1997/04/03 by D. Han, Y. S. Kim, Han, D. +3
Chemistry · Physics and Astronomy · #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Inorganic Fluorides and Related Compounds #Molecular Spectroscopy and Structure #Quantum Physics (quant-ph) #hep-th #quant-ph
paper · pdf · doi:10.48550/arxiv.quant-ph/9704005
RevTeX, 24 pages, no figures, preliminary version in J. Math. Phys. {\bf 36}, 3940 (1995)
arxiv created 1997/04/03 · openalex publication_date 1997/04/03 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In classical mechanics, the system of two coupled harmonic oscillators is shown to possess the symmetry of the Lorentz group O(3,3) applicable to a six-dimensional space consisting of three space-like and three time-like coordinates, or SL(4,r) in the four-dimensional phase space consisting of two position and two momentum variables. In quantum mechanics, the symmetry is reduced to that of O(3,2) or Sp(4), which is a subgroup of O(3,3) or SL(4,r) respectively. It is shown that among the six Sp(4)-like subgroups, only one possesses the symmetry which can be translated into the group of unitary transformations in quantum mechanics. In quantum mechanics, there is the lower bound in the size of phase space for each mode determined by the uncertainty principle while there are no restriction on the phase-space size in classical mechanics. This is the reason why the symmetry is smaller in quantum mechanics.