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Notions of Fermionic Entropies of a Causal Fermion System

Felix Finster, Robert H. Jonsson, Magdalena Lottner, Albert Much, Simone Murro

TL;DR

The paper develops a unified framework for fermionic entropies inside causal fermion systems (CFS), expressing the fermionic von Neumann entropy, entanglement entropy, and relative entropy through the reduced one-particle density operator $D$ and its CFS generalizations. It shows that these definitions reproduce standard quasi-free fermion entropies in diverse settings, including two- and four-dimensional Minkowski space, curved spacetime horizons, and fermionic lattices, and connects them to modular theory via the modular operator and Araki–Uhlmann relative entropy. The authors prove an enhanced area law for entanglement entropy in a two-dimensional causal diamond and a non-enhanced area law for spatial subregions in four dimensions, with precise UV-regulation scaling, and extend the formalism to relative entropies between different CF systems. Overall, the work positions causal fermion systems as a universal, model-agnostic platform for analyzing fermionic entropies across relativistic and condensed-matter contexts, linking geometric, algebraic, and information-theoretic perspectives.

Abstract

The fermionic von Neumann entropy, the fermionic entanglement entropy and the fermionic relative entropy are defined for causal fermion systems. Our definition makes use of entropy formulas for quasi-free fermionic states in terms of the reduced one-particle density operator. Our definitions are illustrated in various examples for Dirac spinors in two- and four-dimensional Minkowski space, in the Schwarzschild black hole geometry and for fermionic lattices. We review area laws for the two-dimensional diamond and a three-dimensional spatial region in Minkowski space. The connection is made to the computation of the relative entropy using modular theory.

Notions of Fermionic Entropies of a Causal Fermion System

TL;DR

The paper develops a unified framework for fermionic entropies inside causal fermion systems (CFS), expressing the fermionic von Neumann entropy, entanglement entropy, and relative entropy through the reduced one-particle density operator and its CFS generalizations. It shows that these definitions reproduce standard quasi-free fermion entropies in diverse settings, including two- and four-dimensional Minkowski space, curved spacetime horizons, and fermionic lattices, and connects them to modular theory via the modular operator and Araki–Uhlmann relative entropy. The authors prove an enhanced area law for entanglement entropy in a two-dimensional causal diamond and a non-enhanced area law for spatial subregions in four dimensions, with precise UV-regulation scaling, and extend the formalism to relative entropies between different CF systems. Overall, the work positions causal fermion systems as a universal, model-agnostic platform for analyzing fermionic entropies across relativistic and condensed-matter contexts, linking geometric, algebraic, and information-theoretic perspectives.

Abstract

The fermionic von Neumann entropy, the fermionic entanglement entropy and the fermionic relative entropy are defined for causal fermion systems. Our definition makes use of entropy formulas for quasi-free fermionic states in terms of the reduced one-particle density operator. Our definitions are illustrated in various examples for Dirac spinors in two- and four-dimensional Minkowski space, in the Schwarzschild black hole geometry and for fermionic lattices. We review area laws for the two-dimensional diamond and a three-dimensional spatial region in Minkowski space. The connection is made to the computation of the relative entropy using modular theory.
Paper Structure (28 sections, 8 theorems, 174 equations, 2 figures)

This paper contains 28 sections, 8 theorems, 174 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Entanglement Entropy of a Quasi-Free Fermionic State
  4. Causal Fermion Systems and the Causal Action Principle
  5. Basic Definitions
  6. Spacetime and Causal Structure
  7. Spinors and Physical Wave Functions
  8. Surface Layer Integrals and the Commutator Inner Product
  9. Connection to Modular Theory
  10. The Fermionic Von Neumann Entropy of a Causal Fermion System
  11. The Fermionic Entanglement Entropy of a Subsystem
  12. Example: Two-Dimensional Minkowski Space
  13. Construction of the Causal Fermion System
  14. The Fermionic Von Neumann Entropy
  15. The Fermionic Entanglement Entropy of a Causal Diamond
  16. ...and 13 more sections

Key Result

Theorem 5.1

(Area law for the entanglement entropy of a causal diamond) The entanglement entropy of the causal diamond SVD obeys the enhanced area law

Figures (2)

  • Figure 1: Spatial localization of the scalar product.
  • Figure 2: A causal diamond.

Theorems & Definitions (20)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 3.1
  • Definition 4.1
  • Theorem 5.1
  • Definition 9.1
  • Theorem A.1
  • Lemma A.2
  • ...and 10 more