Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The Relative Fermionic Entropy in Two-Dimensional Rindler Spacetime

Felix Finster, Albert Much

TL;DR

This work analyzes fermionic relative entropy in a two‑dimensional Rindler setting via two complementary approaches: modular theory and reduced one‑particle density operators. It derives a general Gaussian‑state relative entropy formula and applies it to both unitary vacuum excitations and non‑unitary Gaussian excitations, demonstrating consistency between the modular and one‑particle methods where their domains overlap. The results yield explicit expressions such as $S(\tilde{\omega}\Vert \omega)=2\pi\langle f|H_{\mathcal{R}}\tanh(2\pi H_{\mathcal{R}})|f\rangle$ (and equivalent Gaussian forms) and show how non‑unitary excitations extend the analysis beyond the reach of modular theory. The work clarifies the relationship between modular flow and one‑particle covariance, highlighting a complementary toolkit for studying entropic quantities in relativistic quantum fields and opening paths to treat more general quasi‑free states and other spacetimes.

Abstract

The fermionic relative entropy in two-dimensional Rindler spacetime is studied using both modular theory and the reduced one-particle density operators. The methods and results are compared. A formula for the relative entropy for general Gaussian states is derived. As an application, the relative entropy is computed for a class of non-unitary excitations.

The Relative Fermionic Entropy in Two-Dimensional Rindler Spacetime

TL;DR

This work analyzes fermionic relative entropy in a two‑dimensional Rindler setting via two complementary approaches: modular theory and reduced one‑particle density operators. It derives a general Gaussian‑state relative entropy formula and applies it to both unitary vacuum excitations and non‑unitary Gaussian excitations, demonstrating consistency between the modular and one‑particle methods where their domains overlap. The results yield explicit expressions such as (and equivalent Gaussian forms) and show how non‑unitary excitations extend the analysis beyond the reach of modular theory. The work clarifies the relationship between modular flow and one‑particle covariance, highlighting a complementary toolkit for studying entropic quantities in relativistic quantum fields and opening paths to treat more general quasi‑free states and other spacetimes.

Abstract

The fermionic relative entropy in two-dimensional Rindler spacetime is studied using both modular theory and the reduced one-particle density operators. The methods and results are compared. A formula for the relative entropy for general Gaussian states is derived. As an application, the relative entropy is computed for a class of non-unitary excitations.

Paper Structure

This paper contains 17 sections, 13 theorems, 192 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and general setup
  3. Majorana spinors in two-dimensional Minkowski space
  4. The Rindler wedge and the Rindler Hamiltonian
  5. The Rindler vacuum and its excitations
  6. Computation via the reduced one-particle density operator
  7. The reduced one-particle density operator in Rindler spacetime
  8. The relative entropy of vacuum excitations
  9. Alternative computation via general Gaussian states
  10. Computation using modular theory
  11. Non-unitary excitations of more general states
  12. Comparison of the methods and outlook
  13. The relative entropy of vacuum excitations
  14. Fermionic entropies of general Gaussian states
  15. Bogoliubov transformations
  16. ...and 2 more sections

Key Result

Lemma 3.1

The scalar products of the plane wave solutions in $\mycal M$ and $\mycal R$ are given in the distributional sense by where (as in rindler) we set

Theorems & Definitions (27)

  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Theorem 3.5
  • proof
  • Definition 4.1
  • ...and 17 more