3,514 research outputs found
Normal Order: Combinatorial Graphs
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.Comment: 7 pages, 15 references, 2 figures. Presented at "Progress in
Supersymmetric Quantum Mechanics" (PSQM'03), Valladolid, Spain, July 200
Wick's theorem for q-deformed boson operators
In this paper combinatorial aspects of normal ordering arbitrary words in the
creation and annihilation operators of the q-deformed boson are discussed. In
particular, it is shown how by introducing appropriate q-weights for the
associated ``Feynman diagrams'' the normally ordered form of a general
expression in the creation and annihilation operators can be written as a sum
over all q-weighted Feynman diagrams, representing Wick's theorem in the
present context.Comment: 9 page
Heisenberg-Weyl algebra revisited: Combinatorics of words and paths
The Heisenberg-Weyl algebra, which underlies virtually all physical
representations of Quantum Theory, is considered from the combinatorial point
of view. We provide a concrete model of the algebra in terms of paths on a
lattice with some decomposition rules. We also discuss the rook problem on the
associated Ferrers board; this is related to the calculus in the normally
ordered basis. From this starting point we explore a combinatorial underpinning
of the Heisenberg-Weyl algebra, which offers novel perspectives, methods and
applications.Comment: 5 pages, 3 figure
Dobinski-type relations: Some properties and physical applications
We introduce a generalization of the Dobinski relation through which we
define a family of Bell-type numbers and polynomials. For all these sequences
we find the weight function of the moment problem and give their generating
functions. We provide a physical motivation of this extension in the context of
the boson normal ordering problem and its relation to an extension of the Kerr
Hamiltonian.Comment: 7 pages, 1 figur
Combinatorics and Boson normal ordering: A gentle introduction
We discuss a general combinatorial framework for operator ordering problems
by applying it to the normal ordering of the powers and exponential of the
boson number operator. The solution of the problem is given in terms of Bell
and Stirling numbers enumerating partitions of a set. This framework reveals
several inherent relations between ordering problems and combinatorial objects,
and displays the analytical background to Wick's theorem. The methodology can
be straightforwardly generalized from the simple example given herein to a wide
class of operators.Comment: 8 pages, 1 figur
Combinatorially interpreting generalized Stirling numbers
Let be a word in alphabet with 's and 's.
Interpreting "" as multiplication by , and "" as differentiation with
respect to , the identity , valid
for any smooth function , defines a sequence , the terms of
which we refer to as the {\em Stirling numbers (of the second kind)} of .
The nomenclature comes from the fact that when , we have , the ordinary Stirling number of the second kind.
Explicit expressions for, and identities satisfied by, the have been
obtained by numerous authors, and combinatorial interpretations have been
presented. Here we provide a new combinatorial interpretation that retains the
spirit of the familiar interpretation of as a count of
partitions. Specifically, we associate to each a quasi-threshold graph
, and we show that enumerates partitions of the vertex set of
into classes that do not span an edge of . We also discuss some
relatives of, and consequences of, our interpretation, including -analogs
and bijections between families of labelled forests and sets of restricted
partitions.Comment: To appear in Eur. J. Combin., doi:10.1016/j.ejc.2014.07.00
Normally ordered forms of powers of differential operators and their combinatorics
We investigate the combinatorics of the general formulas for the
powers of the operator h∂k, where h is a central element of a ring
and ∂ is a differential operator. This generalizes previous work on
the powers of operators h∂. New formulas for the generalized Stirling
numbers are obtained.Ministerio de EconomÃa y competitividad MTM2016-75024-PJunta de AndalucÃa P12-FQM-2696Junta de AndalucÃa FQM–33
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