## Advances in Theoretical and Mathematical Physics

### 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.

#### Article information

Source
Adv. Theor. Math. Phys., Volume 14, Number 4 (2010), 1209-1243.

Dates
First available in Project Euclid: 10 August 2011

Permanent link to this document
https://projecteuclid.org/euclid.atmp/1312998219

Mathematical Reviews number (MathSciNet)
MR2821397

Zentralblatt MATH identifier
1229.81349

#### Citation

Blasiak, Pawel; Duchamp, Gerard H.E.; Solomon, Allan I.; Horzela, Andrzej; Penson, Karol A. Combinatorial Algebra for second-quantized Quantum Theory. Adv. Theor. Math. Phys. 14 (2010), no. 4, 1209--1243. https://projecteuclid.org/euclid.atmp/1312998219