$SU(1,1)$ covariant $s$-parametrized maps

TL;DR

This paper introduces a method to compute s-parametrized phase-space maps for systems with SU(1,1) symmetry, linking different symbols via an invariant operator, with specific focus on Wigner functions on hyperbolic geometries.

Contribution

It provides a practical recipe for calculating s-parametrized maps in SU(1,1) symmetric systems, connecting Q and P symbols through an invariant operator.

Findings

Derived a connection between Q and P symbols using an invariant operator.

Analyzed the case of Wigner functions on hyperbolic geometries.

Provided a computational method for s-parametrized maps in SU(1,1) systems.

Abstract

We propose a practical recipe to compute the ${s}$-parametrized maps for systems with $SU(1,1)$ symmetry using a connection between the ${Q}$ and ${P} $ symbols through the action of an operator invariant under the group. The particular case of the self-dual (Wigner) phase-space functions, defined on the upper sheet of the two-sheet hyperboloid (or, equivalently, inside the Poincar\'{e} disc) are analyzed.