9,989 research outputs found
On quantum symmetries of ADE graphs
The double triangle algebra(DTA) associated to an ADE graph is considered. A
description of its bialgebra structure based on a reconstruction approach is
given. This approach takes as initial data the representation theory of the DTA
as given by Ocneanu's cell calculus. It is also proved that the resulting DTA
has the structure of a weak *-Hopf algebra. As an illustrative example, the
case of the graph A3 is described in detail.Comment: 15 page
A generic Hopf algebra for quantum statistical mechanics
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.Comment: 8 pages/(4 pages published version), 1 Figure. arXiv admin note: text
overlap with arXiv:1011.052
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Hopf algebra structure of a model quantum field theory
Recent elegant work[1] on the structure of Perturbative Quantum Field Theory (PQFT) has revealed an astonishing interplay between analysis( Riemann Zeta functions), topology (Knot theory), combinatorial graph theory (Feynman Diagrams) and algebra (Hopf structure). The difficulty inherent in the complexities of a fully-fledged field theory such as PQFT means that the essential beauty of the relationships between these areas can be somewhat obscured. Our intention is to display some, although not all, of these structures in the context of a simple zero-dimensional field theory; i.e. a quantum theory of non-commuting operators which do not depend on spacetime. The combinatorial properties of these boson creation and annihilation operators, which is our chosen example, may be described by graphs [2, 3], analogous to the Feynman diagrams of PQFT, which we show possess a Hopf algebra structure[4]. Our approach is based on the partition function for a boson gas. In a subsequent note in these Proceedings we sketch the relationship between the Hopf algebra of our simple model and that of the PQFT algebra
Hopf algebras and the combinatorics of connected graphs in quantum field theory
In this talk, we are concerned with the formulation and understanding of the
combinatorics of time-ordered n-point functions in terms of the Hopf algebra of
field operators. Mathematically, this problem can be formulated as one in
combinatorics or graph theory. It consists in finding a recursive algorithm
that generates all connected graphs in their Hopf algebraic representation.
This representation can be used directly and efficiently in evaluating Feynman
graphs as contributions to the n-point functions.Comment: 10 pages, 2 figures, LaTeX + AMS + eepic; to appear in the
proceedings of the Conference on Combinatorics and Physics, MPIM Bonn, March
19-23, 200
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