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A new interacting Fock space, the Quon algebra with operator parameter and its Wick's theorem

Yungang Lu

Abstract

Motivated by the creation-annihilation operators in a newly defined interacting Fock space, we initiate the introduction and the study of the Quon algebra. This algebra serves as an extension of the conventional quon algebra, where the traditional constant parameter $q$ found in the $q$--commutation relation is replaced by a specific operator. Importantly, our investigation aims to establish Wick's theorem in the Quon algebra, offering valuable insights into its properties and applications.

A new interacting Fock space, the Quon algebra with operator parameter and its Wick's theorem

Abstract

Motivated by the creation-annihilation operators in a newly defined interacting Fock space, we initiate the introduction and the study of the Quon algebra. This algebra serves as an extension of the conventional quon algebra, where the traditional constant parameter found in the --commutation relation is replaced by a specific operator. Importantly, our investigation aims to establish Wick's theorem in the Quon algebra, offering valuable insights into its properties and applications.
Paper Structure (4 sections, 7 theorems, 169 equations)

This paper contains 4 sections, 7 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Interacting Fock space, the $(q,m)$--Fock space and the Quon algebra
  3. Normally ordered form of a general word of Quon algebra
  4. The case of ${\bf p}_k$'s being identity

Key Result

Proposition 2.6

For any $m\in\mathbb{N}^*$ and $q\in[-1,1]$, for any (pre--)Hilbert space $\mathcal{H}$, the following statements are true for all $f$ belonging to $\mathcal{H}$ when considering the $(q,m)$--Fock space $\Gamma\left( \mathcal{H},\{\lambda_n\}_n\right)$: 1) $(q,m)$--creation operator $A^+(f)$ is boun In particular, its restriction to any $n$--particles space $\mathcal{H}_n$ is bounded. 2) $A(f):=(A

Theorems & Definitions (23)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • Remark 2.8
  • proof
  • ...and 13 more