Tests of restricted Quantum Focusing and a new CFT bound

TL;DR

The paper investigates the validity and consequences of the restricted quantum focusing conjecture (rQFC) in semiclassical gravity. It employs two complementary approaches: (i) a $d=2$ JT gravity toy model with many matter fields to prove the rQFC and to exhibit explicit violations of the original QFC in regimes where matter quantum effects compete with the dilaton scale; (ii) a $d>2$ analysis showing that, in a near-vacuum, large-$c$ CFT context, the rQFC implies a bound stronger than the QNEC for states accessible via a conformal map to a Rindler wedge and a special $G o 0$, $oldsymbol{ extSigma} o 0$ limiting procedure. The authors then conjecture a universal strengthened QNEC of the form $2oldsymbol{ extpi}raket{T_{ij}}k^ik^j - rac{1}{oldsymbol{ extsqrt{h}_ ext{λ}}}rac{oldsymbol{ extdelta S_{ m ren}}}{oldsymbol{ extdelta V}}|_ ext{λ} oldsymbol{ extGeq} oldsymbol{oldsymbol{ extappa}} oldsymbol{ extSigma}^{d-2} ig(rac{1}{oldsymbol{ extsqrt{h}_ ext{λ}}}rac{oldsymbol{ extdelta S_{ m ren}}}{oldsymbol{ extdelta V}}|_ ext{λ}ig)^2$ as $oldsymbol{ extSigma} o 0$, tying quantum information inequalities to energy conditions. These results illuminate non-trivial links between QFT entropy flux, light-ray operators, and semiclassical gravitational constraints, and they motivate further exploration of universal bounds in quantum gravity regimes.

Abstract

The restricted quantum focusing conjecture (rQFC) plays a central role in an axiomatic formulation of semiclassical gravity. Since much hinges on its validity, it is imperative to subject the rQFC to rigorous tests in novel settings. Here we do so in two independent directions. First, we prove rQFC in a class of spacetime dimension $d=2$ toy models, JT gravity coupled to a QFT. We also construct explicit counter-examples to the original and stronger Quantum Focusing Conjecture in a regime where matter quantum effects are comparable to the total dilaton value. Second, for $d>2$, we derive from the rQFC a constraint stronger than the Quantum Null Energy Condition (QNEC). In a broad class of states, this bound forbids the QNEC from saturating faster than $O(\mathcal{A})$ as the transverse area $\mathcal{A}$ of a certain null deformation shrinks to zero. We speculate about a universal strengthened QNEC holding across all QFT states.

Figures (12)

• Figure 1: A spacetime wedge $\mathcal{W}$ with future and past boundaries $\partial^\pm \mathcal{W}$ is shown. The (restricted) QFC, roughly, constrains the second derivative of the generalized entropy of spacetime wedges under deformations in which the edge, $\eth \mathcal{W} = \partial^+\mathcal{W} \cap \partial^- \mathcal{W}$, is deformed along the generators of $\partial^+\mathcal{W}$ to a new location $v = V_\lambda(y^a)$, as shown in shaded blue.
• Figure 2: Application of the rQFC to the domain of dependence $\mathcal{W}$ of a ball in Minkowski space (left). We consider null deformations of the edge $\eth \mathcal{W}$, parametrized by $y^a$, to $v = V_\lambda(y^a)$ with width $\Sigma$. A new non-gravitational limit with $G, \Sigma \to 0$ and $\Sigma^{d-2} \gg G$ yields a stronger-than-QNEC bound in CFTs. As part of the derivation, $\mathcal{W}$ is conformally mapped to a Rindler wedge $\tilde{\mathcal{W}}$ with deformation $\tilde{v} = \tilde{V}_\lambda(\tilde{y}^a)$, where von Neumann entropy derivatives relate to the coincident limit of two $\mathcal{E}$ insertions.
• Figure 3: Let $\mathcal{W}$ be a wedge (light blue) whose edge is the union of points $P_i$. We consider variations of $S_{\rm gen}$ at $P_1$. The quantum expansion $\Theta\vert_{\mathcal{W}}(P_1)$ is defined as the variation of $S_{\rm gen}(\mathcal{W})$ with respect to $\lambda$, the affine parameter of a lightray $L^+(P_1)$ emanating from $P_1$ and included in $\partial^+\mathcal{W}$ (black lines). In this section we focus on $\Theta_{\lambda}=\Theta\vert_{\mathcal{W}_{\lambda}}$ with the $1$-parameter family of wedges $\mathcal{W}_{\lambda}$ defined as $\eth \mathcal{W_{\lambda}}= \tilde{P}(\lambda)\cup P_2\cup P_3$, where $\tilde{P}(\lambda)$ is a point on $L^+(P_1)$ at $\lambda$. The wedge $\mathcal{W}_{\lambda+\delta\lambda}$ is schematically pictured in dark blue.
• Figure 4: Penrose diagram of a two-dimensional extremal black hole in equilibrium with a bath at zero temperature. The blue region is half of Minkowski space and the red region is the Poincaré patch of the black hole. The future and past horizons are located at $U\rightarrow\infty$ and $V\rightarrow -\infty$ and the boundary of AdS is located at $U=V$. An spacelike slice with free endpoint $P$ and fixed endpoint at $(0,0)$ is depicted. The region where $\theta\Theta <0$ associated with $P$ along the $V$ direction, bounded by the curve which saturates the inequality \ref{['eq:bdy']}, is depicted in dark red. In the limit $\phi_r\gg \ell_S$, it shrinks to a narrow strip along the future horizon and the null line $V=0$. The QFC violation region is shaded in purple.
• Figure 5: Plot of $\Theta'$ at fixed $U=1$, along the $-V$ direction. We have taken the constants $\phi_0=1,\phi_r=100,\ell_S=1000$.