Combinatorial Algebra for second-quantized Quantum Theory

Abstract

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