General Principles of Quantum Field Theory (Mathematical Physics and Applied Mathematics)

The majority of the “memorable” results of relativistic quantum theory were obtained within the framework of the local quantum field approach. The explanation of the basic principles of the local theory and its mathematical structure has left its mark on all modern activity in this area. Originally, the axiomatic approach arose from attempts to give a mathematical meaning to the quantum field theory of strong interactions (of Yukawa type). The fields in such a theory are realized by operators in Hilbert space with a positive Poincare-invariant scalar product. This “classical” part of the axiomatic approach attained its modern form as far back as the sixties. * It has retained its importance even to this day, in spite of the fact that nowadays the main prospects for the description of the electro-weak and strong interactions are in connection with the theory of gauge fields. In fact, from the point of view of the quark model, the theory of strong interactions of Wightman type was obtained by restricting attention to just the “physical” local operators (such as hadronic fields consisting of ”fundamental” quark fields) acting in a Hilbert space of physical states. In principle, there are enough such “physical” fields for a description of hadronic physics, although this means that one must reject the traditional local Lagrangian formalism. (The connection is restored in the approximation of low-energy “phe nomenological” Lagrangians. I Elements of Functional Analysis and the Theory of Functions.- Synopsis.- 1. Preliminaries on Functional Analysis.- 1.1. Normed Spaces.- A. Linear spaces (3). B. Direct sum and tensor product of linear spaces (5). C. Normed spaces (7). D. Hilbert spaces (8). E. Direct sum and tensor product of Hilbert spaces (12). F. Linear functionals and dual spaces (14).- 1.2. Locally Convex Spaces.- A. Equivalent systems of seminorms. Structure of LCS’s (16). B. Frchet Spaces (17). C. Examples (18).- 1.3. Linear Operators and Linear Functionals in Frchet Spaces.- A. Continuous maps of LCS’s (20). B. The uniform boundedness principle. The weak and weak* topologies (22). C. The closed graph and open mapping theorems (23).- 1.4. Operators in Hilbert Space.- A. The notion of an (unbounded) self-adjoint operator (25). B. Isometric, unitary and anti-unitary operators (28). C. The spectral theory of self-adjoint and unitary operators (29).- 1.5. Algebras with Involution. C*-Algebras.- A. Definition and elementary properties (31). B. The spectrum (33). C. Positive functionals (34). D. Representations (36). E. Trace class operators (41). F. Von Neumann algebras (43).- 2. The Technique of Generalized Functions.- 2.1. The Concept of a Generalized Function.- A. Functional definition (46). B. Definition in terms of fundamental sequences (49). C. Local properties of generalized functions (51).- 2.2. Transformation of Arguments and Differentiation.- A. Change of variables in a generalized function (53). B. Differentiation of generalized functions. Examples (54).- 2.3. Multiplication of a Generalized Function by a Smooth Function.- A. The problem underlying multiplication of generalized functions. The concept of a multiplicator (56). B. The division problem (58).- 2.4. The Kernel Theorem. Tensor Products of Generalized Functions.- A. Bilinear functionals on spaces of type S (61). B. Tensor products (62).- 2.5. Fourier Transform and Convolution.- A. Fourier transform of test functions (63). B. Fourier transform of generalized functions (65). C. Convolutes (66). D. Generalized functions of integrable type (67). E. Convolution of generalized functions (70).- 2.6. Generalized Functions Dependent on a Parameter.- A. General information (72). B. Restriction of generalized functions (74).C. More on the multiplication of generalized functions (76).- 2.7. Vector- and Operator-Valued Generalized Functions.- A. Generalized functions with values in Hilbert space (78). B. Operator-valued generalized functions (80). C. The notion of a generalized eigenvector (82).- Appendix A. Generalized Functions on Subsets of Rn.- A.1. Generalized functions on an open subset (83). A.2. Generalized functions on canonically closed regular subsets (84). A.3. Application: generalized functions on the compactified sets [A, ?], R?, [??, +?] (86).