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Journal of Physics A Mathematical and Theoretical

A probabilistic approach of sum rules for heat polynomials

Christophe Vignat, O Lévêque

Published Nov 29, 2011• 0 citations
Publisher

Abstract

In this paper, we show that the sum rules for generalized Hermite polynomials derived by Daboul and Mizrahi (2005 J. Phys. A: Math. Gen. 38 427-48) and by Graczyk and Nowak (2004 C. R. Acad. Sci., Ser. 1 338 849) can be interpreted and easily recovered using a probabilistic moment representation of these polynomials. The covariance property of the raising operator of the harmonic oscillator, which is at the origin of the identities proved in Daboul and Mizrahi and the dimension reduction effect expressed in the main result of Graczyk and Nowak are both interpreted in terms of the rotational invariance of the Gaussian distributions. As an application of these results, we uncover a probabilistic moment interpretation of two classical integrals of the Wigner function that involve the associated Laguerre polynomials.

DOI
10.1088/1751-8113/45/1/015001

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In this paper, we show that the sum rules for generalized Hermite polynomials derived by Daboul and Mizrahi (2005 J. Phys. A: Math. Gen. 38 427-48) and by Graczyk and Nowak (2004 C. R. Acad. Sci., Ser. 1 338 849) can be interpreted and easily recovered using a probabilistic moment representation of these polynomials. The covariance property of the raising operator of the harmonic oscillator, which is at the origin of the identities proved in Daboul and Mizrahi and the dimension reduction effect expressed in the main result of Graczyk and Nowak are both interpreted in terms of the rotational invariance of the Gaussian distributions. As an application of these results, we uncover a probabilistic moment interpretation of two classical integrals of the Wigner function that involve the associated Laguerre polynomials.
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 , which include as particular cases, the Hermite polynomials as well as the heat polynomials. Using generalized raising operators, we derive new sum rules for the , which are covariant under O ( N ) transformations. The 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.
Hagedorn wave packets appear as the eigenstates of multidimensional harmonic oscillators, and are given by a polynomial times a Gaussian. We recognise the arising polynomials for normalised and unnormalised Hagedorn wave packets as generalised multivariate Hermite polynomials, which are not tensor products in general. We provide formulas for generating functions and ladder operators, and prove a direct connection to the Laguerre polynomials. As our main result, we show that the class of Hagedorn wave packets is invariant under the Wigner transform, and present an explanation for the product structure of normalised Hagedorn wave packets in phase space, which has recently been discovered.
By virtue of the coherent state representation and the operator ordering method we find a new approach for transiting Hermite polynomials to Laguerre polynomials.We also derive the new reciprocal relation of Laguerre polynomials n=0(1) n l n L n(x) = x l l!,and its application in deriving the sum rule of the Wingner function of Fock states is demonstrated.Some new expansion identities about the operator Laguerre polynomial are also derived.This opens a new route of deriving mathematical polynomials formulas by virtute of the quantum mechanical representations and operator ordering technique.
Hermite polynomials with many variables and many indices play a crucial role within the framework of phase-space formulation of classical or quantum mechanics. A generalized operational formalism useful to handle these polynomials is discussed and also a new set of creation–annihilation operators associated to the phase-space harmonic oscillator orthogonal functions is introduced. The Fourier transform of these orthogonal functions is studied by discussing the analogies with the ordinary case. Finally, the integral representation of the generalized Hermite polynomials and a simple technique to deal with the generalized heat equation are discussed.
Here we explore the link between the moments of the Laguerre polynomials or Laguerre moments and the generalized functions (as the Dirac delta-function and its derivatives), presenting several interesting relations. A useful application is related to a procedure for calculating mean values in quantum optics that makes use of the so-called quasi-probabilities. One of them, the P-distribution , can be represented by a sum over Laguerre moments when the electromagnetic field is in a photon-number state. Consequently, the P-distribution can be expressed in terms of Dirac delta-function and derivatives. More specifically, we found a direct relation between P-distributions and the Laguerre factorial moments .
A set of orthonormalized and complete functions of two real variables in complex representation involving Laguerre polynomials as a substantial part is introduced and is referred to as the set of Laguerre two-dimensional (2D) functions. The properties of the set of Laguerre 2D-functions are discussed and the Fourier and Radon transforms of these functions are calculated. The Laguerre 2D-functions form a basis for a realization of the five-dimensional Lie algebra to the Heisenberg-Weyl group for a two-mode system. Real representation of this set of functions by a sum over products of Hermite functions involving Jacobi polynomials at the zero argument as coefficients is derived and it leads to new connections between Laguerre and Hermite polynomials in both directions. The set of Laguerre 2D-functions is the most appropriate set of functions for the Fock-state representation of quasi-probabilities in quantum optics. The Wigner quasi-probability in Fock-state representation is up to a factor and an argument scaling directly given for each matrix element by a corresponding Laguerre 2D-function. The properties of orthonormality and completeness of the Laguerre 2D-functions provide the Fock-state matrix elements of the density operator directly from the quasi-probabilities. The Perina-Mista representation of the Glauber-Sudarshan quasi-probability can be represented with advantage by the Laguerre 2D-functions. The Fock-state matrix elements of the displacement operator and the scalar product of displaced Fock states are closely related to Laguerre 2D-functions.