Geometric information flows and G. Perelman entropy for relativistic classical and quantum mechanical systems

Bubuianu³

2021

Self Cite

We elaborate on nonassociative differential geometry of phase spaces endowed with nonholonomic (non-integrable) distributions and frames, nonlinear and linear connections, symmetric and nonsymmetric metrics, and correspondingly adapted quasi-Hopf algebra structures. The approach is based on the concept of nonassociative star product introduced for describing closed strings moving in a constant R-flux background. Generalized Moyal-Weyl deformations are considered when, for nonassociative and noncommutative terms of star deformations, there are used nonholonomic frames (bases) instead of local partial derivatives. In such modified nonassociative and nonholonomic spacetimes and associated complex/real phase spaces, the coefficients of geometric and physical objects depend both on base spacetime coordinates and conventional (co) fiber velocity/momentum variables like in (non) commutative Finsler-Lagrange-Hamilton geometry. For nonassociative and (non) commutative phase spaces modelled as total spaces of (co) tangent bundles on Lorentz manifolds enabled with star products and nonholonomic frames, we consider associated nonlinear connection, N-connection, structures determining conventional horizontal and (co) vertical (for instance, 4+4) splitting of dimensions and N-adapted decompositions of fundamental geometric objects. There are defined and computed in abstract geometric and N-adapted coefficient forms the torsion, curvature and Ricci tensors. We extend certain methods of nonholonomic geometry in order to construct R-flux deformations of vacuum Einstein equations for the case of N-adapted linear connections and symmetric and nonsymmetric metric structures. Introduction, Motivations and GoalsWe initialize a research on nonassociative nonholonomic geometry induced by R-flux backgrounds in string gravity and classical and quantum models on phase spaces modelled as (co) tangent Lorentz bundles endowed with nonholonomic (non-integrable, equivalently, anholonomic) structures and nonsymmetric metrics. Such a nonholonomic geometric formalism was elaborated for various types of (non) commutative gravity and matter field theories, modified (relativistic) geometric flows, and classical and quantum information theories, see details in a series of our previous works [1][2][3][4][5][6][7][8][9][10][11] and references therein. In this article, nonassociative geometric and physical models are formulated in a form which is accessible to researchers in particle physics and modern cosmology. There are provided nonholonomic generalizations of two recent models of nonassociative geometry and vacuum gravity due to [12] and, with

Section: Nonassociative Nonholonomic Star Products On (Co) Tangent Lorentz Bundlesmentioning

confidence: 99%

Section: Nonassociative Nonholonomic Star Products On (Co) Tangent Lorentz Bundlesmentioning

confidence: 99%

Section: The Universal N-adapted R-matrix and The Associatormentioning

confidence: 99%

Section: Star D-torsions For D-connections and Quasi-hopf Structuresmentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Nonassociative Geometry of Nonholonomic Phase Spaces with Star R‐flux String Deformations and (non) Symmetric Metrics

Bubuianu³

2021

Self Cite

Decoupling and Integrability of Nonassociative Vacuum Phase Space Gravitational Equations With Star and R‐flux Parametric Deformations

Bubuianu²,

2021

Self Cite

113

We prove that nonassociative star deformed vacuum Einstein equations can be decoupled and integrated in certain general forms on phase spaces involving real R-flux terms induced as parametric corrections on base Lorentz manifold spacetimes. The geometric constructions are elaborated with parametric (on respective Planck, ℏ, and string, := 3 s ∕6ℏ, constants) and nonholonomic dyadic decompositions of fundamental geometric and physical objects. This is our second partner work on elaborating nonassociative geometric and gravity theories with symmetric and nonsymmetric metrics, (non) linear connections, star deformations defined by generalized Moyal-Weyl products, endowed with quasi-Hopf algebra, or other type algebraic and geometric structures, and all adapted to nonholonomic distributions and frames. We construct exact and parametric solutions for nonassociative vacuum configurations (with nontrivial or effective cosmological constants) defined by star deformed generic off-diagonal (non) symmetric metrics and (generalized) nonlinear and linear connections. The coefficients of geometric objects defining such solutions are determined by respective classes of generating and integration functions and constants and may depend on all phase space coordinates [spacetime ones, (x i , t); and momentum like variables, (p a , E)]. Quasi-stationary configurations are stated by solutions with spacetime Killing symmetry on a time like vector t but with possible dependencies on momentum like coordinates on star deformed phase spaces. This geometric techniques of decoupling and integrating nonlinear systems of physically important partial differential equations (the anholonomic frame and connection deformation method, AFCDM) is applied in our partner works for constructing nonassociative and locally anisotropic generalizations of black hole and cosmological solutions and elaborating geometric flow evolution and classical and quantum information theories.

Nonassociative Ricci Flows, Star Product and R‐flux Deformed Black Holes, and Swampland Conjectures

Bubuianu¹,

2023

Self Cite

A theory of nonassociative geometric flows is formulated as an extension of a string‐inspired model of nonassociative gravity determined by star product. The nonassociative Ricci tensor and curvature scalar defined by (non) symmetric metric structures and generalized (non) linear connections are used for defining nonassociative versions of Grigori Perelman F‐ and W‐functionals for Ricci flows and computing associated thermodynamic variables. The anholonomic frame and connection deformation method, AFCDM, which allows us to construct exact and parametric solutions describing nonassociative geometric flow evolution scenarios and modified Ricci soliton configurations with quasi‐stationary generic off‐diagonal metrics are developed and applied. There are provided explicit examples of solutions modeling geometric and statistical thermodynamic evolution on a temperature‐like parameter of modified black hole configurations encoding nonassociative star‐product and R‐flux deformation data. Further perspectives of the paper are motivated by nonassociative off‐diagonal geometric flow extensions of the swampland program, related conjectures and claims on geometric and physical properties of new classes of quasi‐stationary Ricci flow and black hole solutions.