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A geometric approach to general quantum statistical systems (including theharmonic oscillator) is presented. It is applied to Casimir energy and thedissipative system with friction. We regard the (N+1)-dimensional Euclidean{\it coordinate} system (X$^i$,$\tau$) as the quantum statistical system of Nquantum (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 mechanicalsystem. This is a new view of (statistical) quantization of the {\itmechanical} system. The system Hamiltonian appears as the {\it area}. We showquantization is realized by the {\it minimal area principle} in the presentgeometric approach. When we take a {\it line} as the path, the path-integralexpressions of the free energy are shown to be the ordinary ones (such as Nharmonic oscillators) or their simple variation. When we take a {\ithyper-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-stringconfiguration} 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 thevisco-elastic system, and present a new method using the path-integral for thecalculation of the dissipative properties.

Geometric Approach to Quantum Statistical Mechanics and Application to Casimir Energy and Friction Properties