Combinatorial Algebra for second-quantized Quantum Theory

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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(Lₕ), the enveloping algebra of the Heisenberg Lie algebra Lₕ. We show explicitly how G may be endowed with the structure of a Hopf algebra, which is also mirrored in the structure of U(Lₕ). While both H and U(Lₕ) 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.