  We consider an optomechanical system comprising a single cavity mode and a
dense spectrum of acoustic modes and solve for the quantum dynamics of initial
cavity mode Fock (i.e., photon number) superposition states and thermal
acoustic states. The optomechanical interaction results in dephasing without
damping and bears some analogy to gravitational decoherence. For a cavity mode
locally coupled to a one-dimensional (1D) elastic string-like environment or
two-dimensional (2D) elastic membrane-like environment, we find that the
dephasing dynamics depends respectively on the string length and membrane
area--a consequence of an infrared divergence in the limit of an infinite-sized
string or membrane. On the other hand, for a cavity mode locally coupled to a
three-dimensional (3D) bulk elastic solid, the dephasing dynamics is
independent of the solid volume (i.e., is infrared finite), but dependent on
the local geometry of the coupled cavity--a consequence of an ultraviolet
divergence in the limit of a "pointlike" coupled cavity. We consider as
possible respective realizations for the cavity-coupled-1D and 2D acoustic
environments, an LC oscillator capacitively coupled to a partially metallized
strip and a cavity light mode interacting via light pressure with a membrane.
