Canonical Noncommutativity Algebra for the Tetrad Field in General Relativity

TL;DR

This paper explores a noncommutative algebraic framework for the tetrad field in general relativity, deriving generalized field equations that suggest a semiclassical approach to quantum gravity using coherent states.

Contribution

It introduces a novel noncommutative algebra for the tetrad field and derives generalized Einstein equations within this framework, linking noncommutative geometry and quantum gravity.

Findings

Derived generalized field equations with noncommutative tetrad algebra

Proposed a semiclassical approximation using coherent states

Connected noncommutative geometry with quantum gravity concepts

Abstract

General relativity under the assumption of noncommuting components of the tetrad field is considered in this paper. Since the algebraic properties of the tetrad field representing the gravitational field are assumed to correspond to the noncommutativity algebra of the coordinates in the canonical case of noncommutative geometry, this idea is closely related to noncommutative geometry as well as to canonical quantization of gravity. According to this presupposition generalized field equations for general relativity are derived which are obtained by replacing the usual tetrad field by the tetrad field operator within the actions and then building expectation values of the corresponding field equations between coherent states. These coherent states refer to creation and annihilation operators created from the components of the tetrad field operator. In this sense the obtained theory could be regarded as a kind of semiclassical approximation of a complete quantum description of gravity. The consideration presupposes a special choice of the tensor determining the algebra providing a division of space-time into two two-dimensional planes.