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On Equivalence of Quantum Liouville Equation and Metric Compatibility Condition, a Ricci Flow Approach
Version 1
: Received: 8 December 2016 / Approved: 9 December 2016 / Online: 9 December 2016 (02:13:06 CET)
Version 2 : Received: 23 February 2022 / Approved: 25 February 2022 / Online: 25 February 2022 (02:34:16 CET)
Version 2 : Received: 23 February 2022 / Approved: 25 February 2022 / Online: 25 February 2022 (02:34:16 CET)
How to cite: Amiri, M. On Equivalence of Quantum Liouville Equation and Metric Compatibility Condition, a Ricci Flow Approach. Preprints 2016, 2016120049. https://doi.org/10.20944/preprints201612.0049.v1 Amiri, M. On Equivalence of Quantum Liouville Equation and Metric Compatibility Condition, a Ricci Flow Approach. Preprints 2016, 2016120049. https://doi.org/10.20944/preprints201612.0049.v1
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 ∇0gij = 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.
Keywords
Quantum Liouville equation; metric compatibility condition; Joint probability; Binary Data Matrix; Ricci flow
Subject
Physical Sciences, Quantum Science and Technology
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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