Emergence of Time

TL;DR

This paper explores how the concept of time emerges from a noncommutative quantum gravity framework based on groupoid C*-algebras, linking modular groups to the transition from quantum to classical spacetime.

Contribution

It provides a detailed analysis of conditions under which time can be defined in noncommutative quantum gravity using modular theory.

Findings

Conditions for defining modular groups in the von Neumann algebra

Interpretation of modular groups as a state-dependent time flow

Quantum gravitational dynamics expressed via modular groups

Abstract

In the groupoid approach to noncommutative quantization of gravity, gravitational field is quantized in terms of a C*-algebra A of complex valued funcions on a groupoid G (with convolution as multiplication). In the noncommutative quantum gravitational regime the concepts of space and time are meaningless. We study the "emergence of time" in the transition process from the noncommutative regime to the standard space-time geometry. Precise conditions are specified under which modular groups of the von Neumann algebra generated by A can be defined. These groups are interpreted as a state depending time flow. If the above conditions are further refined one obtains a state independent time flow. We show that quantum gravitational dynamics can be expressed in terms of modular groups.