Infinite-Dimensional Dirac Operators and Supersymmetric Quantum Fields: An Introduction to Analysis on Boson–Fermion Fock Spaces

Description

This book explains the mathematical structures of supersymmetric quantum field theory (SQFT) from the viewpoints of functional and infinite-dimensional analysis. The main mathematical objects are infinite-dimensional Dirac operators on the abstract Boson–Fermion Fock space. The target audience consists of graduate students and researchers who are interested in mathematical analysis of quantum fields, including supersymmetric ones, and infinite-dimensional analysis. The major topics are the clarification of general mathematical structures that some models in the SQFT have in common, and the mathematically rigorous analysis of them. The importance and the relevance of the subject are that in physics literature, supersymmetric quantum field models are only formally (heuristically) considered and hence may be ill-defined mathematically. From a mathematical point of view, however, they suggest new aspects related to infinite-dimensional geometry and analysis. Therefore, it is important to show the mathematical existence of such models first and then study them in detail. The book shows that the theory of the abstract Boson–Fermion Fock space serves this purpose. The analysis developed in the book also provides a good example of infinite-dimensional analysis from the functional analysis point of view, including a theory of infinite-dimensional Dirac operators and Laplacians. Preface Contents Symbols Numbers Operational Symbols Other Symbols 1 Abstract Supersymmetric Quantum Mechanics 1.1 Definition and Basic Properties 1.2 Reflection Symmetry of the Spectrum of a Self-adjoint Supercharge 1.3 Orthogonal Decomposition of State Vectors 1.4 Operator Matrix Representations 1.5 Construction of SQM 1.5.1 Method I 1.5.2 Method II 1.6 Spectral Supersymmetry 1.7 Ground States 1.8 Spontaneous Supersymmetry Breaking and an Index Formula 2 Elements of the Theory of Fock Spaces 2.1 Full Fock Space 2.2 Boson Fock Space 2.3 Fermion Fock Space 2.4 Second Quantization Operators on the Full Fock Space 2.5 Boson Second Quantization Operators 2.6 Fermion Second Quantization Operators 2.7 Infinite Determinants 2.8 Boson Creation and Annihilation Operators 2.9 Segal Field Operator 2.10 Isomorphisms Among Boson Fock Spaces 2.11 Fermion Creation and Annihilation Operators 2.12 Fermion Quadratic Operators 3 Q-space Representation of Boson Fock Space 3.1 Gaussian Random Process 3.2 Natural Isomorphism of Boson Fock Spaces 3.3 Gradient Operator 3.4 More General Natural Isomorphisms of Boson Fock Spaces 4 Boson–Fermion Fock Spaces and Abstract Supersymmetric Quantum Field Models 4.1 Boson–Fermion Fock Space 4.2 Q-Space Representation of Boson–Fermion Fock Space 4.3 Exterior Differential Operators 4.3.1 Definitions and Basic Properties 4.3.2 A Cochain Complex 4.4 Operators in the Q-space Representation 4.5 Hilbert Complex 4.5.1 Definitions and Basic Facts 4.5.2 Laplace–Beltrami Operators of Finite Order and de Rham–Hodge–Kodaira Decompositions 4.5.3 The Dirac and the Laplace–Beltrami Operators Associated With a Hilbert Complex 4.5.4 Supersymmetric Structure 4.6 Hilbert Complexes Associated With Boson–Fermion Fock Space 4.7 Laplace–Beltrami Operators on Boson–Fermion Fock Space 4.8 Dirac Operators on Boson–Fermion Fock Space 4.9 Strong Anti-Commutativity of Dirac Operators 4.10 Perturbations of Dirac Operator QS 4.10.1 Witten Deformation 4.10.2 More General Perturbations 4.11 Explicit Form of Supersymmetric Hamiltonian HS(F) 4.12 Path Integral Representation of the Index of QS,+(F) 4.12.1 Path Integral Representations of Pure Imaginary Time Correlation Functions of Bose Fields 4.12.2 Index Formula 5 Models in Supersymmetric Quantum Field Theory 5.1 Preliminaries 5.1.1 Momentum Operator of a Quantum Particle in Tell 5.1.2 The Free Quantum Klein–Gordon Field on Mell 5.1.3 The Majorana Field on Mell 5.2 The N=1 Wess–Zumino Model on Mell 5.2.1 The Free Case 5.2.2 The Interacting Case 5.3 The N=2 Wess–Zumino Model on Mell 5.4 Other Models Appendix A Self-adjoint Extensions of a Symmetric Operator Matrix Appendix B Construction of an Infinite-Dimensional Gaussian Measure on a Path Space Appendix References Index