  A new idea for the quantization of dynamic systems, as well as space time
itself, using a stochastic metric is proposed. The quantum mechanics of a mass
point is constructed on a space time manifold using a stochastic metric. A
stochastic metric space is, in brief, a metric space whose metric tensor is
given stochastically according to some appropriate distribution function. A
mathematically consistent model of a space time manifold equipping a stochastic
metric is proposed in this report. The quantum theory in the local Minkowski
space can be recognized as a classical theory on the stochastic
Lorentz-metric-space. A stochastic calculus on the space time manifold is
performed using white noise functional analysis. A path-integral quantization
is introduced as a stochastic integration of a function of the action integral,
and it is shown that path-integrals on the stochastic metric space are
mathematically well-defined for large variety of potential functions. The
Newton--Nelson equation of motion can also be obtained from the Newtonian
equation of motion on the stochastic metric space. It is also shown that the
commutation relation required under the canonical quantization is consistent
with the stochastic quantization introduced in this report.
  The quantum effects of general relativity are also analyzed through natural
use of the stochastic metrics. Some example of quantum effects on the universe
is discussed.
