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Arai A. Analysis on Fock Spaces and Mathematical Theory of Quantum Fields

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Arai A. Analysis on Fock Spaces and Mathematical Theory of Quantum Fields
World Scientific, 2018. — 882 p. — ISBN-10 9813207116.
This book provides a comprehensive introduction to Fock space theory and its applications to mathematical quantum field theory. The first half of the book, Part I, is devoted to detailed descriptions of analysis on abstract Fock spaces (full Fock space, boson Fock space, fermion Fock space and boson-fermion Fock space). It includes the mathematics of second quantization, representation theory of canonical commutation relations and canonical anti-commutation relations, Bogoliubov transformations, infinite-dimensional Dirac operators and supersymmetric quantum field in an abstract form. The second half of the book, Part II, covers applications of the mathematical theories in Part I to quantum field theory. Four kinds of free quantum fields are constructed and detailed analyses are made. A simple interacting quantum field model, called the van Hove model, is fully analyzed in an abstract form. Moreover, a list of interacting quantum field models is presented and a short description to each model is given. To graduate students in mathematics or physics who are interested in the mathematical aspects of quantum field theory, this book is a good introductory text. It is also well suited for self-study and will provide readers a firm foundation of knowledge and mathematical techniques for reading more advanced books and current research articles in the field of mathematical analysis on quantum fields. Also, numerous problems are added to aid readers to develop a deeper understanding of the field
Contents
Theory of Linear Operators
Tensor Product Hilbert Spaces
Tensor Product of Linear Operators
Full Fock Spaces and Second Quantization Operators
Boson Fock Spaces
Fermion Fock Spaces
Boson-Fermion Fock Spaces and Infinite Dimensional Dirac Type Operators
General Theory of Quantum Fields
Non-relativistic QFT
Relativistic Free Quantum Scalar Fields
Quantum Theory of Electromagnetic Fields
Free Quantum Dirac Field
Van Hove–Miyatake Model
Models in QFT
Appendixes
Weak Convergence of Vectors and Strong
Convergence of Bounded Linear Operators in Hilbert Spaces
Operators on a Direct Sum Hilbert Space
Absolutely Continuous Spectrum and Singular Continuous Spectrum of a Self-adjoint Operator
Elements of the Theory of Distributions
Integrations of Functions with Values in a Hilbert Space
Representations of Linear Lie Groups and Lie Algebras
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