Noncommutative Unification of General Relativity with Quantum Mechanics and Canonical Gravity Quantization

TL;DR

This paper compares a noncommutative algebraic approach to unifying general relativity and quantum mechanics with canonical gravity quantization, revealing that classical spacetime emerges when restricting to a commutative subalgebra.

Contribution

It introduces an algebraic framework linking noncommutative geometry with canonical gravity, showing how classical spacetime arises from noncommutative structures.

Findings

Emergence of classical spacetime from noncommutative algebra

Identification of algebraic counterpart to superspace

Universe is in a commutative regime when space-time slicing appears

Abstract

The groupoid approach to noncommutative unification of general relativity with quantum mechanics is compared with the canonical gravity quantization. It is shown that by restricting the corresponding noncommutative algebra to its (commutative) subalgebra, which determines the space-time slicing, an algebraic counterpart of superspace (space of 3-metrics) can be obtained. It turns out that when this space-time slicing emerges the universe is already in its commutative regime. We explore the consequences of this result.