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Gaussian Transforms in 2N-dimensional Phase Space

Tan Si Do

Published Apr 8, 2023• 0 citations
Publisher

Abstract

In order to obtain with simplicity the properties of linear canonical transformations (LCTs), it is introduced the notion of dual couple of operators having commutator identical to unity and the property that all relations between them are valuable for any other dual couple. It follows that from the translation operator exp (a\({\partial}\)x) which transforms the dual couple (\({\partial}\)x, \(\hat{X}\) ) into (\({\partial}\)x, \(\hat{X}\) + al) one obtains the dilatation operator exp (aBA) which transforms the dual couple (A, B) into (e -al A, e -al B) then the operator exp (aA2) exp (aB2) and so all, leading to the construction of general linear and linear canonical transformations in phase spaces. Moreover, are also obtained the LCT transforms of functions. By this way different kinds of LCTs such as Fast Fourier, Fourier, Laplace, Xin Ma and Rhodes, Baker-Campbell-Haussdorf, Bargman transforms are found again. We also obtain a clear relationship between linear and linear canonical transforms from the formula representing the aforementioned integral realisation. Numerous LCT examples are provided to highlight the method's ease of use.

DOI
10.9734/bpi/rhmcs/v8/18145d

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In order to obtain with simplicity the properties of linear canonical transformations (LCTs), it is introduced the notion of dual couple of operators having commutator identical to unity and the property that all relations between them are valuable for any other dual couple. It follows that from the translation operator exp (a({\partial})x) which transforms the dual couple (({\partial})x, (\hat{X}) ) into (({\partial})x, (\hat{X}) + al) one obtains the dilatation operator exp (aBA) which transforms the dual couple (A, B) into (e -al A, e -al B) then the operator exp (aA2) exp (aB2) and so all, leading to the construction of general linear and linear canonical transformations in phase spaces. Moreover, are also obtained the LCT transforms of functions. By this way different kinds of LCTs such as Fast Fourier, Fourier, Laplace, Xin Ma and Rhodes, Baker-Campbell-Haussdorf, Bargman transforms are found again. We also obtain a clear relationship between linear and linear canonical transforms from the formula representing the aforementioned integral realisation. Numerous LCT examples are provided to highlight the method's ease of use.
In order to obtain with simplicity the known and new properties of linear canonical transformations (LCTs), we show that any relation between a couple of operators (A,B) having commutator identical to unity, called dual couple in this work, is valuable for any other dual couple, so that from the known translation operator exp(a∂x) one may obtain the explicit form and properties of a category of linear and linear canonical transformations in 2N-phase spaces. Moreover, other forms of LCTs are also obtained in this work as so as the transforms by them of functions by integrations as so as by derivations. In this way, different kinds of LCTs such as Fast Fourier, Fourier, Laplace, Xin Ma and Rhodes, Baker-Campbell-Haussdorf, Bargman transforms are found again.
The main objective of the paper is to generalize and enrich the Weyl transform by introducing the Weyl correspondence in the linear canonical transform (LCT) domain. In this paper, we propose the linear canonical-Wigner transform in harmonic analysis of phase space along with the admissible Wigner-Ville distribution (WVD) and Weyl transform in the LCT domain and discuss some useful results. Further we establish the relationship between the Wigner-Ville distribution and the Weyl transform in the LCT domain.
The geometry of classical phase space is determined by the Poisson bracket of functions on phase space. Canonical transformations of the classical coordinates and momenta are defined by the property that the fundamental Poisson brackets be preserved. In this section we shall study in detail the linear canonical transformations in phase space and their representations in quantum mechanics. The reason for developing these concepts is twofold. First of all there are a number of geometric transformations common to classical and quantum mechanics which appear naturally in the analysis of many-body systems. Examples are rotations of the three space coordinates, permutations of single-particle coordinates and transformations to various types of relative vectors. To these we shall add translations of coordinates and momenta and construct the operators which represent these geometric transformations in the Hilbert space of quantum mechanics. The second reason for dealing with these transformations is the fact that certain operators encountered in quantum mechanics may be interpreted as representatives of underlying geometric transformations in classical phase space. This applies in particular to dilatation operators and to Gaussian interactions. Recognition of the underlying geometric transformation allows us to reduce operator multiplication to combination on the geometric level like matrix multiplication.
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.
In the last few years an extensive literature has developed on linear canonical tranformations and their representation in quantum mechanics. Applications of these results have been made to clustering theory in nuclei, problems of accidental degeneracy, etc. In the present paper we wish to turn our attention to nonlinear canonical transformations. We show that by dealing with appropriate functions fα (α=1,...,2n) of xi, pi (i=1,...,n) rather than with these variables themselves, we can in principle set unambiguously the equations that determine the representation in quantum mechanics of the canonical transformation under study. This result holds when the old and new functions fα have the same spectrum. We discuss specific examples when this last condition is satisfied: nonlinear canonical transformations in the radial variable that were obtained from projection of linear ones in higher-dimensional spaces; canonical transformations that take us from one Hamiltonian to another with the same spectrum, be this one continuous or discrete; canonical transformations that relate two sets of integrals of motion (which include the Hamiltonians) when we are dealing with phase spaces of dimensions higher than 2, etc. We discuss briefly, in the concluding section, the possibility of extending our analysis to canonical tranformations that do not conserve the spectrum of the relevant operators.
Relative to an irreducible representation of the canonical commutation relations, convolutions between quantum mechanical operators and between functions and operators are defined, for which the usual Weyl transform acts as a Fourier transform. Basic properties of these operations are developed in close analogy to harmonic analysis on R2n. Using the quantum version of Wiener’s approximation theorem, a natural one-to-one correspondence between the closed, phase-space translation invariant subspaces of classical and quantum observables is established.