Phase Space Representations of Discrete Quantum Systems and Unitary Group Representations

Final Defense
Phase Space Representations of Discrete Quantum Systems and Unitary Group Representations

by Francis D. Delloro
PhD Mathematics-Straight Candidate

Date: Monday, 28 October 2024
Time: 5 pm
Venue: SECA 321 (MJR room)

Advisers:
Job A. Nable, PhD
Ateneo de Manila University

Panelists:
Julius M. Basilla, PhD (Critic Reader)
University of the Philippines Diliman
Reginaldo M. Marcelo, PhD (Critic Reader)
Ateneo de Manila University
Raphael A. Guerrero, PhD
Ateneo de Manila University
Eden Delight P. Miro, PhD
Ateneo de Manila University
Adrian Roy L. Valdez, PhD
University of the Philippines Diliman

Technologies involving photonics, quantum computers, quantum information and many others require deep understanding of discrete and finite quantum systems. On the other hand, as is the case throughout history of quantum mechanics, such technological dis- coveries and innovations motivated many important mathematical developments. Fur- thermore, the technologies have allowed for the manipulation of small finite quantum systems making it possible to investigate foundational questions and problems in quan- tum mechanics, again requiring the appropriate mathematical set-up. In this work, a certain picture of quantum mechanics, called the phase space representation, will be presented in a fairly systematic way using harmonic analysis on finite groups. Basic ob- jects such as phase space, the phase space representation of quantum states and quan- tum observables will be constructed using group frames in Hilbert space and their math- ematical properties will be investigated.

Key Words: Star-product, Quantum Mechanics, Finite Groups, Frames, Measurement algebra