A geometric approach to the friction phenomena is presented. It is based on the holographic view which has recently been popular in the theoretical physics community. We see the system in one-dimension-higher space. The heat-producing phenomena are m
ost widely treated by using the non-equilibrium statistical physics. We take 2 models of the earthquake. The dissipative systems are here formulated from the geometric standpoint. The statistical fluctuation is taken into account by using the (generalized) Feynmans path-integral.
Boltzmann equation describes the time development of the velocity distribution in the continuum fluid matter. We formulate the equation using the field theory where the {it velocity-field} plays the central role. The matter (constituent particles) fi
elds appear as the density and the viscosity. {it Fluctuation} is examined, and is clearly discriminated from the quantum effect. The time variable is {it emergently} introduced through the computational process step. The collision term, for the (velocity)**4 potential (4-body interaction), is explicitly obtained and the (statistical) fluctuation is closely explained. The present field theory model does {it not} conserve energy and is an open-system model. (One dimensional) Navier-Stokes equation or Burgers equation, appears. In the latter part, we present a way to directly define the distribution function by use of the geometry, appearing in the mechanical dynamics, and Feynmans path-integral.
A new field theory formulation is presented for the analysis of the CMB power spectrum distribution in the cosmology. The background-field formalism is fully used. Stimulated by the recent idea of the {it emergent} gravity, the gravitational (metric)
field $g_mn$ is not taken as the quantum-field, but as the background field. The statistical fluctuation effect of the metric field is taken into account by the path (hyper-surface)-integral over the space-time. Using a simple scalar model on the curved (dS$_4$) space-time, we explain the above things with the following additional points: 1) Clear separate treatment of the classical effect, the statistical effect and the quantum effect; 2) The cosmological fluctuation comes not from the quantum gravity but from the unkown microscopic movement; 3) IR parameter ($ell$) is introduced for the time axis as the periodicity. Time reversal(Z$_2$)-symmetry is introduced in order to treat the problem separately with respect to the Z$_2$ parity. This procedure much helps both UV and IR regularization to work well.
In order to understand the dynamical mechanism of the friction phenomena, we heavily rely on the numerical analysis using various methods: molecular dynamics, Langevin equation, lattice Boltzmann method, Monte Carlo, e.t.c.. We propose a new method w
hich has the following characteristic points: 1) the geometrical approach to the statistical mechanical system; 2) the continuum approach using Feynmans path integral (generalized version); 3) the holographic (higher-dimensional) approach; 4) the renormalization phenomenon takes place in order to treat the statistical fluctuation.
Boltzmann equation describes the time development of the velocity distribution in the continuum fluid matter. We formulate the equation using the field theory where the {it velocity-field} plays the central role. The properties of the fluid matter (f
luid particles) appear as the density and the viscosity. {it Statistical fluctuation} is examined, and is clearly discriminated from the quantum effect. The time variable is {it emergently} introduced through the computational process step. Besides the ordinary potential, the general velocity potential is introduced. The collision term, for the Higgs-type velocity potential, is explicitly obtained and the (statistical) fluctuation is closely explained. The system is generally {it non-equilibrium}. The present field theory model does {it not} conserve energy and is an open-system model. One dimensional Navier-Stokes equation, i.e., Burgers equation, appears. In the latter part of the text, we present a way to directly define the distribution function by use of the geometry, appearing in the energy expression, and Feynmans path-integral.
We regard the Casimir energy of the universe as the main contribution to the cosmological constant. Using 5 dimensional models of the universe, the flat model and the warped one, we calculate Casimir energy. Introducing the new regularization, called
{it sphere lattice regularization}, we solve the divergence problem. The regularization utilizes the closed-string configuration. We consider 4 different approaches: 1) restriction of the integral region (Randall-Schwartz), 2) method of 1) using the minimal area surfaces, 3) introducing the weight function, 4) {it generalized path-integral}. We claim the 5 dimensional field theories are quantized properly and all divergences are renormalized. At present, it is explicitly demonstrated in the numerical way, not in the analytical way. The renormalization-group function ($be$-function) is explicitly obtained. The renormalization-group flow of the cosmological constant is concretely obtained.
Electromagnetism in substance is characterized by permittivity (dielectric constant) and permeability (magnetic permeability). They describe the substance property {it effectively}. We present a {it geometric} approach to it. Some models are presente
d, where the two quantities are geometrically defined. Fluctuation due to the micro dynamics (such as dipole-dipole interaction) is taken into account by the (generalized) path-integral. Free energy formula (Lifshitz 1954), for the material composed of three regions with different permittivities, is explained. Casimir energy is obtained by a new regularization using the path-integral. Attractive force or repulsive one is determined by the sign of the {it renormalization-group} $beta$-function.
Casimir energy is calculated for 5D scalar theory in the {it warped} geometry. A new regularization, called {it sphere lattice regularization}, is taken. The regularized configuration is {it closed-string like}. We numerically evaluate $La$(4D UV-cut
off), $om$(5D bulk curvature, extra space UV-boundary parameter) and $T$(extra space IR-boundary parameter) dependence of Casimir energy. 5D Casimir energy is {it finitely} obtained after the {it proper renormalization procedure.} The {it warp parameter} $om$ suffers from the {it renormalization effect}. Regarding Casimir energy as the main contribution to the cosmological term, we examine the dark energy problem.
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.
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) var
iables (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.
