Wigner Functions for the Pair Angle and Orbital Angular Momentum

TL;DR

This paper develops well-defined Wigner functions for the angle and angular momentum pair using the Euclidean group E(2), revealing structural similarities to planar phase space and employing sinc functions for interpolation.

Contribution

It introduces a construction of Wigner functions for cylindrical phase space based on E(2) group representations, addressing the challenge of angle and angular momentum quantization.

Findings

Wigner functions for (theta,p) are structurally similar to planar phase space.

Sinc functions interpolate discrete angular momenta from continuous classical values.

Marginal distributions for angle and angular momentum are obtained via integration.

Abstract

The problem of constructing physically and mathematically well-defined Wigner functions for the canonical pair angle and angular momentum is solved. While a key element for the construction of Wigner functions for the planar phase space (q,p) in R^2 is the Heisenberg-Weyl group, the corresponding group for the cylindrical phase space (theta,p) in S^1 x R is the Euclidean group E(2) of the plane and its unitary representations. Here the angle theta is replaced by the pair (cos theta, sin 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 plays the sinc 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 angle (continuous) and angular momentum (discontinuous) are as usual uniquely obtained by appropriate integrations of the (theta,p) Wigner function. Among the examples discussed is an elementary system of simple "cat" states.