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Boundary-to-bulk maps for AdS causal wedges and the Reeh-Schlieder property in holography

Ian A. Morrison

Abstract

In order to better understand how AdS holography works for sub-regions, we formulate a holographic version of the Reeh-Schlieder theorem for the simple case of an AdS Klein-Gordon field. This theorem asserts that the set of states constructed by acting on a suitable vacuum state with boundary observables contained within any subset of the boundary is dense in the Hilbert space of the bulk theory. To prove this theorem we need two ingredients which are themselves of interest. First, we prove a purely bulk version of Reeh-Schlieder theorem for an AdS Klein-Gordon field. This theorem relies on the analyticity properties of certain vacuum states. Our second ingredient is a boundary-to-bulk map for local observables on an AdS causal wedge. This mapping is achieved by simple integral kernels which construct bulk observables from convolutions with boundary operators. Our analysis improves on previous constructions of AdS boundary-to-bulk maps in that it is formulated entirely in Lorentz signature without the need for large analytic continuation of spatial coordinates. Both our Reeh-Schlieder theorem and boundary-to-bulk maps may be applied to globally well-defined states constructed from the usual AdS vacuum as well more singular states such as the local vacuum of an AdS causal wedge which is singular on the horizon.

Boundary-to-bulk maps for AdS causal wedges and the Reeh-Schlieder property in holography

Abstract

In order to better understand how AdS holography works for sub-regions, we formulate a holographic version of the Reeh-Schlieder theorem for the simple case of an AdS Klein-Gordon field. This theorem asserts that the set of states constructed by acting on a suitable vacuum state with boundary observables contained within any subset of the boundary is dense in the Hilbert space of the bulk theory. To prove this theorem we need two ingredients which are themselves of interest. First, we prove a purely bulk version of Reeh-Schlieder theorem for an AdS Klein-Gordon field. This theorem relies on the analyticity properties of certain vacuum states. Our second ingredient is a boundary-to-bulk map for local observables on an AdS causal wedge. This mapping is achieved by simple integral kernels which construct bulk observables from convolutions with boundary operators. Our analysis improves on previous constructions of AdS boundary-to-bulk maps in that it is formulated entirely in Lorentz signature without the need for large analytic continuation of spatial coordinates. Both our Reeh-Schlieder theorem and boundary-to-bulk maps may be applied to globally well-defined states constructed from the usual AdS vacuum as well more singular states such as the local vacuum of an AdS causal wedge which is singular on the horizon.

Paper Structure

This paper contains 13 sections, 6 theorems, 80 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The bulk Reeh-Schlieder theorem
  3. Bulk theory basics
  4. Locally analytic states and the RS theorem
  5. Boundary-to-bulk maps
  6. Boundary theory basics
  7. The case of the Poincaré chart
  8. AdS causal wedges
  9. The case of the AdS-Rindler wedge
  10. Discussion
  11. Wave front sets
  12. An example transformation to an AdS causal wedge
  13. WKB approximations of AdS-Rindler Klein-Gordon modes

Key Result

Theorem 2.2

Let ${\mathcal{M}}$ be a real analytic connected manifold and $u$ a distribution (in the distribution space dual to that of smooth functions) with the property that Then for each non-void open subset ${\mathcal{O}} \subset {\mathcal{M}}$ if the restriction of $u$ to ${\mathcal{O}}$ vanishes then $u = 0$.

Figures (4)

  • Figure 1: An an equal-time surface of global AdS. A bulk observable is at the black dot. The yellow, red, and blue shaded regions denote AdS causal wedges which include the support of the observable. The boundaries of these wedges have no mutual intersection. The observable may be computed from CFT correlation functions on the boundary of any one of these wedges.
  • Figure 2: Depictions of the domains of analyticity referenced in the text.
  • Figure 3: Left: an AdS causal wedge. Right: global AdS with the AdS-Rindler wedge highlighted.
  • Figure 4: Conformal diagram of a simple AdS wormhole geometry. There are two disconnected conformal boundaries. Each blue shaded region has causal contact with the boundary only through one of it's past or future domains of influence. The gray shaded region has no causal contact with the boundaries.

Theorems & Definitions (10)

  • Definition 2.1
  • Theorem 2.2: Edge of the wedge theorem (Proposition 5.3 of Strohmaier:2002aa)
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Theorem 2.6: Reeh-Schlieder theorem
  • Theorem 2.7
  • Definition A.1
  • Example A.2
  • Example A.3