Fundamental Phase Space Formula for the Similitude Group

• Main Authors: Natividad, Laarni B., Nable, Job A.
• Format: text
• Published: Archīum Ateneo 2024
• Subjects: Mathematics; Physical Sciences and Mathematics
• Online Access: https://archium.ateneo.edu/mathematics-faculty-pubs/249; https://doi.org/10.1063/5.0192115
• Institution: Ateneo De Manila University

In this work, the statement and proof of a fundamental formula in the phase space representation of quantum systems will be carried out for the similitude group, Sim(2). This formula takes the form ∫ a(Y)P(Y)d(Y) = {A}, where Y is the phase space variable and {A} is a linear operator on Hilbert space representing a quantum dynamical observable. {A} is the quantum expected value of the observable in a state of the system. The focus on the similitude group is due to current interest in signal analysis, localization operators and pseudo-differential operators. The fundamental formula states that this may be computed in a classical manner, as an integral against a probability distribution. The formula is intimately related to the quantization-dequantization problem a(Y) ↔ A which assigns a quantum operator to the classical phase space function a(Y).