Geometric entropy, area, and strong subadditivity

TL;DR

This paper shows that the geometric entropy of a vacuum state, defined by tracing over degrees of freedom in a spatial region, obeys an area law in relativistic quantum field theory, under general principles such as strong subadditivity, causality, and Poincaré symmetry. By analyzing Euclidean and Minkowski settings and introducing causally closed sets, the authors derive that the entropy for flat and polyhedral regions is governed by a boundary term proportional to area plus a possible constant; in higher dimensions the area term dominates, while the constant can be subtracted and does not affect the leading behavior. The results unify and generalize previous QFT computations, showing that the area-proportional growth is a universal feature of geometric entropy in the continuum limit, with potential implications for black hole entropy and covariant entropy bounds. The work delineates a rigorous, model-independent route to the area law via strong subadditivity and relativistic causality, clarifying the structure of divergences and offering a framework for broader region classes beyond spheres or half-spaces.

Abstract

The trace over the degrees of freedom located in a subset of the space transforms the vacuum state into a density matrix with non zero entropy. This geometric entropy is believed to be deeply related to the entropy of black holes. Indeed, previous calculations in the context of quantum field theory, where the result is actually ultraviolet divergent, have shown that the geometric entropy is proportional to the area for a very special type of subsets. In this work we show that the area law follows in general from simple considerations based on quantum mechanics and relativity. An essential ingredient of our approach is the strong subadditive property of the quantum mechanical entropy.

Figures (14)

• Figure 1: (a)- A causally closed set $\hat{A}$ in $1+1$ dimensions. It is the causal development of the spatial surfaces $A$ and $A^\prime$. We say that $A$ and $A^\prime$ are Cauchy surfaces for $\hat{A}$. Both of these sets can be continued to a Cauchy surface for the Minkowski space using the same spatial set ${\cal C}_1$. The marked points on the left and right corners of the diamond shaped set $\hat{A}$ represent the spatial corner of $\hat{A}$. (b)- Two commuting causally closed sets in $2+1$ represented here by their Cauchy surfaces representatives $A$ and $B$. These surfaces intersect in the dotted line in this example, dividing $A$ into $A_1$ and $A_2$ and $B$ into $B_1$ and $B_2$. The commutation imposes that $A_2\subseteq \hat{B}$ and $B_2\subseteq \hat{A}$. Note that the spatial corners of $\hat{A}$ and $\hat{B}$, which are the boundaries of $A$ and $B$ respectively, are spatial to each other. The causally closed sets $\hat{A}\vee \hat{B}$ and $\hat{A}\wedge \hat{B}$ that enter into the SSA relation are generated by $A_1\cup B_1$ and $A_2\cup B_2$ respectively.
• Figure 2: A non decreasing concave function $S(x)$. Two points on the curve $S(x)$ determine a segment that lies below the function graph. The limit of $S(x)$ for $x\rightarrow 0$ is $\gamma$, and the curve slope approaches to $s$ as $x$ goes to infinity (however, $S(x)$ does not necessarily approach any straight line asymptotically).
• Figure 3: Geometric constructions used to prove the formula (\ref{['limite']}) for the entropies of one dimensional sets in the limit of small size.
• Figure 4: Eating from inside a strip $D$ to a smaller strip $D^\prime$ (the right side of $D^\prime$ is the dashed line), by means of $A-B$. One has to position $A$ inside $D$ (shown with the dotted line) and take out from $D$ the sector corresponding to $A-B=C$. Then translating down $A$ inside $D$ and repeating the operation it is possible to eat all $D-D^\prime$.
• Figure 5: The geometric construction used to show that the function $\beta$ is proportional to the volume. The solid line represents a transversal view of the two component set $A$. Drawn with dashed lines are the two component set $B_1$ on the left and the two component set $B_2$ on the right. All the connected components of these sets have the same height $a$.