  A geometric approach to some quantum statistical systems (including the
harmonic oscillator) is presented. We regard the (N+1)-dimensional Euclidean
{\it coordinate} system (X$^i$,$\tau$) as the quantum statistical system of N
quantum (statistical) variables (X$^i$) and one {\it Euclidean time} variable
($\tau$). Introducing a path (line or hypersurface) in this space
(X$^i$,$\tau$), we adopt the path-integral method to quantize the mechanical
system. This is a new view of (statistical) quantization of the {\it
mechanical} system. It is inspired by the {\it extra dimensional model},
appearing in the unified theory of forces including gravity, using the
bulk-boundary configuration. The system Hamiltonian appears as the {\it area}.
We show quantization is realized by the {\it minimal area principle} in the
present geometric approach. When we take a {\it line} as the path, the
path-integral expressions of the free energy are shown to be the ordinary ones
(such as N harmonic oscillators) or their simple variation. When we take a {\it
hyper-surface} as the path, the system Hamiltonian is given by the {\it area}
of the {\it hyper-surface} which is defined as a {\it closed-string
configuration} in the bulk space. In this case, the system becomes a O(N)
non-linear model. The two choices,\ (1) the {\it line element} in the bulk
($X^i,\tau $) and (2) the Hamiltonian(defined as the damping functional in the
path-integral) specify the system dynamics. After explaining this new approach,
we apply it to a topic in the 5 dimensional quantum gravity. We present a {\it
new standpoint} about the quantum gravity: (a)\ The metric (gravitational)
field is treated as the background (fixed) one;\ (b)\ The space-time
coordinates are not merely position-labels but are quantum (statistical)
variables by themselves. We show the recently-proposed 5 dimensional Casimir
energy is valid.
