Holography versus correspondence principle: Eternal Schwarzschild–anti–de Sitter geometry

It is shown that the correspondence principle and the holographic principle are incompatible in the background of an eternal Schwarzschild–anti–de Sitter geometry. The argument is based on the observation that algebraic structures of local quantum field and conformal field theory (CFT) operators are not equivalent. This implies, in particular, that the bulk CFT must be singular near the black-hole horizon. A CFT Hilbert space representation is elaborated which may correspond to the AdS black hole in the dual theory.

Figure 1

A sketch of the Kruskal extension of the Schwarzschild-AdS black hole.

Figure 2

“Left” and “right” AdS-Rindler (RAdS) patches in ${\mathrm{AdS}}_3$ and their boundary $\partial {\mathrm{AdS}}_3$. The CFT algebra ${\mathcal{O}}_{\partial \mathrm{AdS}}$ can be represented as a product of two factor subalgebras, i.e. ${\mathcal{O}}_{\text{Rindler-L}}\otimes {\mathcal{O}}_{\text{Rindler}\text{−}\mathrm{R}}$. The CFT or Minkowski vacuum probed by either ${\mathcal{O}}_{\text{Rindler}\text{−}\mathrm{L}}$ or ${\mathcal{O}}_{\text{Rindler}\text{−}\mathrm{R}}$ responses as a thermal state, i.e. the vacuum two-point function for each of the factor subalgebras satisfies the Kubo-Martin-Schwinger condition with respect to the boost generators of the conformal group.

Figure 3

A sketch of the Hawking-Page phase transition. Before the transition ${\mathcal{M}}_1$ is filled by a thermal gas of particles that are states in the CFT Hilbert space $\partial {\mathcal{H}}_R$. If the gas temperature $T$ exceeds the critical temperature $T_c$, then the nucleation of the black hole occurs. This is described at the quantum level by ${\mathcal{O}}_{{\mathcal{M}}_1^{\prime }}\otimes {\mathcal{O}}_{{\mathcal{M}}_1}\to {\mathcal{O}}_{{\mathcal{M}}_2^{\prime }}\otimes {\mathcal{O}}_{{\mathcal{M}}_2}$ and $\partial {\mathcal{H}}_{\mathrm{L}}\otimes \partial {\mathcal{H}}_R\to \partial \mathcal{H}$. The Hilbert space $\partial \mathcal{H}$ is defined with respect to the vacuum $|\partial {\mathrm{\Omega }}_2⟩$ which cannot be understood in quantum field theory as an entangled state in $\partial {\mathcal{H}}_{\mathrm{L}}\otimes \partial {\mathcal{H}}_R$. However, it is a (maximally) entangled state in the sense that it is not a product state for nontrivial field operators that are products of elements from ${\mathcal{O}}_{{\mathcal{M}}_2^{\prime }}$ and ${\mathcal{O}}_{{\mathcal{M}}_2}$. In other words, the state $|\partial {\mathrm{\Omega }}_2⟩$ carries nonlocal (vacuum) correlations. This is in complete analogy with the property of the CFT vacuum which carries nonlocal correlations between the “left” and “right” Rindler patches.

Figure 4

A sketch of the boundary of the eternal Schwarzschild-AdS black hole.