On Equivalence of Quantum Liouville Equation and Metric Compatibility Condition, a Ricci Flow Approach

Abstract

In this paper after introducing a model of binary data matrix for physical measurements of an evolving system (of particles), we develop a Hilbert space as an ambient space to derive induced metric tensor on embedded parametric manifold identified by associated joint probabilities of particles observables (parameters). Parameter manifold assumed as space-like hypersurface evolving along time axis, an approach that resembles 3+1 formalism of ADM and numerical relativity. We show the relation of endowed metric with related density matrix. Identification of system density matrix by this metric tensor, leads to the equivalence of quantum Liouville equation and metric compatibility condition ∇ = 0 while covariant derivative of metric tensor has been calculated respect to Wick rotated time coordinate. After deriving a formula for expected energy of the particles and imposing the normalized Ricci flow as governing dynamics, we prove the equality of this expected energy with local scalar curvature of related manifold. Consistency of these results with Einstein tensor, field equations and Einstein-Hilbert action has been verified. Given examples clarify the compatibility of the results with well-known principles. This model provides a background for geometrization of quantum mechanics compatible with curved manifolds and information geometry.

Generalized Inflation Theory as a solution to Galaxies Rotation Curves And Variation of fine structure constant

“…From the main equations of the BDM model that introduced in previous article [1] we conclude that all base vectors of all phase space parameters inflate through time through the equation:…”

“…This acceleration replaces the Newtonian acceleration around the distance 0 and dominates it at the astronomical distances comparable to it. Therefore, we get two separable acceleration, Newtonian which is proportional to 1 2 and BDM derived acceleration that is proportional to 1 . At the astronomical distances the Newtonian acceleration decreases rapidly and approaches to 0 as the final constant velocity that corresponds to ∞ in MOND theory.…”

“…By introducing the concept of generalized inflation (expansion), I extend the cosmic inflation to a broader concept including all phase space physical parameters. This generalization was achieved by the results of Binary Data Matrix model [1]. Incorporating the linear and angular momentum with spatial parameters in universe expansion leads to a formalism that interprets both rotation curve of galaxies and evolution of fine structure constant.…”

“…The two remaining models i.e. NBDM (Non-Baryonic Dark Matter) and MOND (Modified Newtonian Dynamics) have been challenged for their unsatisfactory results or mysterious nature [1]. Velocity (rotation) curves of distant stars relative to their galaxy's center do not obey the Newtonian dynamics.…”

A Physical Theory of Information vs A Mathematical Theory of Communication

“…If we Consider [ * ] as the dual base vectors in the parameter space (phase space), the inner product (4) gives the dual metric tensor of this space. Author in his previous work [13] proved that the evolution of this metric tensor over time,…”