Entanglement entropy for non-coplanar regions in quantum field theory

TL;DR

This work addresses entanglement entropy for regions not restricted to a single Cauchy surface in quantum field theory, where Euclidean replica techniques fail, by developing a real-time method for a massive free Dirac field in two dimensions. It uses a small-mass expansion via resolvent techniques, exploiting a chiral factorization in the massless limit to obtain explicit expressions for S(V) and the mutual information, and then analyzes the impact of large boosts on information sharing between regions. The main findings include a boost-enhanced entanglement that yields finite mutual information for vanishingly small boosted regions, leading to extensivity of information in deeply Lorentzian regimes and nontrivial tripartite correlations; the results have potential implications for black hole information localization and Hawking radiation, with connections to twist-operator OPEs. Overall, the paper provides quantitative tools and qualitative insights into how spacetime geometry and relativistic kinematics control entanglement structure in QFT, suggesting general conditions under which mutual information becomes extensive in highly boosted, null or near-null configurations.

Abstract

We study the entanglement entropy in a relativistic quantum field theory for regions which are not included in a single spatial hyperplane. This geometric configuration cannot be treated with the Euclidean time method and the replica trick. Instead, we use a real time method to calculate the entropy for a massive free Dirac field in two dimensions in some approximations. We find some specifically relativistic features of the entropy. First, there is a large enhancement of entanglement due to boosts. As a result, the mutual information between relatively boosted regions does not vanish in the limit of zero volume and large relative boost. We also find extensivity of the information in a deeply Lorentzian regime with large violations of the triangle inequalities for the distances. This last effect is relevant to an interpretation of the amount of entropy enclosed in the Hawking radiation emitted by a black hole.

Figures (7)

• Figure 1: The entanglement entropy is a function of pieces of spatial surfaces like the set $A$ in this figure (light like lines are shown at $\pm 45^o$). Spatial sets having the same causal domain of dependence (diamond shaped set) such as $A$ and $A^\prime$, have the same reduced density matrix. In consequence, a regularization independent quantity such as the mutual information $I(A,B)$ coincides with the one between equivalent regions $I(A^\prime,B^\prime)$.
• Figure 2: The projections $V^\pm$ of the set $V$ on the null coordinate axis. $V$ has three connected components in this example. The labeling of the extreme points of the intervals in $V^\pm$ is done in increasing order for the null coordinates. Note for example that $b_3^+$ and $a_1^-$ are null coordinates corresponding to the same point.
• Figure 3: Configuration of two adjacent intervals $B$ and $C$ separated from the interval $A$ by a large distance $d$. All three intervals are collinear.
• Figure 4: Two different limits of vanishing length and divergent relative boost for the intervals $A$ and $B$.
• Figure 5: Two different configurations of three intervals with a large relative boost between $A$ and $B$ or $C$. The case (a) is extensive in the limit of large relative boosts and fixed sizes $a$, $b$ and $c$ while the case (b) does not show extensivity of the mutual information.