transformation Laws of Creation and Annihilation Operators with Respect to Γ-Operators
		5.17 Time Development of Segal Field Operators and Field Equations
		5.18 Abstract Free Bose Fields
		5.19 Vacuum Expectation Values
		5.20 Coherent Vectors
			5.20.1 Definition and basic properties
			5.20.2 Relation with the Segal field operators
			5.20.3 Physical meaning of coherent states
		5.21 Spectra of A(f) and A(f)∗
		5.22 The Boson Fock Space over a Direct Sum Hilbert Space
		5.23 Representations of CCR
		5.24 Representation of CCR over a Vector Space
		5.25 Second Quantization Associated with a Representation of CCR
		5.26 Bosonic Diagonalization
		5.27 Representation of Heisenberg CCR
		5.28 Weyl Representations of CCR
			5.28.1 Definition and remarks
			5.28.2 A class of irreducible Weyl representations on the abstract boson Fock space
			5.28.3 General properties
		5.29 A Class of Representations of CCR
			5.29.1 Symplectic group on a Hilbert space
			5.29.2 Bogoliubov transformations
			5.29.3 Analysis of Bogoliubov translations
			5.29.4 A general case of Bogoliubov transformations
		5.30 Functional Schrödinger Representation of Heisenberg CCR with Infinite Degrees of Freedom
			5.30.1 Heuristic arguments
			5.30.2 Existence of a probability space with an infinite-dimensional measure
			5.30.3 Representation of the Heisenberg CCR with countable infinite degrees of freedom
			5.30.4 Functionalizations
			5.30.5 Lp-integrability of ϕ(f)
			5.30.6 Wick products
			5.30.7 An orthogonal decomposition
			5.30.8 A natural isomorphism with a boson Fock space
		5.31 Concluding Remarks
		5.32 Problems
	6. Fermion Fock Spaces
		6.1 Definitions and Basic Properties
		6.2 Fermion Second Quantization Operators
		6.3 Fermion Γ-Operators
		6.4 Commutation Properties of Fermion Second Quantization Operators
		6.5 Spectral Properties of Fermion Second Quantization Operators
			6.5.1 Point spectrum of T(q)f
			6.5.2 Point spectrum of ∧qT
			6.5.3 Spectrum of T(p)f and ∧pT
			6.5.4 Spectra of dΓf(T) and Γf(T)
		6.6 Fermion Annihilation and Creation Operators
		6.7 Canonical Anti-Commutation Relations
		6.8 Spectra of B(u)#
		6.9 Characterization of the Fermion Fock Vacuum
		6.10 Occupation Number Representation for a System of Finitely Many Fermions
		6.11 Relations Between the Fermion Second Quantization Operators and the Fermion Annihilation–Creation Operators
		6.12 Uniform Differentiability of Operator-Valued Functions
		6.13 The Fermion Fock Space over a Direct Sum Hilbert Space
		6.14 Representations of CAR
		6.15 Second Quantization Associated with a Representation of CAR
		6.16 Fermionic Bogoliubov Transformations
		6.17 Problems
	7. Boson–Fermion Fock Spaces and Infinite-Dimensional Dirac-Type Operators
		7.1 Fundamental Structures of a Boson–Fermion Fock Space
		7.2 Second Quantization Operators on the Boson–Fermion Fock Space F(H, K)
		7.3 Operators of Co-boundary Type
		7.4 Laplace–Beltrami Operators
		7.5 Quotient Hilbert Space
		7.6 Cohomology Groups and De Rham–Hodge–Kodaira Decomposition
		7.7 Identification of the Cohomology Groups
		7.8 Properties of Kernel of Some Operators
		7.9 Kernel of Second Quantization Operators
		7.10 The Kernel of ΔS,p and the Dimension of HpS
		7.11 Infinite-Dimensional Dirac-Type Operators on F(H, K)
		7.12 Anti-Commutation Relations of QS and QT
		7.13 Abstract Dirac Operator
		7.14 Abstract Supersymmetric Quantum Mechanics
		7.15 Fredholm Operators
		7.16 Operator Matrix Representation of a τ-Dirac Operator
		7.17 Self-Adjoint τ-Dirac Operators
		7.18 Index Formula
