Matrix Wigner Function and SU(1,1)
- URL: http://arxiv.org/abs/2306.01238v1
- Date: Fri, 2 Jun 2023 02:05:26 GMT
- Title: Matrix Wigner Function and SU(1,1)
- Authors: P. G. Morrison
- Abstract summary: We give an overview of the technique as it is applied to some simple differential systems.
We show that by expanding the solution space to the hyperbolic plane and utilising some results from matrix calculus, we are able to recover a number of interesting identities.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: This paper contains a brief sketch of some methods that can be used to obtain
the Wigner function for a number of systems. We give an overview of the
technique as it is applied to some simple differential systems related to
diffusion problems in one dimension. We compute the Wigner function for the
harmonic oscillator, the $xp$ interaction, and a hyperbolic oscillator. These
systems are shown to share several properties in common related to the
Whittaker function and various formulae for the Laguerre polynomials. To
contrast with the techniques that are applicable to problems involving
continuous states, we then show that by expanding the solution space to the
hyperbolic plane and utilising some results from matrix calculus, we are able
to recover a number of interesting identities for SU(1,1) and the pseudosphere.
We close with a discussion of some more advanced topics in the theory of the
Wigner function.
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