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 and 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–Miyatake model, is fully analyzed in an abstract form. Moreover, a list of interacting quantum field models is presented and an introductory description to each model is given. In this second edition, a new chapter (Chapter 15) is added to describe a mathematical theory of spontaneous symmetry breaking which is an important subject in modern quantum physics.

This book is a good introductory text for graduate students in mathematics or physics who are interested in the mathematical aspects of quantum field theory. It is also well-suited for self-study, providing readers a firm foundation of knowledge and mathematical techniques for more advanced books and current research articles in the field of mathematical analysis on quantum fields. Numerous problems are added to aid readers in developing a deeper understanding of the field.

Contents:

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  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
  
  
  
  


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  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
  
  
  • Mathematical Formulation of Spontaneous Symmetry Breaking
  
  
  
  


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  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 (the book may be used as a textbook for a one-year course in mathematics or mathematical physics for first year graduate students), researchers in the fields of functional analysis and mathematical quantum field theory.

Key Features:

• The book includes mathematical preliminaries and six appendices to make it as self-contained as possible


• The book presents detailed descriptions on Fock spaces (full Fock space, boson Fock space, fermion Fock space, boson-fermion Fock space) in an abstract and unified way, which may not be seen in other books


• The book presents detailed descriptions of constructions of quantum field models with mathematical rigor as applications of the Fock space theory. In particular, the van Hove model is described in detail in an abstract framework. This may not be seen in other books either


• The book may be the first which presents pedagogical and detailed descriptions of a mathematical theory of spontaneous symmetry breaking


• The book provides 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