Causal Fermion Systems, Non-Commutative Geometry and Generalized Trace Dynamics

TL;DR

This paper compares causal fermion systems with generalized trace dynamics and non-commutative geometry, highlighting their shared focus on fiber bundle structures and the innovative encoding of spacetime relations via generalized correlators.

Contribution

It reveals the common geometric framework and introduces a novel perspective on encoding spacetime relations across these theories.

Findings

All three theories recover a fiber bundle structure in the continuum limit.

Causal fermion systems encode spacetime relations through a generalized two-point correlator.

The approach can be transferred to non-commutative geometry and trace dynamics.

Abstract

We compare the structures and methods in the theory of causal fermion systems with generalized trace dynamics and non-commutative geometry. Although the three theories differ on many aspects, they agree in that the geometric structure to be recovered in the continuum limit is not the bare spacetime but a suitable fiber bundle. Furthermore, the comparison leads us to the conclusion that the key innovation in causal fermion systems lies in the manner in which the relation between different spacetime points is encoded. The role of Synge's classical world function $\sigma(x,y)$ that encodes the geodesic distance between any two points in the manifold, is taken by a generalized two-point correlator. We show that this idea can be transferred to non-commutative geometry and generalized trace dynamics.