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arXiv:2212.04345

A generalized Fourier transform of the P-quasi distribution function

Dušan Popov

Published Dec 6, 2022• 0 citations
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Abstract

In the paper we made a generalization of the Fourier transform in the complex space, applicable to the pair of Husimi and P-quasi distributions, in the representation of nonlinear coherent states. Implicitly, our result is a generalization similar result of Mehta, but which referred only to the canonical coherent states (associated with the one-dimensional harmonic oscillator). Our result is valid for both types of coherent states (defined in the Barut-Girardello, respectively Klauder-Perelomov manner).

DOI
10.48550/arxiv.2212.04345

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In the paper we made a generalization of the Fourier transform in the complex space, applicable to the pair of Husimi and P-quasi distributions, in the representation of nonlinear coherent states. Implicitly, our result is a generalization similar result of Mehta, but which referred only to the canonical coherent states (associated with the one-dimensional harmonic oscillator). Our result is valid for both types of coherent states (defined in the Barut-Girardello, respectively Klauder-Perelomov manner).
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We propose a procedure to generalize the Husimi distribution to systems with continuous spectrum. We start examining a pioneering work, by Gazeau and Klauder, where the concept of coherent states for systems with discrete spectrum was extended to systems with continuous one. In the present article, we see the Husimi distribution as a representation of the density operator in terms of a basis of coherent states. There are other ways to obtain it, but we do not consider here. We specially discuss the problem of the continuous harmonic oscillator.
We construct two-parameters family of nonlinear coherent states by replacing the factorial in coefficients zn/n!z^n/\sqrt{n!}zn/n!​ of the canonical coherent states by a specific generalized factorial xnγ,σ!x_n^{γ,σ}!xnγ,σ​! where parameters γγγ and σσσ satisfy some conditions for which the normalization condition and the resolution of identity are verified. The obtained family is a generalization of the Barut-Girardello coherent states and those of the philophase states. In the particular case of parameters γγγ and σσσ, we attache these states to the pseudo-harmonic oscillator depending on two parameters α,β>0α,β> 0α,β>0. The obtained nonlinear coherent states are superposition of eigenstates of this oscillator. The associated Bargmann-type transform is defined and we derive some results.
Continuing the line of development of Paper I [J. Math. Phys. 15, 1295 (1974)], we enlarge the concept of canonical transformations in quantum mechanics in two directions: first, by allowing the definition of a canonical transformation to be made through the preservation of an so(2,1) algebra, rather than the usual Heisenberg algebra, and providing the bridge between the classical and quantum mechanical descriptions, and, second, through the complexification of the transformation group. In this paper we study specifically the transformations which can be interpreted as the radial part of n-dimensional complex linear transformations in Paper I. We show that we can build Hilbert spaces of analytic functions with a scalar product defined through integration over half the complex plane of a variable which has the meaning of a complex radius. A unitary mapping to the ordinary Hilbert space Lrn−12(0,∞) is provided with a kernel involving a Bessel function. Special cases of this are shown to be the Barut-Girardello, one-dimensional Bargmann and Hankel transforms. The transform kernels provide a series of representations of a subsemigroup of SL(2,C) and allow the construction of coherent states for the harmonic oscillator with an extra centrifugal force. We present a hyperdifferential operator realization of these transforms which yields new Baker-Campbell-Hausdorff and special function relations.