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A generic Hopf algebra for quantum statistical mechanics

Solomon, A. I.; Duchamp, G. H. E.; Blasiak, P.; Horzela, A. and Penson, K. A. (2010). A generic Hopf algebra for quantum statistical mechanics. Physica Scripta, 82(3) 038115.

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DOI (Digital Object Identifier) Link: http://dx.doi.org/10.1088/0031-8949/82/03/038115
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Abstract

In this paper, we present a Hopf algebra description of a bosonic quantum model, using the elementary combinatorial elements of Bell and Stirling numbers. Our objective in doing this is as follows. Recent studies have revealed that perturbative quantum field theory (pQFT) displays an astonishing interplay between analysis (Riemann zeta functions), topology (Knot theory), combinatorial graph theory (Feynman diagrams) and algebra (Hopf structure). Since pQFT is an inherently complicated study, so far not exactly solvable and replete with divergences, the essential simplicity of the relationships between these areas can be somewhat obscured. The intention here is to display some of the above-mentioned structures in the context of a simple bosonic quantum theory, i.e. a quantum theory of non-commuting operators that do not depend on space–time. The combinatorial properties of these boson creation and annihilation operators, which is our chosen example, may be described by graphs, analogous to the Feynman diagrams of pQFT, which we show possess a Hopf algebra structure. Our approach is based on the quantum canonical partition function for a boson gas.

Item Type: Journal Article
Copyright Holders: 2010 The Royal Swedish Academy of Sciences
ISSN: 0031-8949
Academic Unit/Department: Science > Physical Sciences
Item ID: 22894
Depositing User: Astrid Peterkin
Date Deposited: 02 Sep 2010 12:59
Last Modified: 25 Nov 2012 22:35
URI: http://oro.open.ac.uk/id/eprint/22894
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