Boson Normal Ordering via Substitutions and Sheffer-type Polynomials

P Blasiak , A Horzela , K A Penson (1) , G H E Duchamp (2) , a I Solomon

1 LPTL - Laboratoire de Physique Théorique des Liquides
2 LIPN - Laboratoire d'Informatique de Paris-Nord

We solve the boson normal ordering problem for (q(a*)a + v(a*))^n with arbitrary functions q and v and integer n, where a and a* are boson annihilation and creation operators, satisfying [a,a*]=1. This leads to exponential operators generalizing the shift operator and we show that their action can be expressed in terms of substitutions. Our solution is naturally related through the coherent state representation to the exponential generating functions of Sheffer-type polynomials. This in turn opens a vast arena of combinatorial methodology which is applied to boson normal ordering and illustrated by a few examples.

Mots clés: boson normal ordering problem; coherent states; Sheffer-type polynomials; generating functions

Domaines: Physique Quantique [quant-ph]; Combinatoire [math.CO]; Mathématique discrète [cs.DM]

P Blasiak, A Horzela, K A Penson, G H E Duchamp, a I Solomon. Boson Normal Ordering via Substitutions and Sheffer-type Polynomials. 2005. ⟨hal-00004077⟩