Analysis on Fock Spaces and Mathematical Theory of Quantum Fields

An Introduction to Mathematical Analysis of Quantum Fields

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:

• Analysis on Fock Spaces:

• 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

• Mathematical Theory of Quantum Fields:

• 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

• Appendices:

• 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

Readership: Advanced undergraduate and graduate students in mathematics or physics, mathematicians and mathematical physicists.
Key Features:

• Detailed description of the theory of Fock spaces including full Fock spaces, boson Fock spaces, fermion Fock spaces and boson-fermion Fock spaces
• New topics are included, such as the theory of infinite dimensional Dirac operators and an abstract supersymmetric quantum field theory, which have been originally developed by the author
• Detailed treatment of mathematical constructions of free quantum field models as well as a simple interacting model