O(N)symmetries, sum rules for generalized Hermite polynomials and squeezed states

Jamil Daboul, Salomon S Mizrahi
2004 Journal of Physics A: Mathematical and General  
Quantum optics has been dealing with coherent states, squeezed states and many other non-classical states. The associated mathematical framework makes use of special functions as Hermite polynomials, Laguerre polynomials and others. In this connection we here present some formal results that follow directly from the group O(N) of complex transformations. Motivated by the squeezed states structure, we introduce the generalized Hermite polynomials (GHP), which include as particular cases, the
more » ... ular cases, the Hermite polynomials as well as the heat polynomials. Using generalized raising operators, we derive new sum rules for the GHP, which are covariant under O(N) transformations. The GHP and the associated sum rules become useful for evaluating Wigner functions in a straightforward manner. As a byproduct, we use one of these sum rules, on the operator level, to obtain raising and lowering operators for the Laguerre polynomials and show that they generate an sl(2, R) su(1, 1) algebra.
doi:10.1088/0305-4470/38/2/010 fatcat:qtrcvv2cx5etha2q3cuyxznv34