		7.19 Fredholmness and the Γ-Index of the Dirac Operator QS
		7.20 Notes
		7.21 Problems
Part 2: Mathematical Theory of Quantum Fields
	8. General Theory of Quantum Fields
		8.1 Introduction
		8.2 Operator-Valued Distributions
			8.2.1 Operator-valued functions
			8.2.2 Operator-valued distributions
			8.2.3 Transformations of operator-valued distributions
			8.2.4 Partial derivatives of operator-valued distributions
		8.3 General Concept of Quantum Field
			8.3.1 Time-translation covariant quantum field theory
			8.3.2 Uniqueness of Hamiltonians
			8.3.3 Sharp-time quantum fields
		8.4 Equations for Quantum Fields
		8.5 Vacuum Expectation Values and Wightman Distributions
		8.6 A General Form of Canonical Formalism in QFT
		8.7 Example: A Free Neutral Scalar Quantum Field
		8.8 Unitary Representations of Topological Groups
		8.9 Translation Covariant QFT
		8.10 Review of Some Aspects Related to the Theory of Special Relativity
			8.10.1 Minkowski space–time
			8.10.2 Lorentz group
			8.10.3 Dual operators on (M1+d)∗
			8.10.4 The Lie algebra of L↑+
			8.10.5 Poincaré group
		8.11 Axioms for Relativistic QFT
			8.11.1 The Gårding–Wightman axioms
			8.11.2 Angular momentum
			8.11.3 The Wightman axioms
			8.11.4 The Osterwalder–Schrader axioms
		8.12 Euclidean Quantum Fields
		8.13 PCT Theorem
		8.14 Haag’s Theorem
		8.15 Scattering Theory and Spectral Analysis
			8.15.1 Introduction
			8.15.2 Asymptotic annihilation and creation operators
			8.15.3 A vanishing theorem
			8.15.4 Commutation relations and representations of CCR
			8.15.5 Scattering operator
			8.15.6 Spectrum of H
			8.15.7 Existence of asymptotic creation and annihilation operators
		8.16 Concluding Remarks
		8.17 Problems
	9. Non-Relativistic QFT
		9.1 Introduction
		9.2 Classical Field Theory
		9.3 Heuristic Arguments toward Construction of QFT
		9.4 A Bosonic Quantum de Broglie Field
			9.4.1 A free bosonic quantum de Broglie field
			9.4.2 Energy–momentum operator
			9.4.3 CT symmetry and parity symmetry
			9.4.4 Energy–momentum spectrum
			9.4.5 Galilean symmetry
			9.4.6 Interaction with an external field
		9.5 The Operator-Valued Distribution Kernel of the Boson Annihilation Operator
		9.6 Sesquilinear Forms Defined by ψ(x) and ψ(x)∗
		9.7 Hamiltonians as Sesquilinear Forms
			9.7.1 A Hamiltonian with an external field
			9.7.2 A Hamiltonian of a self-interacting quantum de Broglie field
		9.8 A Fermionic Quantum de Broglie Field
			9.8.1 Hilbert space for state vectors
			9.8.2 Annihilation and creation operators
			9.8.3 A fermionic quantum de Broglie field
			9.8.4 The operator-valued distribution kernel of the fermion annihilation operator
		9.9 QFT on a Bounded Space Region
			9.9.1 QFT on a cubic box
			9.9.2 A natural embedding of the finite volume theory into an infinite one
			9.9.3 Infinite volume limit
			9.9.4 Tensor product representation
		9.10 Problems
	10. Relativistic Free Quantum Scalar Fields
		10.1 Free Classical Klein–Gordon Field
		10.2 A Sharp-Time Free Neutral Quantum KG Field
		10.3 Representation Theoretic Aspects
		10.4 PCT Theorem
		10.5 Vacuum Expectation Values
		10.6 Energy–Momentum Spectrum
		10.7 Unitary Representation of the (1 + d)-Dimensional Poincaré Group on the Boson Fock Space Fb(L2(Rdk))
		10.8 A Free Neutral Quantum KG Field Smeared over the Space-Time
		10.9 Cyclicity of the Fock Vacuum
		10.10 Commutation Relations and Microscopic Causality
			10.10.1 Commutation relations
			10.10.2 The Pauli–Jordan distribution in the four-dimensional space-time
			10.10.3 Microscopic causality
