  A geometric approach to general quantum statistical systems (including the
harmonic oscillator) is presented. It is applied to Casimir energy and the
dissipative system with friction. 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 paths (lines or hypersurfaces) 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. 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. We show the recently-proposed 5 dimensional Casimir energy
(ArXiv:0801.3064,0812.1263) is valid. We apply this approach to the
visco-elastic system, and present a new method using the path-integral for the
calculation of the dissipative properties.
