Physics – Quantum Physics
Scientific paper
2004-02-12
Physics
Quantum Physics
7 pages, 15 references, 2 figures. Presented at "Progress in Supersymmetric Quantum Mechanics" (PSQM'03), Valladolid, Spain, J
Scientific paper
A conventional context for supersymmetric problems arises when we consider systems containing both boson and fermion operators. In this note we consider the normal ordering problem for a string of such operators. In the general case, upon which we touch briefly, this problem leads to combinatorial numbers, the so-called Rook numbers. Since we assume that the two species, bosons and fermions, commute, we subsequently restrict ourselves to consideration of a single species, single-mode boson monomials. This problem leads to elegant generalisations of well-known combinatorial numbers, specifically Bell and Stirling numbers. We explicitly give the generating functions for some classes of these numbers. In this note we concentrate on the combinatorial graph approach, showing how some important classical results of graph theory lead to transparent representations of the combinatorial numbers associated with the boson normal ordering problem.
Blasiak Pawel
Duchamp Gérard
Horzela Andrzej
Penson Karol A.
Solomon Allan I.
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