  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.
