Causal Fermion Systems: Spacetime as the web of correlations of a many-body quantum system

TL;DR

This work advances a full ontology for causal fermion systems (CFS) by treating spacetime as a relational web of correlations among a many-body quantum system. It introduces an auxiliary Hilbert space and a probe–background split to define observables, localization, and spacetime superposition, and it recasts the causal action as a variance of the causal correlation strength, linking microscopic correlations to emergent classical spacetime. In the continuum (large $N$) and Minkowski vacuum, the formalism reproduces the standard two-point functions of quantum field theory and general relativity, while numerical results support spacetime emergence via equipartition of correlations. The paper also outlines how experiments can be described relationally within CFS, including a notion of quantum reference frames and a no-go result for macroscopic superpositions of causal structures, and it discusses future directions toward correlation-geometric notions of distance, mesoscopic causality, and connections to dark matter and baryogenesis. Overall, the work provides a concrete, correlation-centric foundation for interpreting CFS beyond the continuum limit and paves the way for testable phenomenology anchored in the web of spacetime correlations.

Abstract

In this paper, we argue that spacetime in causal fermion systems can be understood as the web of correlations of a many-body quantum system.This argument highlights the fact that causal fermion systems is a completely relational theory. We also explain how our perception of a background (spacetime) emerges in the limit where the number of states taken in consideration goes to infinity. This thereby constitutes a complete viable ontology for causal fermion systems which are not reliant on the continuum limit. A key insight is the fact that in a relevant subset of causal fermion systems, which includes the continuum limit of the Minkowski vacuum spacetime, minimization of the causal action can be understood as a minimization of fluctuations in the causal structure of spacetime.

Figures (2)

• Figure 1: This visualization shows associated one particle spacetime $M_u$ of a state $\ket{u} \in H$. The spacetime region $\mathcal{U}$ intersects $M_u$ completely. Thus, by Definition \ref{['def:localization-of-states']}, $\ket{u}$ is localized in $\mathcal{U}$. In addition, $\ket{u}$ is not delocalized in the sense of Definition \ref{['def:delocalized-states']}. To help with intuition we chose the full spacetime to be the continuum limit of Minkowski space. In general $\mathcal{U}$ and $M_u$ are subsets of an abstract cfs spacetime $M$.
• Figure 2: The plot shows the numerical integration of expression of equation \ref{['eq:total-variance']} for different values of $m \varepsilon$. The numerical values are fitted using the function $\varepsilon \mapsto a \varepsilon^b$ with $a = 2.7 \cdot 10^{-8}$ and $b = 8$.

Theorems & Definitions (34)

• Definition 2.1: Causal Fermion System
• Definition 2.2
• Definition 3.1: Auxillary Hilbert space
• Definition 3.2: (Sub)System
• Definition 3.3: Occupation number
• Proposition 3.4: Pauli Exclusion Principle
• proof
• Definition 3.5: Position Observables
• Proposition 3.6
• proof