Pseudo-bosons, so far

TL;DR

This paper reviews recent mathematical developments and examples of pseudo-bosons, a deformation of canonical commutation relations where the creation and annihilation operators are not necessarily adjoints, highlighting their significance in physics and mathematics.

Contribution

It provides a comprehensive review of the mathematical properties and examples of pseudo-bosons, expanding understanding of deformed canonical commutation relations.

Findings

Pseudo-bosons arise from a deformation of $[a,a^\u00d7]=\1$ to $[a,b]=\1$

Mathematical properties of pseudo-bosons are systematically analyzed

Several physical and mathematical examples of pseudo-bosons are discussed

Abstract

In the past years several extensions of the canonical commutation relations have been proposed by different people in different contexts and some interesting physics and mathematics have been deduced. Here, we review some recent results on the so-called {\em pseudo-bosons}. They arise from a special deformation of the canonical commutation relation $[a,a^\dagger]=\1$, which is replaced by $[a,b]=\1$, with $b$ not necessarily equal to $a^\dagger$. We start discussing some of their mathematical properties and then we discuss several examples.