Partition Functions, the Bekenstein Bound and Temperature Inversion in Anti-de Sitter Space and its Conformal Boundary

TL;DR

We reformulate the Bekenstein bound as $L=E-\frac{S}{2\pi R}$ and show the minimum occurs at $T_L=\frac{1}{2\pi R}$; in AdS$_n$ the radius is fixed by $R=\frac{l}{n-2}$, enabling a precise bulk test, and the paper shows the bound holds for known AdS black holes but can be violated by free quantum fields in AdS, even when the Casimir energy $E_c$ is included. Using canonical and grand canonical partition functions encoded by the one-particle function $Y(\beta,\Omega)$ and the Hamiltonian zeta function $\zeta_H(s)$, the authors compute bulk energies and entropies (including $E_c$) for free fields in AdS$_4$, AdS$_5$, and AdS$_7$, and explore rotating cases and matrix-valued field statistics. They find that the Bekenstein bound is violated for several free fields in AdS with explicit negative $L_{\min}$ in multiple dimensions, rotation can drive violations arbitrarily large, and large-$N$ matrix sectors with multi-trace dynamics can also breach the bound, while temperature-inversion relations reveal deep modular-like structure in AdS/CFT contexts. The work highlights the nonuniversality of the bound in quantum field theory on curved backgrounds, clarifying the roles of boundary conditions, Casimir energy, rotation, and higher-spin multiplets, and connects to Hawking-Page physics through marginal bounded states.

Abstract

We reformulate the Bekenstein bound as the requirement of positivity of the Helmholtz free energy at the minimum value of the function L=E- S/(2πR), where R is some measure of the size of the system. The minimum of L occurs at the temperature T=1/(2πR). In the case of n-dimensional anti-de Sitter spacetime, the rather poorly defined size R acquires a precise definition in terms of the AdS radius l, with R=l/(n-2). We previously found that the Bekenstein bound holds for all known black holes in AdS. However, in this paper we show that the Bekenstein bound is not generally valid for free quantum fields in AdS, even if one includes the Casimir energy. Some other aspects of thermodynamics in anti-de Sitter spacetime are briefly touched upon.

Figures (1)

• Figure 1: A plot of $L=E-S/\pi$ for a scalar field in the $D(2,0)$ representation of $SO(3,2)$ in AdS$_4$, as a function of $\beta=1/T$. The function attains its minimum at $\beta=1/T=\pi$. It is positive at high temperature, but it is negative at sufficiently low temperature.