- Appendix B. The Laplace Transform of Generalized Functions.- B.1. The Laplace transform as an analytic function in the complex plane (89). B.2. The case of a generalized function with support in a pointed cone (96). B.3. Example: generalized functions of retarded type (98). B.4. Boundary values of the Laplace transform (99). B.S. Example: the “mathematics” of dispersion relations (103). B.6. Restriction of the Laplace transform (105).- Appendix C. Homogeneous Generalized Functions.- C.1. Homogeneous generalized functions in $${{mathop{R}limits^{^circ } }^{n}}$$ (106). C.2. The single real variable case (109). C.3. Extension of homogeneous generalized functions (110). C.4. Application to covariant homogeneous generalized functions (113). C.S. Homogeneous generalized functions in the complex plane (114).- 3. Lorentz-Covariant Generalized Functions.- 3.1. The Lorentz Group.- A. The geometry of Minkowski space (118). B. Definition of the general Lorentz group and its connected components (119). C. The universal covering of the group L+? (121). D. Finite-dimensional representations of the group SL(2,C) (125). E. Simply reducible finite-dimensional representations of SL(2, C). Spatial reflection (128).- 3.2. Lorentz-Invariant Generalized Functions in Minkowski Space.- A. Definition (131). B. Even invariant generalized functions. Invariant generalized functions with support at a point (132). C. Odd invariant generalized functions (136).- 3.3. Lorentz-Covariant Generalized Functions in Minkowski Space.- A. Definition (138). B. Structure of covariant generalized functions (139).- 3.4. The Case of Several Vector Variables.- A. Generalized functions that are invariant with respect to a compact group (143). B. Generalized functions that are covariant with respect to a compact group (149). C. Applications to Lorentz-invariant and Lorentz-covariant generalized functions (155).- Appendix D. Vocabulary of Lie Groups and their Representations.- D.1. Abstract groups. Algebraic properties (159). D.2. Lie groups (160). D.3. Lie algebras (162). D.4. Relation between Lie groups and Lie algebras D.5. Local Lie groups. Canonical parametrization. Lie’s theorems D.6. Linear representations (166). D.7. Adjoint and co-adjoint representations. Killing forms (167).- 4. The Jost-Lehmann-Dyson Representation.- 4.1. Relation between the JLD Representation and the Wave Equation.- A. Preliminary remarks (170). B. Outline of the derivation (171). C. Departure into six-dimensional space (173).- 4.2. Properties of Solutions of the d’Alembert Equation in S’.- A. Notation (175). B. Fundamental Solution of the Cauchy Problem (176). C. Cauchy problem on a spacelike hypersurface; Huygens’ principle (179). D. The Asgeirsson formula and its applications (183).- 4.3. Derivation of the Jost-Lehmann-Dyson Formula.- A. Construction of the spectral function (185). B. Further properties of the support of the spectral function (188). C. Examples (192). D. Representations for generalized functions of retarded and advanced types (193).- 5. Analytic Functions of Several Complex Variables.- 5.1. Properties of Holomorphic Functions. Plurisubharmonic Functions.- A. Space of holomorphic functions (197). B. Holomorphy and analyticity (199). C. Analytic continuation (200). D. Generalized principle of analytic continuation; “edge of the wedge” theorem (204). E. Holomorphic distributions (207). F. Invariant and covariant analytic functions (209). G. Plurisubharmonic functions (211).- 5.2. Domains of Holomorphy.- A. Holomorphic convexity (215). B. Pseudo-convexity (217). C. Modified principle of continuity (219). D. Single-sheeted envelopes of holomorphy (221). E. Invariant domains (223). F. An example of holomorphic extension (226).- II Relativistic Quantum Systems.- Synopsis.- 6. Algebra of Observables and State Space.- 6.1. Algebraic Formulation of Quantum Theory.- A. Algebra of observables. States (233). B. Transition probability (235). C. Relationship to representations (236).- 6.2. Superselection Rules.- A. The role of pure vector states (239). B. Standard superselection rules (243). C. Connection with gauge groups (245). D. Example of non-abelian gauge groups (247).- 6.3. Symmetries in the Algebraic Approach.- A. The concept of symmetry (249). B. Proof and discussion of Wigner’s theorem (252). C. Symmetry groups (256).