We consider an algebraic formulation of Quantum Theory and develop a combinatorial model of the Heisenberg-Weyl algebra structure. It is shown that by lifting this structure to the richer algebra of graph operator calculus, we gain a simple interpret
ation involving, for example, the natural composition of graphs. This provides a deeper insight into the algebraic structure of Quantum Theory and sheds light on the intrinsic combinatorial underpinning of its abstract formalism.
Non-commutativity is ubiquitous in mathematical modeling of reality and in many cases same algebraic structures are implemented in different situations. Here we consider the canonical commutation relation of quantum theory and discuss a simple urn mo
del of the latter. It is shown that enumeration of urn histories provides a faithful realization of the Heisenberg-Weyl algebra. Drawing on this analogy we demonstrate how the operator normal forms facilitate counting of histories via generating functions, which in turn yields an intuitive combinatorial picture of the ordering procedure itself.
Quantum physics has revealed many interesting formal properties associated with the algebra of two operators, A and B, satisfying the partial commutation relation AB-BA=1. This study surveys the relationships between classical combinatorial structure
s and the reduction to normal form of operator polynomials in such an algebra. The connection is achieved through suitable labelled graphs, or diagrams, that are composed of elementary gates. In this way, many normal form evaluations can be systematically obtained, thanks to models that involve set partitions, permutations, increasing trees, as well as weighted lattice paths. Extensions to q-analogues, multivariate frameworks, and urn models are also briefly discussed.
We consider a general concept of composition and decomposition of objects, and discuss a few natural properties one may expect from a reasonable choice thereof. It will be demonstrated how this leads to multiplication and co- multiplication laws, the
reby providing a generic scheme furnishing combinatorial classes with an algebraic structure. The paper is meant as a gentle introduction to the concepts of composition and decomposition with the emphasis on combinatorial origin of the ensuing algebraic constructions.
We describe an algebra G of diagrams which faithfully gives a diagrammatic representation of the structures of both the Heisenberg-Weyl algebra H - the associative algebra of the creation and annihilation operators of quantum mechanics - and U(L_H),
the enveloping algebra of the Heisenberg Lie algebra L_H. We show explicitly how G may be endowed with the structure of a Hopf algebra, which is also mirrored in the structure of U(L_H). While both H and U(L_H) are images of G, the algebra G has a richer structure and therefore embodies a finer combinatorial realization of the creation-annihilation system, of which it provides a concrete model.
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 decompositi
on 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.
We consider properties of the operators D(r,M)=a^r(a^dag a)^M (which we call generalized Laguerre-type derivatives), with r=1,2,..., M=0,1,..., where a and a^dag are boson annihilation and creation operators respectively, satisfying [a,a^dag]=1. We o
btain explicit formulas for the normally ordered form of arbitrary Taylor-expandable functions of D(r,M) with the help of an operator relation which generalizes the Dobinski formula. Coherent state expectation values of certain operator functions of D(r,M) turn out to be generating functions of combinatorial numbers. In many cases the corresponding combinatorial structures can be explicitly identified.
We show that the formalism of hybrid polynomials, interpolating between Hermite and Laguerre polynomials, is very useful in the study of Motzkin numbers and central trinomial coefficients. These sequences are identified as special values of hybrid po
lynomials, a fact which we use to derive their generalized forms and new identities satisfied by them.
