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The Schrödinger representation of Heisenberg group in terms of linear canonical transform

Amit Kumar, Akhilesh Prasad

Published Dec 14, 2022• 0 citations
Publisher

Abstract

Abstract In this paper, we deduce the Heisenberg group by using the translation and modulation operators of the linear canonical transform(LCT) in place of the Fourier transform and corresponding Schr\"{o}dinger representation is given. Apart from this the linear canonical Wigner transform is defined. Moreover, various analytic expressions, such as Schr\"{o}dinger representation is a unitary operator on $ L^{2}(\mathbb{R}^{n}) $ are discussed.

DOI
10.21203/rs.3.rs-2356122/v1

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Abstract In this paper, we deduce the Heisenberg group by using the translation and modulation operators of the linear canonical transform(LCT) in place of the Fourier transform and corresponding Schr"{o}dinger representation is given. Apart from this the linear canonical Wigner transform is defined. Moreover, various analytic expressions, such as Schr"{o}dinger representation is a unitary operator on L2(Rn)L^{2}(\mathbb{R}^{n})L2(Rn) are discussed.
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.
A classical theorem of Stone and von Neumann says that the Schrödinger representation is, up to unitary equivalences, the only irreducible representation of the Heisenberg group on the Hilbert space of square-integrable functions on configuration space. Using the Wigner-Moyal transform we construct an irreducible representation of the Heisenberg group on a certain Hilbert space of square-integrable functions defined on phase space. This allows us to extend the usual Weyl calculus into a phase-space calculus and leads us to a quantum mechanics in phase space, equivalent to standard quantum mechanics. We also briefly discuss the extension of metaplectic operators to phase space and the probabilistic interpretation of the solutions of the phase space Schrödinger equation
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 algebra of generalized linear quantum canonical transformations is examined in the prespective of Schwinger's unitary-canonical basis. Formulation of the quantum phase problem within the theory of quantum canonical transformations and in particular with the generalized quantum action-angle phase space formalism is established and it is shown that the conceptual foundation of the quantum phase problem lies within the algebraic properties of the quantum canonical transformations in the quantum phase space. The representations of the Wigner function in the generalized action-angle unitary operator pair for certain Hamiltonian systems with the dynamical symmetry are examined. This generalized canonical formalism is applied to the quantum harmonic oscillator to examine the properties of the unitary quantum phase operator as well as the action-angle Wigner function.