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SpringerBriefs in Mathematical Physics

Ground States of Quantum Field Models

Perturbation of Embedded Eigenvalues

Authors: Hiroshima, Fumio

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  • Is a comprehensive survey of recent results of the existence of ground states of models in quantum field theory
  • Focuses on non-perturbative methods, independent of coupling constants and other factors
  • Begins with fundamentals of quantum field theory and takes the reader to a higher level of appreciation of that theory
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  • ISBN 978-981-329-305-2
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About this book

This book provides self-contained proofs of the existence of ground states of several interaction models in quantum field theory. Interaction models discussed here include the spin-boson model, the Nelson model with and without an ultraviolet cutoff, and the Pauli–Fierz model with and without dipole approximation in non-relativistic quantum electrodynamics. These models describe interactions between bose fields and quantum mechanical matters.A ground state is defined as the eigenvector associated with the bottom of the spectrum of a self-adjoint operator describing the Hamiltonian of a model. The bottom of the spectrum is however embedded in the continuum and then it is non-trivial to show the existence of ground states in non-perturbative ways. We show the existence of the ground state of the Pauli–Fierz mode, the Nelson model, and the spin-boson model, and several kinds of proofs of the existence of ground states are explicitly provided. Key ingredients are compact sets and compact operators in Hilbert spaces. For the Nelson model with an ultraviolet cutoff and the Pauli–Fierz model with dipole approximation we show not only the existence of ground states but also enhanced binding. The enhanced binding means that a system for zero-coupling has no ground state but it has a ground state after turning on an interaction.The book will be of interest to graduate students of mathematics as well as to students of the natural sciences who want to learn quantum field theory from a mathematical point of view. It begins with abstract compactness arguments in Hilbert spaces and definitions of fundamental facts of quantum field theory: boson Fock spaces, creation operators, annihilation operators, and second quantization. This book quickly takes the reader to a level where a wider-than-usual range of quantum field theory can be appreciated, and self-contained proofs of the existence of ground states and enhanced binding are presented.

About the authors

The author was an invited speaker at the ICMP 2015. 

Table of contents (6 chapters)

Table of contents (6 chapters)
  • Introduction

    Pages 1-14

    Hiroshima, Fumio

  • Preliminaries

    Pages 15-40

    Hiroshima, Fumio

  • The Pauli–Fierz Model

    Pages 41-74

    Hiroshima, Fumio

  • The Nelson Model

    Pages 75-102

    Hiroshima, Fumio

  • Spin-Boson Model

    Pages 103-107

    Hiroshima, Fumio

Buy this book

eBook n/a
  • ISBN 978-981-329-305-2
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
Softcover n/a
  • ISBN 978-981-329-304-5
  • Free shipping for individuals worldwide
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Bibliographic Information

Bibliographic Information
Book Title
Ground States of Quantum Field Models
Book Subtitle
Perturbation of Embedded Eigenvalues
Authors
Series Title
SpringerBriefs in Mathematical Physics
Series Volume
35
Copyright
2019
Publisher
Springer Singapore
Copyright Holder
The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd.
eBook ISBN
978-981-329-305-2
DOI
10.1007/978-981-32-9305-2
Softcover ISBN
978-981-329-304-5
Series ISSN
2197-1757
Edition Number
1
Number of Pages
XVI, 136
Number of Illustrations
14 b/w illustrations
Topics