- 6.4. Canonical Commutation Relations.- A. The role of the Schrdinger representation (260). B. Infinite number of degrees of freedom (263). C. Proof of von Neumann’s uniqueness theorem (267).- 7. Relativistic Invariance in Quantum Theory.- 7.1. The Poincar Group.- A. Definition (270). B. Reflections (271). C. The Lie algebra of the Poincar group (272).- 7.2. Unitary Representations of the Proper Poincar Group.- A. Poincar invariance condition (274). B. Classification of irreducible representations of ?0. Spectral principle (275). C. Description of representations corresponding to particles with positive mass (280). D. Manifestly covariant realization of “physical” irreducible representations (284).- 7.3. Fock Space of Relativistic Particles.- A. Second quantization space (288). B. Connection with (anti-)commutation relations (292). C. Covariant creation and annihilation operators (296). D. Symmetries of the general Poincar group (299). E. Relativistic scattering matrix (302). F. Kinematics of two-particle processes (307).- Appendix E. Four-Component Spinors and the Dirac Equation.- E.1. Clifford algebra over Minkowski space (310). E.2. Spinor representation of the Lorentz group; various realizations of the ?-matrices (312). E.3. Dirac equation; representations of the Poincar group with spin 1/2 (314).- III Local Quantum Fields and Wightman Functions.- Synopsis.- 8. The Wightman Formalism.- 8.1. Quantum Field Systems.- A. Concept of localization (321). B. Principle of local commutativity (322). C. “Fundamental” fields and “physical” fields (323).- 8.2. Definition and Properties of a Local Quantum Field.- A. Wightman’s axioms (324). B. Discussion of the axioms (325). C. Irreducibility of fields (329). D. Separating property of the vacuum vector (331).- 8.3. Wightman Functions.- A. Characteristic properties of Wightman functions (332). B. Klln-Lehmann representation for a scalar field (335). C. Reconstruction of the theory from the Wightman functional (337).- 8.4. Examples: Free and Generalized Free Fields.- A. Free scalar neutral field (340). B. Free scalar charged field (345). C. Free Dirac field (348). D. Generalized free fields (351).- Appendix F. Summary of Invariant Solutions and Green’s Functions of the Klein-Gordon Equation.- Appendix G. General Form of the Covariant Two-Point Function.- G.1. Covariant decompositions compatible with locality (355). G.2. Decomposition with respect to spin (356).- 9. Analytic Properties of Wightman Functions in Coordinate Space.- 9.1. Bargmann-Hall-Wightman Theorem and its Corollaries.- A. Complex Lorentz transformations (359). B. Lorentz-covariant analytic functions in the past tube (362). C. Real points of the extended tube (366). D. Analyticity of Wightman functions in a symmetrized tube (368). E. Global nature of locality (371).- 9.2. TCP-Theorem.- A. TCP-invariance (375). B. Weak locality (378). C. Borchers classes; the notion of a local composite field (378).- 9.3. Connection between Spin and Statistics.- A. Statement of the results (381). B. Necessary conditions for anomalous commutation relations (383). C. Reduction of ? to canonical form (385). D. Construction of the Klein transformation (387).- 9.4. Equal-Time Commutation Relations. Haag’s Theorem.- A. Three-dimensional version of Haag’s theorem (388). B. Haag’s theorem in the relativistic theory (391). Comments on Haag’s theorem (392).- 9.5. Euclidean Green’s Functions.- A. Group of rotations of four-dimensional Euclidean space (394). B. Properties of the Schwinger functions (396). C. Reconstruction theorem in terms of Schwinger functions (400).- Appendix H. Parastatistics.- H.1. Free parafields and paracommutation relations (403). H.2. Comment on the TCP-theorem and the connection between spin and parastatistics for local parafields (406).- Appendix I. Infinite-Component Fields.- I.1. Elementary representation of SL(2, C) (407). 1.2. Concept of a quantum IFC (408). I.3. Covariant structure of the two-point function. Infinite degeneracy of mass with respect to spin (410). I.4. Absence of P+-covariance and connection between spin and statistics in ICF models (413).- 10. Fields in an Indefinite Metric.- 10.1. Pseudo-Wightman Formalism.- A. Pseudo-Hilbert space (417). B. Axioms of pseudo-Wightman type (420). C. Vacuum sector and charged states (423). D. Physical subspace of pseudo-Hilbert space (427).