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Physical Review A

Comment on “Full quantum reconstruction of vortex states”

João P. S. Bizarro

Published Jan 13, 2010• 2 citations
Publisher

Abstract

It is pointed out that the bona fide Wigner distribution recently introduced [Phys. Rev. A 78, 060101(R) (2008)] is in fact the rotational Wigner function earlier derived and thoroughly studied [Phys. Rev. A 49, 3255 (1994); 71, 069901(E) (2005)], so the latter remains indeed the natural phase-space representation for the rotation-angle and angular-momentum pair.

DOI
10.1103/physreva.81.016101

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It is pointed out that the bona fide Wigner distribution recently introduced [Phys. Rev. A 78, 060101(R) (2008)] is in fact the rotational Wigner function earlier derived and thoroughly studied [Phys. Rev. A 49, 3255 (1994); 71, 069901(E) (2005)], so the latter remains indeed the natural phase-space representation for the rotation-angle and angular-momentum pair.
It is pointed out that in a recent paper (2012 New J. Phys. 14 103009) in which a Wigner function for a particle in an infinite lattice (a system described by an unbounded discrete coordinate and its conjugate angle-like momentum) has been introduced, no reference is made to previous, pioneering work on discrete Wigner distributions (more precisely, on the rotational Wigner function for a system described by a rotation angle and its unbounded discrete-conjugate angular momentum). Not only has the problem addressed in essence been solved for a long time (the discrete coordinate and angle-like conjugate momentum are the perfect dual of the rotation angle and discrete-conjugate angular momentum), but the solution advanced only in some distorted manner obeys two of the fundamental properties of a Wigner distribution (that, when integrated over one period of the momentum variable, it should yield the correct marginal distribution on the discrete position variable, and that it should be invariant with respect to translation).
The problem of constructing physically and mathematically well-defined Wigner functions for the canonical pair angle $\ensuremath{\theta}$ and angular momentum ppp is solved. While a key element for the construction of Wigner functions for the planar phase space ${(q,p)\ensuremath{\in}{\mathbb{R}}^{2}}$ is the Heisenberg-Weyl group, the corresponding group for the cylindrical phase space ${(\ensuremath{\theta},p)\ensuremath{\in}{S}^{1}\ifmmode\times\else\texttimes\fi{}\mathbb{R}}$ is the Euclidean group E(2)E(2)E(2) of the plane and its unitary representations. Here the angle $\ensuremath{\theta}$ is replaced by the pair $(cos\ensuremath{\theta},sin\ensuremath{\theta})$, which corresponds uniquely to the points on the unit circle. The main structural properties of the Wigner functions for the planar and the cylindrical phase spaces are strikingly similar. A crucial role is played by the sincsincsinc function, which provides the interpolation for the discontinuous quantized angular momenta in terms of the continuous classical ones, in accordance with the famous Whittaker cardinal function well known from interpolation and sampling theories. The quantum mechanical marginal distributions for the angle (continuous) and angular momentum (discontinuous) are, as usual, uniquely obtained by appropriate integrations of the $(\ensuremath{\theta},p)$ Wigner function. Among the examples discussed is an elementary system of simple cat states.
Recently a paper on the construction of consistent Wigner functions for cylindrical phase spaces ${\mathbb{S}}^{1}\ifmmode\times\else\texttimes\fi{}\mathbb{R}$, i.e., for the canonical pair angle and orbital angular momentum, was published [H. A. Kastrup, Phys. Rev. A 94, 062113 (2016)] in which the main properties of these functions are derived and discussed and their usefulness is illustrated with examples. The present paper is a continuation which compares the properties of the Wigner functions for cylindrical phase spaces with those of the well-known Wigner functions for planar phase spaces in more detail. Furthermore, the mutual (Weyl) correspondence between Hilbert space operators and their phase-space functions is discussed. The $\ensuremath{\star}$ product formalism is shown to be completely implementable. In addition, basic dynamical laws for Wigner and Moyal functions are derived as generalized Liouville and energy equations. They are very similar to those in the planar case but also show characteristic differences.
We discuss a family of quasidistributions (sss-ordered Wigner functions of Agarwal and Wolf [Phys. Rev. D 2, 2161 (1970); Phys. Rev. D 2, 2187 (1970); Phys. Rev. D 2, 2206 (1970)]) and its connection to the so-called phase space representation of the Schr\"odinger equation. It turns out that although Wigner functions satisfy the Schr\"odinger equation in phase space, they have a completely different interpretation.