Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Wave Function Renormalization for Particle-Field Interactions

Marco Falconi, Benjamin Hinrichs, Javier Valentín Martín

Abstract

In this paper, we develop a wave function renormalization scheme for models of non-relativistic quantum particles interacting with a quantized relativistic field, in the Hamiltonian formalism of quantum field theory. We construct the interacting Hamilton operator, in its ground state representation, for a large class of particle-field interactions; hereby addressing a number of open problems related to both ultraviolet and infrared singularities in the spin-boson and Nelson models, where the infinite wave function renormalization plays a fundamental role.

Wave Function Renormalization for Particle-Field Interactions

Abstract

In this paper, we develop a wave function renormalization scheme for models of non-relativistic quantum particles interacting with a quantized relativistic field, in the Hamiltonian formalism of quantum field theory. We construct the interacting Hamilton operator, in its ground state representation, for a large class of particle-field interactions; hereby addressing a number of open problems related to both ultraviolet and infrared singularities in the spin-boson and Nelson models, where the infinite wave function renormalization plays a fundamental role.
Paper Structure (21 sections, 55 theorems, 160 equations)

This paper contains 21 sections, 55 theorems, 160 equations.

Table of Contents

  1. Introduction
  2. Wave Function Renormalization of the van Hove--Miyatake Model
  3. The vHM model in the Fock representation
  4. The vHM model with singular interactions
  5. Energy renormalization
  6. Wave function renormalization I: renormalized Hilbert space
  7. Wave function renormalization II: renormalized Hamiltonian
  8. Wave Function Renormalization of the Spin-Boson Model
  9. The spin-boson model in the Fock representation
  10. The SB model with singular interactions and renormalization in the Fock representation
  11. SB wave function renormalization I: renormalized Hilbert space
  12. SB wave function renormalization II: renomalized Hamiltonian
  13. (Un)expected exact diagonalization: the standard and energy-preserving SB models
  14. Wave Function Renormalization of the Nelson Model
  15. The Nelson model in the Fock representation
  16. ...and 6 more sections

Key Result

Proposition 2.2

[proposition]prop:5 Let $v\in \mathfrak{h}$, and $\omega\geq 0$. Then:

Theorems & Definitions (110)

  • Remark
  • Definition 2.1: Fock vHM Hamiltonian
  • Example : Concrete vHM model
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • Definition 2.4: Singular annihilation
  • Definition 2.5: vHM form
  • Proposition 2.6
  • Proposition 2.7
  • ...and 100 more