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Quasi-probability distributions for the simplest dynamical groups.

A B Klimov¹, S M Chumakov

¹Departamento de Física, Universidad de Guadalajara, Jalisco, Mexico.

Journal of the Optical Society of America. A, Optics, Image Science, and Vision

|January 5, 2001

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We show how the Wigner-Stratonovich-Agarwal operator for SU(2) quasi-probability distributions can be integrated over a single variable. This simplifies the distribution and allows for contractions to Heisenberg-Weyl and Euclidean groups.

Area of Science:

Quantum mechanics
Mathematical physics
Group theory

Background:

The Wigner-Stratonovich-Agarwal operator is crucial for defining quasi-probability distributions in quantum mechanics.
The SU(2) dynamical group is fundamental in describing systems with spherical symmetry.

Purpose of the Study:

To reformulate the Wigner-Stratonovich-Agarwal operator for SU(2) quasi-probability distributions.
To establish a connection between SU(2) quasi-probability distributions and simpler group structures.

Main Methods:

Representing the Wigner-Stratonovich-Agarwal operator as an integral of SU(2) representation elements.
Utilizing the coadjoint representation to identify relevant variables for integration.
Applying group contraction techniques.

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Main Results:

The Wigner-Stratonovich-Agarwal operator for SU(2) can be expressed as an integral over a single variable labeling coadjoint orbits.
This integral representation facilitates the contraction of SU(2) quasi-probability distributions.
The study successfully connects SU(2) distributions to those of the Heisenberg-Weyl and two-dimensional Euclidean groups.

Conclusions:

The reformulated operator offers a more tractable approach to SU(2) quasi-probability distributions.
Group contraction provides a pathway to understand simpler quantum systems from more complex ones.
This work bridges advanced group theory with fundamental concepts in quantum probability.