Combinatorial Models of Creation-Annihilation

TL;DR

This paper explores the connection between combinatorial structures and the algebra of operators satisfying a specific quantum commutation relation, using labelled graphs and diagrams to systematically evaluate normal forms.

Contribution

It introduces a combinatorial framework linking classical structures to operator algebra normal form reduction, including extensions to q-analogues and multivariate models.

Findings

Normal form evaluations via labelled graphs and diagrams

Connections to set partitions, permutations, and lattice paths

Extensions to q-analogues and urn models

Abstract

Quantum physics has revealed many interesting formal properties associated with the algebra of two operators, A and B, satisfying the partial commutation relation AB-BA=1. This study surveys the relationships between classical combinatorial structures and the reduction to normal form of operator polynomials in such an algebra. The connection is achieved through suitable labelled graphs, or "diagrams", that are composed of elementary "gates". In this way, many normal form evaluations can be systematically obtained, thanks to models that involve set partitions, permutations, increasing trees, as well as weighted lattice paths. Extensions to q-analogues, multivariate frameworks, and urn models are also briefly discussed.