Exponential Operators, Dobinski Relations and Summability

We investigate properties of exponential operators preserving the particle number using combinatorial methods developed in order to solve the boson normal ordering problem. In particular, we apply generalized Dobinski relations and methods of multivariate Bell polynomials which enable us to understand the meaning of perturbation-like expansions of exponential operators. Such expansions, obtained as formal power series, are everywhere divergent but the Pade summation method is shown to give results which very well agree with exact solutions got for simplified quantum models of the one mode bosonic systems.Comment: Presented at XIIth Central European Workshop on Quantum Optics, Bilkent University, Ankara, Turkey, 6-10 June 2005. 4 figures, 6 pages, 10 reference

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

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

Bernoulli type polynomials on Umbral Algebra

The aim of this paper is to investigate generating functions for modification of the Milne-Thomson's polynomials, which are related to the Bernoulli polynomials and the Hermite polynomials. By applying the Umbral algebra to these generating functions, we provide to deriving identities for these polynomials

Combinatorial approach to generalized Bell and Stirling numbers and boson normal ordering problem

We consider the numbers arising in the problem of normal ordering of expressions in canonical boson creation and annihilation operators. We treat a general form of a boson string which is shown to be associated with generalizations of Stirling and Bell numbers. The recurrence relations and closed-form expressions (Dobiski-type formulas) are obtained for these quantities by both algebraic and combinatorial methods. By extensive use of methods of combinatorial analysis we prove the equivalence of the aforementioned problem to the enumeration of special families of graphs. This link provides a combinatorial interpretation of the numbers arising in this normal ordering problem.Comment: 10 pages, 5 figure

Hierarchical Dobinski-type relations via substitution and the moment problem

We consider the transformation properties of integer sequences arising from the normal ordering of exponentiated boson ([a,a*]=1) monomials of the form exp(x (a*)^r a), r=1,2,..., under the composition of their exponential generating functions (egf). They turn out to be of Sheffer-type. We demonstrate that two key properties of these sequences remain preserved under substitutional composition: (a)the property of being the solution of the Stieltjes moment problem; and (b) the representation of these sequences through infinite series (Dobinski-type relations). We present a number of examples of such composition satisfying properties (a) and (b). We obtain new Dobinski-type formulas and solve the associated moment problem for several hierarchically defined combinatorial families of sequences.Comment: 14 pages, 31 reference

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

Some useful combinatorial formulae for bosonic operators

We give a general expression for the normally ordered form of a function F(w(a,a*)) where w is a function of boson annihilation and creation operators satisfying [a,a*]=1. The expectation value of this expression in a coherent state becomes an exact generating function of Feynman-type graphs associated with the zero-dimensional Quantum Field Theory defined by F(w). This enables one to enumerate explicitly the graphs of given order in the realm of combinatorially defined sequences. We give several examples of the use of this technique, including the applications to Kerr-type and superfluidity-type hamiltonians.Comment: 8 pages, 3 figures, 17 reference

Monomiality principle, Sheffer-type polynomials and the normal ordering problem

We solve the boson normal ordering problem for $(q(a^\dag)a+v(a^\dag))^n$ with arbitrary functions $q(x)$ and $v(x)$ and integer $n$, where $a$ and $a^\dag$ are boson annihilation and creation operators, satisfying $[a,a^\dag]=1$. This consequently provides the solution for the exponential $e^{\lambda(q(a^\dag)a+v(a^\dag))}$ generalizing the shift operator. In the course of these considerations we define and explore the monomiality principle and find its representations. We exploit the properties of Sheffer-type polynomials which constitute the inherent structure of this problem. In the end we give some examples illustrating the utility of the method and point out the relation to combinatorial structures.Comment: Presented at the 8'th International School of Theoretical Physics "Symmetry and Structural Properties of Condensed Matter " (SSPCM 2005), Myczkowce, Poland. 13 pages, 31 reference

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