Phase Space Quantum Mechanics on Groups

The project aims to formulate and compute the important objects of phase space representation of quantum mechanics. These are the Weyl operators, Wigner functions and star- product of phase space functions. The phase spaces we are concerned with are, for the most part, Lie groups and their factor groups. Thus, we want to formulate quantum mechanics on spaces of functions whose parameters are symmetry operations on configuration spaces. Our main tools are the unitary representations of the Lie groups and the harmonic analysis arising from this.

Phase space quantum mechanics may be viewed as harmonic analysis on phase space, and the main tasks are the decomposition of functions and operators in terms of the simplest states. This brings the parallel development of time-frequency analysis and its methods also to our research.

Under our research program, the following finished and successfully defended their research for their Ph.D. Mathematics Degrees at the Ateneo de Manila University:

• Alexander J. Balsomo, West Visayas University, Iloilo City (late 2018)
• Laarni B. Natividad, Saint Louis University, Baguio City (late 2019)
• Daisy A. Romeo, Cebu Institute of Technology-University, Cebu City (February, 2021)
• Luis S. Silvestre, Jr., Ateneo De Manila University (April 2021)

A few scientific papers have been published from our efforts, and a few are currently in various stages of preparation, submitted and under review. One student, Francis D. Delloro, Ateneo De Naga, is currently working on phase space representation of finite and discrete quantum mechanics for his Ph.D. Mathematics Degree. A Ph.D. student from the Department of Physics of ADMU, Franulfo dela Cruz is also trying to use some of our mathematical methods to model the optical systems that he is working on.

Published papers

• Natividad, Laarni B., and Job A. Nable. “Wigner Functions and Weyl Operators on the Euclidean Motion Group.” Journal of Geometry and Symmetry in Physics 54 (2019): 37-54.
• Romeo, Daisy A., and Job A. Nable. “Generalized Weyl Quantization and Time.” Proceedings of the Twenty-Second International Conference on Geometry, Integrability and Quantization. Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, 2021.
• Natividad, Laarni B., and Job A. Nable. “Star Product on The Euclidean Motion Group in the Plane.” Proceedings of the Twenty-Second International Conference on Geometry, Integrability and Quantization. Institute of Biophysics and Biomedical Engineering, Bulgarian Academy of Sciences, 2021.