2006/04/20 by Robert W. Johnson, Johnson, Robert W.
Physics and Astronomy · #FOS: Physical sciences #Quantum Physics (quant-ph) #quant-ph
paper · pdf · doi:10.48550/arxiv.quant-ph/0604153
Submitted to Intern. J. Theor. Phys
arxiv created 2006/04/20 · arxiv updated 2009/12/01
I describe, in the simplified context of finite groups and their representations, a mathematical model for a physical system that contains both its quantum and classical aspects. The physically observable system is associated with the space containing elements fxf for f an element in the regular representation of a given finite group G. The Hermitian portion of fxf is the Wigner distribution of f whose convolution with a test function leads to a mathematical description of the quantum measurement process. Starting with the Jacobi group that is formed from the semidirect product of the Heisenberg group with its automorphism group SL(2,FN) for N an odd prime number I show that the classical phase space is the first order term in a series of subspaces of the Hermitian portion of fxf that are stable under SL(2,FN). I define a derivative that is analogous to a pseudodifferential operator to enable a treatment that parallels the continuum case. I give a new derivation of the Schrodinger-Weil representation of the Jacobi group. Keywords: quantum mechanics, finite group, metaplectic. PACS: 03.65.Fd; 02.10.De; 03.65.Ta.