Combinatorial Algebra for second-quantized Quantum Theory

TL;DR

This paper introduces a diagrammatic algebra G that models the Heisenberg-Weyl algebra and its enveloping algebra, providing a richer combinatorial framework for second-quantized quantum systems.

Contribution

It presents a new diagrammatic algebra G with a Hopf algebra structure that offers a detailed combinatorial representation of quantum creation and annihilation operators.

Findings

G faithfully represents the Heisenberg-Weyl algebra

G has a richer structure than H and U(L_H)

G provides a concrete combinatorial model for quantum systems

Abstract

We describe an algebra G of diagrams which faithfully gives a diagrammatic representation of the structures of both the Heisenberg-Weyl algebra H - the associative algebra of the creation and annihilation operators of quantum mechanics - and U(L_H), the enveloping algebra of the Heisenberg Lie algebra L_H. We show explicitly how G may be endowed with the structure of a Hopf algebra, which is also mirrored in the structure of U(L_H). While both H and U(L_H) are images of G, the algebra G has a richer structure and therefore embodies a finer combinatorial realization of the creation-annihilation system, of which it provides a concrete model.