Kaluza--Klein gravity & cosmology emerging from G. Perelman's entropy functionals and quantum geometric information flows

Abstract

We elaborate on quantum geometric information flows, QGIFs, and emergent (modified) Einstein-Maxwell and Kaluza-Klein, KK, theories formulated in Lagrange-Hamilton and general covariant variables. There are considered nonholonomic deformations of Grigory Perelman's F- and W-functionals (originally postulated for Riemannian metrics) for describing relativistic geometric flows, gravity and matter field interactions, and associated statistical thermodynamic systems. We argue that the concept of Perelman W-entropy presents more general and alternative possibilities to characterize geometric flow evolution, GIF, and gravity models than the Bekenstein-Hawking and another area-holographic type entropies. Formulating the theory of QGIFs, a set of fundamental geometric, probability, and quantum concepts, and methods of computation, are reconsidered for curved spacetime and (relativistic) phase spaces. Such generalized metric-affine spaces are modeled as nonholonomic Lorentz manifolds, (co) tangent Lorentz bundles, and associated vector bundles. Using geometric and entropic and thermodynamic values, we define QGIF versions of the von Neumann entropy, relative and conditional entropy, mutual information, etc. There are analyzed certain important inequalities and possible applications of G. Perelman and related entanglement and Rényi entropies to theories of KK QGIFs and emergent gravitational and electromagnetic interactions. New classes of exact cosmological solutions for GIFs and respective quasiperiodic evolution scenarios are elaborated. We show how classical and quantum thermodynamic values can be computed for cosmological quasiperiodic solutions and speculate how such constructions can be used for explaining structure formation in dark energy and dark matter physics.

latex2e 11pt, v4 on 43 pages, with minor modifications requested by the referee, version accepted by EPJP

References in corpus (11)

• Towards a derivation of holographic entanglement entropy
• Perturbative analysis of the gradient flow in non-abelian gauge theories
• Infinite N phase transitions in continuum Wilson loop operators
• Ricci Flow and the Poincare Conjecture
• Unification of inflation and dark energy {\it à la} quintessential inflation
• Inconsistencies in Verlinde's emergent gravity
• A note on a covariant version of Verlinde's emergent gravity
• Geometric information flows and G. Perelman entropy for relativistic classical and quantum mechanical systems
• Classical and quantum geometric information flows and entanglement of relativistic mechanical systems
• Entropy functionals and thermodynamics of relativistic geometric flows, stationary quasi-periodic Ricci solitons, and gravity
• Off-diagonal cosmological solutions in emergent gravity theories and Grigory Perelman entropy for geometric flows

Cited by in corpus (5)

• Massive quantum systems as interfaces of quantum mechanics and gravity
• The Anholonomic Frame and Connection Deformation Method for constructing off-diagonal solutions in (modified) Einstein gravity and nonassociative geometric flows and Finsler-Lagrange-Hamilton theories
• Constantin Carathéodory axiomatic approach and Grigory Perelman thermodynamics for geometric flows and cosmological solitonic solutions
• Quantum geometric information flows and relativistic generalizations of G. Perelman thermodynamics for nonholonomic Einstein systems with black holes and stationary solitonic hierarchies
• Cosmology in $R^2$-gravity: Effects of a Higher Derivative Scalar Condensate Background