Some Implications of Invariant Boltzmann Statistical Mechanics to Quantum Gravity and Noncommutative Geometry of Physical Space and its Fractal Spectral Dimension.

According to invariant Boltzmann statistical mechanics [1], Kelvin absolute temperature T [K] is identified as Wien wavelength $\lambda _{\mathrm{w\beta -1}}$ [m] of thermal oscillations leading to \textit{internal} \textit{measures} of spacetime $(\lambda_{w\beta -1} ,\tau_{w\beta -1} )$ and \textit{external} \textit{measures} of space and time $(x_{\beta } =N_{x} \lambda_{w\beta -1} ,t_{\beta } =N_{t} \tau _{w\beta -1} )$. Therefore, temperature of space or Casimir vacuum fixes local measures of \textit{spacetime} $(\lambda_{w\beta -1} ,\tau_{w\beta -1} )$that are not \textit{independent} because $v_{ws} =\lambda_{ws} /\tau_{ws} $ must satisfy the vacuum temperature. Since Wien displacement law $\lambda_{w} T=0.29\mbox{\thinspace \thinspace cm-K\thinspace =\thinspace 0.0029\thinspace [m}^{2}]$ requires the change of units [m/cm] $=$ 100, the classical temperature conversion formula becomes $T[m]=^{o}\mbox{C[m]}\mbox{\thinspace +\thinspace 2.731}$ with 2.731 close to Penzias-Wilson [1965] cosmic microwave background radiation temperature $T_{CMB} \simeq 2.73\mbox{\thinspace [m}]$. The role of analytic functions, Cauchy-Riemann conditions, and possible imaginary nature of internal spacetime coordinates, due to connections to Riemann surfaces at lower scale $\beta \quad -$1, on path-independence of trajectories of quantum transitions and Heisenberg equation of motion are discussed. Finally, some implications of the hydrodynamic model to quantum gravity as a dissipative deterministic system [2] and fractal spectral dimension of noncommutative geometry of space [3] are examined. $^{\mathrm{1}}$ Sohrab, S. H.,\textit{ ASME J. Energy Resoures Technology} \textbf{138}, 1-12 (2016). $^{\mathrm{2}}$ `t Hooft, G., \textit{Quantum Grav}. \textbf{16}, 3263 (1999). $^{\mathrm{3\thinspace }}$Connes, A., \textit{Lett. Math. Phys}. \textbf{34}, 238 (1995).