paper

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

arXiv:1010.5558 · doi:10.1088/1742-6596/258/1/012003

Abstract

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,) as the quantum statistical system of N quantum (statistical) variables (X) and one {\it Euclidean time} variable (). Introducing paths (lines or hypersurfaces) in this space (X,), 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.

20 pages, 8 figures, Proceedings of ICFS2010 (2010.9.13-18, Ise-Shima, Mie, Japan)

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