- 10.2. Abelian Models with Gauge Invariance of the 2nd Kind.- A. The field of the dipole ghost and the gradient model (428). B. Local formulation of quantum electrodynamics (434).- 10.3. Internal Symmetries.- A. Symmetries and currents in the Wightman formalism (440). B. Gold-stone’s theorem (443). C. Spontaneous symmetry breaking in abelian gauge theories (446).- 11. Examples: Explicitly Soluble Two-Dimensional Models.- 11.1. Free Scalar Massless Field in Two-Dimensional Space-Time.- A. One-dimensional non-canonical scalar field (450). B. Physical representation (454). C. Free “quark” fields; bosonization of fermions (461). D. Free scalar massless “ghost” field (467).- 11.2. The Thirring Model.- A. Solution of the field equation (469). B. Currents and charges; vacuum representation (473).- 11.3. The Schwinger Model.- A. Solution in the Lorentz gauge (474). B. Vacuum functional (480). C. Physical fields; observables (481).- IV Collision Theory. Axiomatic Theory of the S-Matrix.- Synopsis.- 12. Haag-Ruelle Scattering Theory.- 12.1. Scheme of the Quantum Field Theory of Scattering.- A. The one-particle problem in quantum field theory (486). B. Construction of in- and out-states (488). C. S-matrix and TCP-operators in the asymptotically complete theory (489).- 12.2. Existence of Asymptotic States.- A. Truncated vacuum expectation values (491). B. Strengthened cluster property (495). C. Spread of relativistic wave packets (497). D. Proof of the main result (501).- 13. Lehmann-Symanzik-Zimmermann Formalism.- 13.1. Basic Concepts.- A. T-products of fields (503). B. Retarded products (509). C. LSZ axioms (512).- 13.2. Asymptotic Conditions and Reduction Formulae.- A. LSZ asymptotic conditions (515). B. Yang-Feldman equations (520). C. Partial reduction formulae (522). D. Reduction formulae for the scattering matrix (526).- 14. The S-Matrix Method.- 14.1. S-Matrix Formulation of the Basic Requirements of the Local Theory.- A. The concept of extending the S-matrix beyond the mass shell (530). B. Choice of the class of test functions (534). C. Axioms of the S-matrix approach (535). D. Radiation operators; current (537).- 14.2. Fields in the Asymptotic Representation.- A. Construction of quantum fields and their T-products (540). B. Fulfillment of the LSZ axioms (544).- V Causality and the Spectral Property: The Origins of the Analytic Properties of the Scattering Amplitude.- Synopsis.- 15. Analyticity with respect to Momentum Transfer and Dispersion Relations.- 15.1. The Lehmann Small Ellipse.- A. Introductory remarks (548). B. JLD representation for retarded and advanced (anti)commutators (551). C. Analyticity with respect to t (553).- 15.2. Dispersion Relations.- A. The main steps for the derivation of the dispersion relations (556). B. Passage to the complex domain with respect to the momenta p2, p4 (557). C. Dispersion relation for non-physical “masses” (560). D. Analytic properties of the absorptive part of the amplitude (563). E. Dispersion relation on the mass shell (570).- 16. Analytic Properties of the Four-Point Green’s Function.- 16.1. Generalized Retarded Functions.- A. Generalized retarded products (574). B. Supports in x-space (576).- 16.2. Four-Point Green’s Functions.- A. Notation (579). B. Domains of coincidence in p-space (581). C. Steinmann identities (583). D. Analyticity near physical points (586).- 16.3. Crossing Relation.- A. Statement of the result (589). B. The case of imaginary “masses” (590). C. Analytic continuation with respect to the mass variables (591). D. Passage to the mass shell (596).- Appendix J. The Role of Unitarity.- J.1. Partial wave decomposition of the scattering amplitude of a two-particle process (598). J.2. Analytic continuation of the dispersion relation with respect to t (603).- 17. Consequences for High-Energy Elementary Processes.- 17.1. Restrictions on the Behaviour of Cross Sections at High Energies.- A. Froissart bound (608). B. Comparison of the cross sections of the interaction of a particle and an antiparticle with the same target (613).- 17.2. Inclusive Processes.- A. Physical characteristics of inclusive processes (617). B. Analytic properties of differential cross sections with respect to angular variables (621). C. Asymptotic estimates (624).- Commentary on the Bibliography and References.- References.- Index of Notation.