Combinatorial Algebra for second-quantized Quantum Theory

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.