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arXiv:quant-ph/0510079 · Journal of Physics Conference Series

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

K. A. Penson, Paweł Błasiak, G. Dattoli, Gérard Duchamp, A. Horzela, A. I. Solomon

Published Feb 28, 2006• 8 citations
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Abstract

We solve the boson normal ordering problem for ( q ( a † ) a + v ( a † )) n with arbitrary functions q ( x ) and v ( x ) and integer n , where a and a † are boson annihilation and creation operators, satisfying [ a , a † ] = 1. This consequently provides the solution for the exponential e λ( q ( a † ) a + v ( a † )) 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.

DOI
10.1088/1742-6596/30/1/012

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We solve the boson normal ordering problem for ( q ( a † ) a + v ( a † )) n with arbitrary functions q ( x ) and v ( x ) and integer n , where a and a † are boson annihilation and creation operators, satisfying [ a , a † ] = 1. This consequently provides the solution for the exponential e λ( q ( a † ) a + v ( a † )) 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.
The s-ordered form of any product of single-mode boson creation and annihilation operators, containing only a single annihilator, is computed explicitly. The s-ordering concept originated in quantum optics, but subsumes normal, symmetric (Weyl), and anti-normal ordering for any two operators satisfying a canonical commutation relation. Because the s-ordering map can be viewed as producing a function of a complex variable, its inverse is a quantization map that takes such “classical” functions to quantum operators. The explicit s-ordered expressions are derived with the aid of a parametric family of Sheffer polynomial sequences (or equivalently a parametric exponential Riordan array of polynomial coefficients), called the Hsu–Shiue family. To yield orderings interpolating between normal and anti-normal, this family must be extended.
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 we discuss to a wide class of operators.
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 Dobiński 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 Padé 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.
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.
We derive explicit formulas for the normal ordering of powers of arbitrary monomials of boson operators. These formulas lead to generalisations of conventional Bell and Stirling numbers and to appropriate generalisations of the Dobihski relations. These new combinatorial numbers are shown to be coherent state matrix elements of powers of the monomials in question. It is further demonstrated that such Bell-type numbers, when considered as power moments, give rise to positive measures on the positive half-axis, which in many cases can be written in terms of known functions.