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Boson normal ordering via substitutions and Sheffer-type polynomials

Paweł Błasiak, A. Horzela, K. A. Penson +2 more

arXiv:quant-ph/0501155Physics Letters AMar 1, 200543 citations

Coherent state and normal ordering method for transiting Hermite polynomials to Laguerre polynomials

Hong-Yi Fan, Jun Zhou

Science China Physics Mechanics and AstronomyMar 13, 201212 citations

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

K. A. Penson, Paweł Błasiak, G. Dattoli +3 more

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.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.

arXiv:quant-ph/0510079Journal of Physics Conference SeriesFeb 28, 20068 citations

Normal and antinormal ordering of boson operator functions

B. L. Willis, Paul F. Zweifel

Physics Letters AApr 1, 19851 citation

Sheffer polynomials and the s -ordering of exponential boson operators

Robert S. Maier

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.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.

arXiv:2508.13094Journal of Mathematical PhysicsDec 1, 20250 citations

The Weyl Algebra, Quantum Theory, and Normal Ordering

Chapman & Hall/CRC eBooksAug 20, 20150 citations

Representations of monomiality principle with Sheffer-type polynomials and boson normal ordering

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Representations of monomiality principle with Sheffer-type polynomials and boson normal ordering

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