Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Spectral Spacetime Entropy for Quasifree Theories

Joshua Y. L. Jones, Yasaman K. Yazdi

TL;DR

This work develops a covariant, spacetime formulation of entanglement entropy for quasifree bosonic and fermionic quantum field theories, enabling UV regularisation directly in spacetime regions. By exploiting a spectral construction based on region-restricted two-point functions and generalized eigenvalue problems, it connects entropy to the density matrix and its eigenstructure without relying on a Cauchy hypersurface. The authors validate the approach by reproducing known thermal entropies in the continuum and demonstrate its power through a causal-set example in 1+1 dimensions, where a slightly modified entropy scaling hints at spacetime discreteness. The framework provides a versatile, covariant tool for probing horizon entropy in quantum gravity and offers practical advantages for numerical studies in nontrivial spacetime geometries and discrete spacetimes. Overall, the paper advances a unified, covariant perspective on entanglement entropy that spans bosons, fermions, continuum spacetimes, and causal-set discretisations, with implications for black hole thermodynamics and quantum gravity phenomenology.

Abstract

Motivated by the necessity to UV-regularise entanglement entropy, we present a spectral method for calculating the entropy of quasifree states, for both bosonic and fermionic field theories. This construction is defined in spacetime rather than on a hypersurface, enabling the covariant regularisation of entropies, and its calculation in generic spacetime regions. We derive these formulae, which have previously appeared in the literature, in a new manner and highlight certain aspects of them, such as their connection to the density matrix and its eigenvalues. The spacetime nature of the formulation makes it particularly apt in the context of semiclassical and quantum gravity and in connection to black hole entropy. Another useful property of the formulation is its application to settings where no notion of a Cauchy surface exists, such as in the causal set theory approach to quantum gravity. We show example applications of the formulae which demonstrate their ability to reproduce known results. We also show a calculation in a causal set in $1+1$ dimensions which makes use of several of the unique and useful features of the formalism. In this last example, we obtain a novel result of a slightly modified entanglement entropy scaling coefficient, giving a possible signature of spacetime discreteness.

Spectral Spacetime Entropy for Quasifree Theories

TL;DR

This work develops a covariant, spacetime formulation of entanglement entropy for quasifree bosonic and fermionic quantum field theories, enabling UV regularisation directly in spacetime regions. By exploiting a spectral construction based on region-restricted two-point functions and generalized eigenvalue problems, it connects entropy to the density matrix and its eigenstructure without relying on a Cauchy hypersurface. The authors validate the approach by reproducing known thermal entropies in the continuum and demonstrate its power through a causal-set example in 1+1 dimensions, where a slightly modified entropy scaling hints at spacetime discreteness. The framework provides a versatile, covariant tool for probing horizon entropy in quantum gravity and offers practical advantages for numerical studies in nontrivial spacetime geometries and discrete spacetimes. Overall, the paper advances a unified, covariant perspective on entanglement entropy that spans bosons, fermions, continuum spacetimes, and causal-set discretisations, with implications for black hole thermodynamics and quantum gravity phenomenology.

Abstract

Motivated by the necessity to UV-regularise entanglement entropy, we present a spectral method for calculating the entropy of quasifree states, for both bosonic and fermionic field theories. This construction is defined in spacetime rather than on a hypersurface, enabling the covariant regularisation of entropies, and its calculation in generic spacetime regions. We derive these formulae, which have previously appeared in the literature, in a new manner and highlight certain aspects of them, such as their connection to the density matrix and its eigenvalues. The spacetime nature of the formulation makes it particularly apt in the context of semiclassical and quantum gravity and in connection to black hole entropy. Another useful property of the formulation is its application to settings where no notion of a Cauchy surface exists, such as in the causal set theory approach to quantum gravity. We show example applications of the formulae which demonstrate their ability to reproduce known results. We also show a calculation in a causal set in dimensions which makes use of several of the unique and useful features of the formalism. In this last example, we obtain a novel result of a slightly modified entanglement entropy scaling coefficient, giving a possible signature of spacetime discreteness.
Paper Structure (29 sections, 154 equations, 3 figures)

This paper contains 29 sections, 154 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Quasifree States
  4. Bosonic Entropy
  5. Fermionic Entropy
  6. Thermal Entropy Calculations in the Continuum
  7. Thermal Entropy of a Real Scalar Field in $\textbf{R}^{(3,1)}$
  8. Thermal Entropy of a Dirac Fermion Field in $\textbf{R}^{(3,1)}$
  9. An Entanglement Entropy Calculation in a Discrete Setting
  10. Causal Set Theory
  11. Entanglement Entropy in a Rindler Wedge Causal Set
  12. Spacetime Setup
  13. $W_{SJ}$ in a Causal Diamond
  14. Continuum Theory:
  15. Causal Set Theory:
  16. ...and 14 more sections

Figures (3)

  • Figure 1: (a) The full spacetime under consideration, with the left subdiamond in red, and the right subdiamond in blue. The red and blue subdiamonds are Rindler wedges with an infrared cutoff, and they are causal complements of one another. (b) A sprinkling of the spacetime on the left. The points sprinkled into the left and right subdiamonds are shown in red and blue respectively; we will restrict $W$ and $i\Delta$ to either the red or the blue elements when we calculate the entanglement entropy via \ref{['Sev']} and \ref{['Ssum']}.
  • Figure 2: A plot of the results of our simulations. The raw data points are in orange, and the mean and standard deviations of the bins are in black. We have fitted to the data a curve of the form $S=b \log\left(\sqrt{\langle N_{{2L}}\rangle}\right)+c$, shown in purple, via a least squares regression weighted by the inverse variance.
  • Figure 3: A log-log plot of the positive eigenspectrum of $i\Delta$ in a $1+1$d causal diamond. The causal set eigenvalues from a $900$-element sprinkling, into a diamond of side length $1$, are shown in purple. The asymptotic power law expected from the continuum eigenvalues \ref{['eq: spectral density']}, rescaled by a factor of $\frac{1}{2}\tilde{\rho}=450$, is shown in orange (the factor of half is necessary to account for the two families of eigenfunctions). (a) A first order (Delaunay) interpolation of the real part of the eigenfunction corresponding to the $4^{\text{th}}$ largest positive eigenvalue of $i\Delta$. This eigenfunction shows the typical characteristics of a continuum eigenfunction. (b) A first order (Delaunay) interpolation of the real part of the eigenfunction corresponding to the $225^{\text{th}}$ largest positive eigenvalue. This eigenfunction shows the typical characteristics of a non-continuumlike eigenfunction.