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AIP conference proceedings

Quantization of time of arrival functions via harmonic analysis on the Euclidean motion group on the plane

Daisy A. Romeo, Job A. Nable

Published Jan 1, 2024• 0 citations
Publisher

Abstract

This work presents a generalized Weyl quantization for the time of arrival function for a system consisting of a free particle confined to move on a line segment.The generalized Weyl quantization relies crucially on the phase space of the classical system being quantized and on the choice of the group of symmetries of this space, which is non-unique.The Hilbert space of the corresponding quantum system is the representation Hilbert space of the unitary irreducible representations of the symmetry group.For the system considered here, the phase space of our time of arrival function is the cylinder × ℝ and the symmetry group (2) is the Euclidean motion group of the plane.We consider two coordinate systems leading to two distinct quantization.

DOI
10.1063/5.0192163

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This work presents a generalized Weyl quantization for the time of arrival function for a system consisting of a free particle confined to move on a line segment.The generalized Weyl quantization relies crucially on the phase space of the classical system being quantized and on the choice of the group of symmetries of this space, which is non-unique.The Hilbert space of the corresponding quantum system is the representation Hilbert space of the unitary irreducible representations of the symmetry group.For the system considered here, the phase space of our time of arrival function is the cylinder × ℝ and the symmetry group (2) is the Euclidean motion group of the plane.We consider two coordinate systems leading to two distinct quantization.
This work presents coherent state quantization f↦∫f(x)|X〉〈X|dμ(X) applied to the classical time of arrival functions of the form T(q,p)=-μqp, expressed in different coordinate systems, corresponding to phase spaces of the p confined and free particle, respectively. The most important ingredient in this technique is the coherent state. Two ways of constructing coherent states will be adopted:via unitary group representations and via superposition method, applied to time of arrival function of one – dimensional confined and free particles.
The Weyl–Heisenberg symmetries originate from translation invariances of various manifolds viewed as phase spaces, e.g. Euclidean plane, semi-discrete cylinder, torus, in the two-dimensional case, and higher-dimensional generalizations. In this paper, we describe, on an elementary level, how this symmetry emerges through displacement operators and standard Fourier analysis, and how their unitary representations are used both in Signal Analysis (time–frequency techniques, Gabor transform) and in quantum formalism (covariant integral quantizations and semi-classical portraits). An example of application of the formalism to the Majorana stellar constellation in the plane is presented.
The Weyl-Heisenberg symmetries originate from translation invariances of various manifolds viewed as phase spaces, e.g. Euclidean plane, semi-discrete cylinder, torus, in the two-dimensional case, and higher-dimensional generalisations. In this review we describe, on an elementary level, how this symmetry emerges through displacement operators and standard Fourier analysis, and how their unitary representations are used both in Signal Analysis (time-frequency techniques, Gabor transform) and in quantum formalism (covariant integral quantizations and semi-classical portraits). An example of application of the formalism to the Majorana stellar constellation in the plane is presented.
The Wigner distribution function is one of the pillars of the phase space formulation of quantum mechanics. Its original formulation may be cast in terms of the unitary representations of the Weyl–Heisenberg group. Following the construction proposed by Wolf and coworkers in constructing the Wigner functions for general Lie groups using the irreducible unitary representations of the groups, we develop here the Wigner functions and Weyl operators on the Euclidean motion group of rank three. We give complete derivations and proofs of their important properties.
Covariant integral quantizations are based on the resolution of the identity by continuous or discrete families of normalized positive operator valued measures (POVM), which have appealing probabilistic content and which transform in a covariant way. One of their advantages is their ability to circumvent problems due to the presence of singularities in the classical models. In this paper, we implement covariant integral quantizations for systems whose phase space is Z×S1, i.e., for systems moving on the circle. The symmetry group of this phase space is the discrete & compact version of the Weyl–Heisenberg group, namely the central extension of the abelian group Z×SO(2). In this regard, the phase space is viewed as the right coset of the group with its center. The non-trivial unitary irreducible representation of this group, as acting on L2(S1), is square integrable on the phase space. We show how to derive corresponding covariant integral quantizations from (weight) functions on the phase space and resulting resolution of the identity. As particular cases of the latter we recover quantizations with de Bièvre-del Olmo–Gonzales and Kowalski–Rembielevski–Papaloucas coherent states on the circle. Another straightforward outcome of our approach is the Mukunda Wigner transform. We also look at the specific cases of coherent states built from shifted gaussians, Von Mises, Poisson, and Fejér kernels. Applications to stellar representations are in progress.
Any quantization maps linearly functions on a phase space to symmetric operators in a Hilbert space. Covariant integral quantization combines operator-valued measure with symmetry group of the phase space. Covariant means that the quantization map intertwines classical (geometric operation) and quantum (unitary transformations) symmetries. Integral means that we use all ressources of integral calculus, in order to implement the method when we apply it to singular functions, or distributions, for which the integral calculus is an essential ingredient. In this paper we emphasize the deep connection between Fourier transform and covariant integral quantization when the Weyl-Heisenberg and affine groups are involved. We show with our generalisations of the Wigner-Weyl transform that many properties of the Weyl integral quantization, commonly viewed as optimal, are actually shared by a large family of integral quantizations.