		10.11 Sharp-Time Distributions and Feynman Propagator
		10.12 Microscopic Causality in the (1 + d)-Dimensional Space-Time and a Summary
		10.13 Vacuum Expectation Values
		10.14 Quantum Field with Momentum Cutoff
		10.15 Schwinger Functions
		10.16 Charged Quantum Scalar Field
			10.16.1 Heuristic arguments
			10.16.2 Outline of a mathematically rigorous construction
			10.16.3 Unitary equivalence to the system of two neutral quantum scalar fields
			10.16.4 PCT theorem
		10.17 Finite Volume Approximations and Infinite Volume Limit
		10.18 Problems
	11. Quantum Theory of Electromagnetic Fields
		11.1 Classical Theory of Electromagnetic Fields
			11.1.1 The Maxwell equations and gauge invariance
			11.1.2 A difficulty in constructing a quantum electromagnetic field
		11.2 Free Electromagnetic Potentials in the Coulomb Gauge
		11.3 Quantum Radiation Field
		11.4 Commutation Relations
		11.5 Vacuum Expectation Values
		11.6 Quantum Electromagnetic Fields
		11.7 Sharp-Time Fields
		11.8 Canonical Conjugate Field
		11.9 Hamiltonian and Momentum Operator
		11.10 Quantum Radiation Field with Momentum Cutoff
		11.11 Equivalent Representation: A Natural Isomorphism
		11.12 Problems
	12. Free Quantum Dirac Field
		12.1 Classical Theory of the Free Dirac Field
			12.1.1 Free Dirac equation
			12.1.2 A remark in connection with the quantum theory of a Dirac particle
			12.1.3 Discrete symmetries in the free Dirac equation
			12.1.4 The special linear group SL(2, C) and the four-dimensional proper Lorentz group
			12.1.5 Relativistic covariance of the free Dirac equation
			12.1.6 Hamiltonian and momentum
			12.1.7 Charge density
			12.1.8 Plane wave expansion of the free Dirac field
			12.1.9 Spectral properties of the free Dirac operator
			12.1.10 Spin angular momentum
		12.2 Parities and a Massless Dirac Field
			12.2.1 A space-inversion operator
			12.2.2 Left-handed and right-handed fields
			12.2.3 The massless free Dirac field
		12.3 Construction of a Free Quantum Dirac Field
			12.3.1 Definition of a sharp-time free quantum Dirac field
			12.3.2 Heuristic arguments
			12.3.3 Rigorous construction
			12.3.4 The free quantum Dirac field smeared over the whole space-time
			12.3.5 Anti-commutation relations at different space-time points and microscopic causality
			12.3.6 Vacuum expectation values
			12.3.7 Feynman propagator
		12.4 Hamiltonian and Momentum Operator of the Free Quantum Dirac Field
		12.5 Total Charge Operator
		12.6 Poincaré Covariance
		12.7 The Free Quantum Dirac Field with Momentum Cutoff
		12.8 PCT Theorem
		12.9 Notes
		12.10 Problems
	13. Van Hove–Miyatake Model
		13.1 Introduction
		13.2 A Realization of the vHM Model
		13.3 Abstract Form of Bose Field Models in the Canonical Formalism in QFT
		13.4 Definition of the Abstract vHM Model
		13.5 The Heisenberg Fields
		13.6 Spectral Properties of HT(g) (I): The Case g ∈ D(T−1)
		13.7 Spectral Properties of HT(g) (II): The Case g ∉ D(T−1)
			13.7.1 Spectrum of HT(g)
			13.7.2 Absence of eigenvalues of HT(g)
		13.8 Application to the Concrete vHM Model
		13.9 Infrared Catastrophe
		13.10 A General Structure behind the Infrared Catastrophe
		13.11 Van Hove–Miyatake Phenomena and Inequivalent Representations of CCR
			13.11.1 The infrared case
			13.11.2 The ultraviolet case
		13.12 Heisenberg Field and VEVs
			13.12.1 An explicit form of Heisenberg field of the abstract vHM model
			13.12.2 VEVs
		13.13 Removal of Cutoffs
		13.14 Construction of the vHM Model Without Cutoffs
		13.15 Point Source Limit of the Ground State Energy of the Concrete vHM Model
		13.16 Scattering Theory
