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August 2010 Combinatorial Algebra for second-quantized Quantum Theory
Pawel Blasiak, Gerard H.E. Duchamp, Allan I. Solomon, Andrzej Horzela, Karol A. Penson
Adv. Theor. Math. Phys. 14(4): 1209-1243 (August 2010).


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.


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Pawel Blasiak. Gerard H.E. Duchamp. Allan I. Solomon. Andrzej Horzela. Karol A. Penson. "Combinatorial Algebra for second-quantized Quantum Theory." Adv. Theor. Math. Phys. 14 (4) 1209 - 1243, August 2010.


Published: August 2010
First available in Project Euclid: 10 August 2011

zbMATH: 1229.81349
MathSciNet: MR2821397

Rights: Copyright © 2010 International Press of Boston

Vol.14 • No. 4 • August 2010
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