# Wigner functions for angle and orbital angular momentum: Operators and dynamics

## TL;DR

This paper extends the theory of Wigner functions to cylindrical phase spaces involving angle and angular momentum, comparing their properties with planar cases and deriving dynamical laws.

## Contribution

It introduces a detailed comparison of Wigner functions for cylindrical and planar phase spaces and develops their dynamical equations and operator correspondences.

## Key findings

- Wigner functions for cylindrical phase spaces are consistent and comparable to planar cases.
- The star product formalism is fully applicable to cylindrical phase spaces.
- Derived generalized quantum Liouville and energy equations for these Wigner functions.

## Abstract

Recently a paper on the construction of consistent Wigner functions for cylindrical phase spaces S^1 x R, i.e. for the canonical pair angle and angular momentum, was presented (arXiv:1601.02520), main properties of those functions derived, discussed and their usefulness illustrated by examples. The present paper is a continuation which compares properties of the new Wigner functions for cylindrical phase spaces with those of the well-known Wigner functions on planar ones in more detail. Furthermore, the mutual (Weyl) correspondence between Hilbert space operators and their phase space functions is discussed. The star product formalism is shown to be completely implementable. In addition basic dynamical laws for the new Wigner and Moyal functions are derived as generalized quantum Liouville and energy equations. They are very similar to those of the planar case, but also show characteristic differences.