  We consider the classical dynamics of bosonic and fermionic matrix variables
in complex Hilbert space, defined by a trace action, assuming cyclic invariance
under the trace and the presence of a global unitary invariance. With plausible
and explicitly stated assumptions, including the existence of a large hierarchy
of scale between the underlying dynamics and observed physics, we show that (1)
the equilibrium statistical mechanics of this matrix dynamics, in the canonical
ensemble, gives rise to an emergent quantum mechanics for many degrees of
freedom, including the standard canonical commutation/anticommutation relations
and the usual unitary Heisenberg and Schr\"odinger picture time evolutions of
operators and states, and (2) the fluctuation or Brownian motion corrections to
this thermodyamics lead to an energy-driven stochastic modification of the
Schr\"odinger equation, which is known to imply state vector reduction with
Born rule probabilities. Thus, quantum mechanics and its probabilistic
interpretation both arise as emergent phenomena in the underlying trace
dynamics.
