Entropy on a null surface for interacting quantum field theories and the Bousso bound

TL;DR

The paper extends the quantum Bousso bound to interacting quantum field theories in the weak gravity regime by showing that the entropy on a null interval equals a modular energy-like expression, ΔS = ΔK, with ΔK given by a g(x^+)–weighted integral of the stress tensor T_{++}. The authors derive the light-like OPE structure of defect operators realizing Rényi entropies, constrain the function g(x^+) using relative entropy and strong subadditivity, and demonstrate that the bound holds for interacting theories in d>2. They complement the field theory analysis with holographic calculations, obtaining an explicit g(x^+) for theories with gravity duals via minimal-surface areas, and show that g differs from the free-field result in higher dimensions. The work clarifies why ΔS = ΔK holds on null surfaces in interacting theories and discusses broader implications, including phase transitions in holographic settings and open problems in extending these results beyond the weak-gravity limit.

Abstract

We study the vacuum-subtracted von Neumann entropy of a segment on a null plane. We argue that for interacting quantum field theories in more than two dimensions, this entropy has a simple expression in terms of the expectation value of the null components of the stress tensor on the null interval. More explicitly $ΔS = 2π\int d^{d-2}y \int_0^1 dx^+\, g(x^+)\, \langle T_{++}\rangle$, where $g(x^+)$ is a theory-dependent function. This function is constrained by general properties of quantum relative entropy. These constraints are enough to extend our recent free field proof of the quantum Bousso bound to the interacting case. This unusual expression for the entropy as the expectation value of an operator implies that the entropy is equal to the modular Hamiltonian, $ΔS = \langle ΔK \rangle $, where $K$ is the operator in the right hand side. We explain how this equality is compatible with a non-zero value for $ΔS$. Finally, we also compute explicitly the function $g(x^+)$ for theories that have a gravity dual.

Figures (6)

• Figure 1: The Rényi entropies for an interval $A$ involve the two point function of defect operators $D$ inserted at the endpoints of the interval. An operator in the $i^{th}$ CFT becomes an operator in the $(i+1)^{th}$ CFT when we go around the defect.
• Figure 2: The functions $g(v)$ in the expression for the modular Hamiltonian of the null slab, for conformal field theories with a bulk dual. Here $d=2,3,4,8,\infty$ from bottom to top. Near the boundaries ($v\to 0$, $v\to 1$), we find $g\to 0$, $g'\to \pm 1$, in agreement with the modular Hamiltonian of a Rindler wedge. We also note that the functions are concave (see Sec. 6). In particular, we see that $|g'|\leq 1$, in agreeement with our general argument of Sec. 3.
• Figure 3: Operator algebras associated to various regions. (a) Operator algebra associated to the domain of dependence (yellow) of a spacelike interval. (b) The domain of dependence of a boosted interval. (c) In the null limit, the domain of dependence degenerates to the interval itself.
• Figure 4: The maximum value $E_{\textrm{max}}(p)$ of $E$ for getting a surface that returns to the boundary (solid line). For comparison, we also plotted the line $E=p-1$ (dashed line). The extremal surface solutions of interest appear in the region $p>1$, $0<E<E_{\textrm{max}}(p)$. Here, we have taken $d=3$.
• Figure 5: Curves of constant $\Delta x^+$ (black solid curves) and $\Delta x^-$ (blue dashed curves), in the logarithmic parameter space defined by $(\log(p-1),-\log(E_{\textrm{max}}(p)-E)/E_{\textrm{max}}(p))$. The value $p = 1$ maps to $-\infty$ and $p = \infty$ maps to $+\infty$ on the horizontal axis, while $E = 0$ maps to 0 and $E = E_\textrm{max}(p)$ maps to $+ \infty$ on the vertical axis. The thick blue contour represent the null solutions with $\Delta x^-=0$. Above this contour, the boundary interval is time-like. If $\Delta x^+\gtrsim 15$ and we follow a contour of constant $\Delta x^+$, we find two solutions with exact $\Delta x^-=0$. For all contours of fixed $\Delta x^+$, there exists an asymptotic null solution in the limit $p\rightarrow \infty$.