		13.17 Notes
		13.18 Problems
	14. Models in QFT
		14.1 Introduction
		14.2 Purely Bosonic Field Models
			14.2.1 P(ϕ)1+d model
			14.2.2 Scalar quantum electrodynamics
			14.2.3 Non-Relativistic Bose field theories
		14.3 Purely Fermionic Field Models
			14.3.1 A model of a self-interacting quantum Dirac field
			14.3.2 A non-relativistic self-interacting Fermi field model: The BCS model
		14.4 Models of Bose Fields Interacting with Fermi Fields
			14.4.1 Yukawa model
			14.4.2 Quantum electrodynamics
			14.4.3 Supersymmetric quantum fields
		14.5 Particle–Field Interaction Models
			14.5.1 Nelson model
			14.5.2 Pauli–Fierz model in non-relativistic QED
			14.5.3 Generalized spin-boson model
			14.5.4 Pauli–Fierz Hamiltonian
			14.5.5 A particle–field interaction model in relativistic QED
		14.6 Problems
	15. Mathematical Formulation of Spontaneous Symmetry Breaking
		15.1 ∗-Algebras and Representations
			15.1.1 ∗-algebras
			15.1.2 C∗-algebra and von Neumann algebra
			15.1.3 O∗-algebra
			15.1.4 States on ∗-algebras
			15.1.5 Representations of ∗-algebras
			15.1.6 GNS representation
			15.1.7 Weyl algebras
			15.1.8 Schrödinger representation of Weyl algebra with finite degrees of freedom and uniqueness theorem
			15.1.9 Fock representation
		15.2 Spontaneous Symmetry Breaking
			15.2.1 Algebraic symmetry
			15.2.2 Definition of spontaneous symmetry breaking
			15.2.3 Absence of spontaneous symmetry breaking in quantum systems with finite degrees of freedom
		15.3 Example: A Free Bose Field
			15.3.1 A finite volume theory
			15.3.2 Infinite volume limit
			15.3.3 States of the infinite volume system
			15.3.4 Time development
			15.3.5 Representation of the translation group R1+d
		15.4 Order Parameter
		15.5 Mathematical Form of Nambu–Goldstone Theorem
			15.5.1 Basic assumptions
			15.5.2 Heuristic arguments and remarks
			15.5.3 Mathematical formulation and proof of Nambu–Goldstone theorem
		15.6 Problems
Appendix A Weak Convergence of Vectors and Strong Convergence of Bounded Linear Operators in Hilbert Spaces
Appendix B Operators on a Direct Sum Hilbert Space
Appendix C Absolutely Continuous Spectrum and Singular Continuous Spectrum of a Self-Adjoint Operator
Appendix D Elements of the Theory of Distributions
	D.1 Distributions
		D.1.1 Definitions and examples
		D.1.2 Multiplication by C∞-functions
		D.1.3 Complex conjugation, reality and positivity
		D.1.4 Partial derivatives of distributions
	D.2 Tempered Distributions
	D.3 Examples of Tempered Distributions
		D.3.1 Lp-functions
		D.3.2 Polynomially bounded functions
		D.3.3 Delta distributions
	D.4 Some Operations on Tempered Distributions
		D.4.1 Product with polynomially bounded C∞-functions
		D.4.2 Transformations
	D.5 Rd-Translation Invariant Tempered Distributions
	D.6 Convergence in S′(Rd)
	D.7 The Schwartz Nuclear Theorem
	D.8 Fourier Transform
		D.8.1 Fourier transform of rapidly decreasing functions
		D.8.2 Fourier transform of tempered distributions
	D.9 Convolutions
		D.9.1 Convolution of functions in S(Rd)
		D.9.2 Convolution of a tempered distribution and a rapidly decreasing function
		D.9.3 Applications: Fundamental solutions for partial differential operators
Appendix E Integrations of Functions with Values in a Hilbert Space
	E.1 Strong Riemann Integral
	E.2 Bochner Integral
Appendix E Integrations of Functions with Values in a Hilbert Space
Appendix F Representations of Linear Lie Groups and Lie Algebras
	F.1 A Linear Lie Group and Its Lie Algebra
	F.2 Differential Representation of Lie Algebras
Bibliography
Index