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arXiv · 2508.13094

Sheffer Polynomials and the s-ordering of Exponential Boson Operators

Robert S. Maier
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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.

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Robert S. Maier. 2025-08-18. Sheffer Polynomials and the s-ordering of Exponential Boson Operators. https://doi.org/10.1063/5.0297108

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