# Proposal for a new quantum theory of gravity II: Spectral equation of motion for the atom of space-time-matter

## TL;DR

This paper develops a spectral equation of motion for the atom of space-time-matter using non-commutative geometry and Connes time, linking quantum gravity concepts with the Dirac equation.

## Contribution

It introduces a novel spectral equation of motion for the STM atom, integrating gravity into trace dynamics via non-commutative geometry and Connes time.

## Key findings

- Derivation of the spectral equation of motion as the Dirac equation on non-commutative space
- Integration of gravity into trace dynamics using spectral action and Connes time
- Establishment of a link between quantum gravity and non-commutative geometry

## Abstract

In the first paper of this series, we have introduced the concept of an atom of space-time-matter [STM], which is described by the spectral action of non-commutative geometry, corresponding to a classical theory of gravity. In the present work, we use the Connes time parameter along with the spectral action, to incorporate gravity into trace dynamics. We then derive the spectral equation of motion for the STM atom, which turns out to be the Dirac equation on a non-commutative space.