We review some highlights from the String theory, the black hole physics and At temperatures exceeding the Planck energy, we obtain the gas’ maximum internal energy N∕2β and recover the zero heat capacity of an ideal gas. The partition function of a gas of noninteracting particles is evaluated. Finally, we obtain a path-integral expression for the partition function of a thermodynamic system using the maximally-localized states.
#Coherence principle free
The free particle momentum-space propagator and the propagator for maximal localization are analytically evaluated up to quadratic-order in β. As application, we use the maximally-localized position in developing the Euclidean path-integral and introduce the compact form of the propagator for maximal localization. Furthermore, we show that the maximally-localized position is diagonal in momentum-space and that the maximally-localized position and its adjoint satisfy commutation and anti-commutation relations reminiscent of the harmonic oscillator commutation and anti-commutation relations. We calculate the overlap between maximally-localized states and show that the identity operator can be expressed in terms of the maximally-localized states. In this paper, we look at maximally-localized states as eigenstates of the operator ξ=X+iβP that we refer to as the maximally-localized position. In dealing with quantum mechanics at very high energies, it is essential to adapt to a quasiposition representation using the maximally-localized states because of the generalized uncertainty principle. Using the conformal gravity theory we provide a first principles derivation of the empirical Tully–Fisher relation. We present a new empirical plot of the difference between the observed centripetal accelerations and the luminous Newtonian expectations as a function of distance from the centers of galaxies, and show that at distances greater than 10 kpc the plot also occupies a remarkably narrow band, one even close to constant. We show that at very low luminous Newtonian accelerations the plot can become independent of the luminous Newtonian contribution altogether, but still be non-trivial due to the contribution of matter outside of the galaxies (viz. While one could summarize the mean properties of the band by drawing a single mean curve through it, by fitting the band with the illustrative conformal gravity theory with fits that fill out the width of the band we show here that the width of the band is just as physically significant. In a recent paper McGaugh, Lelli, and Schombert showed that in an empirical plot of the observed centripetal accelerations in spiral galaxies against those predicted by the Newtonian gravity of the luminous matter in those galaxies the data points occupied a remarkably narrow band.
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