Einstein gravity in almost Kähler and Lagrange–Finsler variables and deformation quantization

Abstract

A geometric procedure is elaborated for transforming (pseudo) Riemanian metrics and connections into canonical geometric objects (metric and nonlinear and linear connections) for effective Lagrange, or Finsler, geometries which, in turn, can be equivalently represented as almost Kähler spaces. This allows us to formulate an approach to quantum gravity following standard methods of deformation quantization. Such constructions are performed not on tangent bundles, as in usual Finsler geometry, but on spacetimes enabled with nonholonomic distributions defining 2+2 splitting with associate nonlinear connection structure. We also show how the Einstein equations can be written in terms of Lagrange-Finsler variables and corresponding almost symplectic structures and encoded into the zero-degree cohomology coefficient for a quantum model of Einstein manifolds.

Einstein Gravity in Almost Kähler Variables and Stability of Gravity with Nonholonomic Distributions and Nonsymmetric Metrics

“…This way we can solve the Janssen-Prokopec stability problem in nonsymmetric gravity theory [31][32][33] and develop a new (nonholonomic) direction in (non) symmetric gravity and related spacetime geometry. Here we also note that nonsymmetric components of metrics arise naturally as generalized almost symplectic structures in deformation quantization of gravity [4,13] when corresponding almost Kähler models are elaborated for quantum models. It was proved how general relativity can be represented equivalently in nonsymmetric almost symplectic variables for a canonical model on a corresponding almost Kähler spaces.…”

“…To consider such formal Euclidean coordinates is useful for some purposes of analogous modelling of gravity theories as effective Lagrange mechanics geometries, but this does not mean that we introduce any complexification of classical spacetimes. In this section, we outline the constructions for classical gravity from [3,4,12].…”

“…In our recent papers on quantum gravity [3,4], we worked with almost Kähler (canonical almost symplectic) variables, see a review of results and applications of the geometric formalism for constructing exact solutions in gravity [5] and modelling locally anisotropic interactions in standard theories of physics [6]. In this article, we shall construct such symplectic (nonsymmetric) nonholonomic variables for classical Einstein gravity and possible generalizations to nonsymmetric gravity theories.…”

“…For instance, we shall construct explicit "ellipsoidal" stationary solutions in nonsymmetric gravity theories to which a static Schwarzschild metric is deformed by very small nonsymmetric metric components and nonholonomic distributions and which seem to be stable for geometric distorsions in Einstein gravity [41,42]. 4 Such metrics were constructed for different models of metricaffine, generalized Finsler on nonholonomic manifolds and noncommutative gravity [5,40,43]) and can be included in nonsymmetric gravity theories both by nonsymmetric metric components and/or as a nonholonomic symmetric background, see examples from [2].…”

Fractional almost Kähler–Lagrange geometry

“…Nevertheless, there is not a general/systematic approach to quantization of fractional field/evolution theories. The aim of our third partner work [30] is to prove that the strategy used in [31][32][33][34] allows us to quantize also a class of fractional geometries following Fedosov method [35,36]. To perform such a program is necessary to elaborate (in this paper) a general fractional version of almost Kähler geometry and then to apply (in the next paper [30]) the Karabegov-Schlichenmaier deformation quantization scheme [37].…”