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Register a new userGeometric Approach to Quantum Statistical Mechanics and Minimal Area Principle
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Authors
Shoichi Ichinose
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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 $) and (2) the Hamiltonian(defined as the damping functional in the path-integral) specify the system dynamics. After explaining this new approach, we apply it to a topic in the 5 dimensional quantum gravity. We present a {it new standpoint} about the quantum gravity: (a) The metric (gravitational) field is treated as the background (fixed) one; (b) The space-time coordinates are not merely position-labels but are quantum (statistical) variables by themselves. We show the recently-proposed 5 dimensional Casimir energy is valid.
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A geometric approach to general quantum statistical systems (including the harmonic oscillator) is presented. It is applied to Casimir energy and the dissipative system with friction. We regard the (N+1)-dimensional Euclidean {it coordinate} system (X$^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.
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