Number Operator Algebras

TL;DR

This paper classifies all quadratic algebras generated by creation and destruction operators with number operators, including bosons, fermions, and their generalizations, under certain symmetry and confluence conditions.

Contribution

It provides a complete enumeration of such algebras and introduces pseudo-bosons and pseudo-fermions as new generalizations.

Findings

Identifies all algebras of this type under given hypotheses

Shows these include known particles and their generalizations

Recovers q-bosons through algebra completion

Abstract

Under some hypotheses (symmetry, confluence), we enumerate all quadratically presented algebras, generated by creation and destruction operators, in which number operators exist. We show that these are algebras of bosons, fermions, their immediate generalizations that we call pseudo-bosons and pseudo-fermions, and also matrix algebras, in the finitely generated case. We then recover q-bosons (and pseudo-q-bosons) by a completion operation.