Geometric Approach to Quantum Statistical Mechanics and Minimal Area Principle

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.