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Infinite-Dimensional Dirac Operators and Supersymmetric Quantum Fields

An Introduction to Analysis on Boson–Fermion Fock Spaces

  • Book
  • © 2022

Overview

Authors:
  1. Asao Arai

    1. Department of Mathematics, Hokkaido University, Sapporo, Japan

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  • Provides an introductory description of the theory of mathematical supersymmetric quantum field theory
  • Clarifies general mathematical structures that some supersymmetric quantum field models have in common
  • Describes a theory of infinite dimensional Dirac operators on the abstract Boson–Fermion Fock space

Part of the book series: SpringerBriefs in Mathematical Physics (BRIEFSMAPHY, volume 46)

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Table of contents (5 chapters)

  1. Front Matter

    Pages i-xiii
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  2. Abstract Supersymmetric Quantum Mechanics

    • Asao Arai
    Pages 1-11

  3. Elements of the Theory of Fock Spaces

    • Asao Arai
    Pages 13-25

  4. Q-space Representation of Boson Fock Space

    • Asao Arai
    Pages 27-33

  5. Boson–Fermion Fock Spaces and Abstract Supersymmetric Quantum Field Models

    • Asao Arai
    Pages 35-73

  6. Models in Supersymmetric Quantum Field Theory

    • Asao Arai
    Pages 75-98

  7. Back Matter

    Pages 99-117
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Keywords

About this book

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.

Reviews

“The book under review is a well-organized readable text in mathematical studies of supersymmetric quantum field theory. … The book is almost self-contained.” (Farhang Loran, Mathematical Reviews, June, 2023)

Authors and Affiliations

  • Department of Mathematics, Hokkaido University, Sapporo, Japan

    Asao Arai

About the author

Professor Asao Arai is a Professor Emeritus of Hokkaido University. 

Bibliographic Information

  • Book Title: Infinite-Dimensional Dirac Operators and Supersymmetric Quantum Fields

  • Book Subtitle: An Introduction to Analysis on Boson–Fermion Fock Spaces

  • Authors: Asao Arai

  • Series Title: SpringerBriefs in Mathematical Physics

  • DOI: https://doi.org/10.1007/978-981-19-5678-2

  • Publisher: Springer Singapore

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2022

  • Softcover ISBN: 978-981-19-5677-5Published: 19 October 2022

  • eBook ISBN: 978-981-19-5678-2Published: 18 October 2022

  • Series ISSN: 2197-1757

  • Series E-ISSN: 2197-1765

  • Edition Number: 1

  • Number of Pages: XIII, 117

  • Number of Illustrations: 1 b/w illustrations

  • Topics: Mathematical Physics, Functional Analysis, Quantum Physics

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