Bound of Casimir Effect by Holography

Abstract

Inspired by the Kovtun-Son-Starinet bound, we propose that holography imposes a lower bound on the Casimir effect. For simplicity, we focus on the Casimir effect between parallel planes for three-dimensional conformal field theories and briefly comment on the generalizations to other boundary shapes and higher dimensions. Remarkably, the ghost-free holographic models impose a universal lower bound on the Casimir effect. We verify the holographic bound by free theories, the Ising model, and $O(N)$ models with $N=2,3$ at critical points and prove it for the two-dimensional case. Remarkably, a general class of quantum field theories without conformal symmetries also obeys the holographic bound.

Figures (3)

• Figure 1: $-\kappa_1/C_D$ in normal phase with $0\le \lambda< 1/2$ (top) and singular phase with $1/2< \lambda\le1$ (bottom) for DGP gravity. Einstein gravity corresponds to $\lambda=0$. The top figure shows all holographic curves approach the same limit (red line) from above for $\rho\to -\infty$ ($T\to -2$) in the normal phase. The bottom figure shows $-\kappa_1/C_D$ increases with $\lambda$ and approaches zero for $\lambda\to 1$ in the singular phase. Together, it shows all examples obey the holographic bound eq.(\ref{['3d Casimir bound']}) (red line).
• Figure 2: Geometry of holographic strip: a portion of AdS soliton. The region between red/blue curves (branes) and black line is the bulk dual of strip I with negative brane tension $T\le 0$; its complement in bulk is the gravity dual for strip II (green line) with $T> 0$. The gravity dual of strip II contains the 'horizon' $z=z_h$. To remove the conical singularity on it, the angle $\theta$ should be periodic. Thus green lines for strip II are connected. Without loss of generality, we focus on strip I with $T\le 0$. We have $(\rho<0)$ and $(\rho>0)$ for the blue and red curves, respectively. $z_{\text{max}}$ and $z_c$ denote the turning points with $\theta'(z_{\text{max}})=\infty$ and $\theta'(z_{c})=0$.
• Figure 3: Strip widths in normal phase with $0\le \lambda< 1/2$ (top) and in singular phase with $1/2< \lambda\le 1$ (bottom). Note that $L$ can be larger than $\beta$ for negative enough $\rho$ in the normal phase. We show only range of $L\le \beta$ for simplicity. On the other hand, $L$ is always smaller than $\beta$ in the singular phase. $0\le L$ imposes a lower bound of $\rho$ in the singular phase, and a upper bound of $\rho$ in both phases.