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

Felix Finster, Shane Farnsworth, Claudio F. Paganini, Tejinder P. Singh

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 $σ(x,y)$ 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.

Causal Fermion Systems, Non-Commutative Geometry and Generalized 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 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.
Paper Structure (60 sections, 2 theorems, 93 equations)

This paper contains 60 sections, 2 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Overview of NCG, CFS, and GTD
  3. NCG Overview
  4. NCG Overview: Fundamental Elements
  5. NCG Overview: The Spectral Action Principle
  6. CFS Overview
  7. CFS Overview: Fundamental Elements
  8. CFS Overview: Dynamics
  9. Spacetime, Causal Structure and the Physical Wave Functions
  10. Analyzing the causal action principle in classical spacetimes
  11. GTD Overview
  12. TD Overview: Fundamental Elements
  13. Trace dynamics
  14. Generalized trace dynamics
  15. GTD Overview: Dynamics
  16. ...and 45 more sections

Key Result

Lemma 3.1

For a CFS with spin dimension $n$, the causal Lagrangian evaluated on the diagonal can be written as If the spacetime point operator $x$ is regular (in the sense that its rank is $2n$), we have where $\pi_x$ is the orthogonal projection operator onto the image of $x$.

Theorems & Definitions (7)

  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • proof
  • Definition 3.2
  • Lemma 3.3
  • proof