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Ground States of Quantum Field Models

Perturbation of Embedded Eigenvalues

  • Fumio Hiroshima

Part of the SpringerBriefs in Mathematical Physics book series (BRIEFSMAPHY, volume 35)

Table of contents

  1. Front Matter
    Pages i-xvi
  2. Fumio Hiroshima
    Pages 1-14
  3. Fumio Hiroshima
    Pages 15-40
  4. Fumio Hiroshima
    Pages 41-74
  5. Fumio Hiroshima
    Pages 75-102
  6. Fumio Hiroshima
    Pages 103-107
  7. Fumio Hiroshima
    Pages 109-129
  8. Back Matter
    Pages 131-136

About this book

Introduction

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.

Keywords

mathematical quantum field theory spin-boson (SB) model the Nelson model the Pauli-Fierz (PF) model ground states

Authors and affiliations

  • Fumio Hiroshima
    • 1
  1. 1.Faculty of MathematicsKyushu UniversityFukuokaJapan

Bibliographic information

  • DOI https://doi.org/10.1007/978-981-32-9305-2
  • Copyright Information The Author(s), under exclusive licence to Springer Nature Singapore Pte Ltd. 2019
  • Publisher Name Springer, Singapore
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-981-32-9304-5
  • Online ISBN 978-981-32-9305-2
  • Series Print ISSN 2197-1757
  • Series Online ISSN 2197-1765
  • Buy this book on publisher's site