2010/04/15 by Shoichi Ichinose, Ichinose, Shoichi · 3 citations
Physics and Astronomy · #Black Holes and Theoretical Physics #Noncommutative and Quantum Gravity Theories #Cosmology and Gravitation Theories
paper · pdf · doi:10.48550/arxiv.1004.2573
A geometric approach to some quantum statistical systems (including the harmonic oscillator) is presented. We regard the (N+1)-dimensional Euclidean \it coordinate system (Xi,τ) as the quantum statistical system of N quantum (statistical) variables (Xi) and one \it Euclidean time variable (τ). Introducing a path (line or hypersurface) in this space (Xi,τ), 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 (Xi,τ) 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.