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# Advances in Theoretical and Mathematical Physics

## Volume 14 (2010)

### Number 4

### Combinatorial Algebra for second-quantized Quantum Theory

Pages: 1209 – 1243

DOI: http://dx.doi.org/10.4310/ATMP.2010.v14.n4.a5

#### Authors

#### 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.