Sheffer Polynomials and the s-ordering of Exponential Boson Operators
Robert S. Maier
TL;DR
This paper develops a comprehensive, algebraic framework for s-ordering of single-mode exponential boson operators by marrying quantum-optical orderings with the theory of Sheffer polynomials and Riordan arrays.The authors first recap s-ordering, its special cases (normal, Weyl, anti-normal), and transform relations, then introduce a two-fold mathematical machinery—exponential Riordan arrays and HS polynomials—to systematize orderings of exponentials of boson strings ${a^ dag}^L a {a^ dag}^R$.They introduce a new parametric two-point Hsu–Shiue family that yields explicit, s-dependent representations of e^{λ(a†^L a a†^R)} for arbitrary s, with the main result unifying and extending known s-orderings and delivering Weyl-order formulas that are, in general, nontrivial to obtain.The results are illustrated through detailed examples for e=0,1,2, showing closed-form expressions for normal, Weyl, and anti-normal orderings, and highlighting the complexity that arises at higher e when solving the associated inverse Riordan functions.
Abstract
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.