Surface/State Correspondence as a Generalized Holography (1503.03542v1)

Abstract: We propose a new duality relation between codimension two space-like surfaces in gravitational theories and quantum states in dual Hilbert spaces. This surface/state correspondence largely generalizes the idea of holography such that we do not need to rely on any existence of boundaries in gravitational spacetimes. The present idea is motivated by the recent interpretation of AdS/CFT in terms of the tensor networks so called MERA. Moreover, we study this correspondence from the viewpoint of entanglement entropy and information metric. The Cramer-Rao bound in quantum estimation theory implies that the quantum fluctuations of radial coordinate of the AdS is highly suppressed in the large N limit.

• The paper introduces a generalized holography by associating every convex surface with a quantum state, thereby transcending traditional boundary constraints.
• The paper applies tensor network frameworks like MERA and cMERA to establish a connection between quantum entanglement and the emergence of spacetime geometry.
• The paper leverages the Cramer-Rao bound to show that quantum fluctuations decrease in the large N limit, offering insights into cosmological spacetimes such as de Sitter spaces.

An Analysis of the Surface/State Correspondence in Generalized Holography

The paper "Surface/State Correspondence as a Generalized Holography" by Masamichi Miyaji and Tadashi Takayanagi presents an ambitious extension of the holographic principle, which traditionally relates gravitational theories to equivalent quantum field theories (QFTs) on boundary surfaces. Known for its implications in the AdS/CFT correspondence, the holographic principle posits that every region of space can be described by information encoded on its boundary, akin to a hologram.

The authors propose a more generalized framework that is not constrained by the existence of boundaries, thus broadening the applicability of holography beyond Anti-de Sitter (AdS) spaces to other cosmological spacetimes including de Sitter (dS) spaces. This novel duality asserts a correspondence between codimension two space-like surfaces in a gravitational theory and quantum states within a suitably defined Hilbert space. This relation is encapsulated by what the authors term the "surface/state correspondence."

Conceptual Framework and Proposals

The key theoretical leap from standard holography to this surface/state correspondence is motivated by tensor network theories, especially the Multi-scale Entanglement Renormalization Ansatz (MERA) and its continuous counterpart cMERA. These frameworks are compelling because they suggest a structural link between quantum entanglement and spacetime geometry, wherein the distribution of quantum entanglement governs the emergence of spacetime structure.

In the proposed framework, every convex surface $\Sigma$ in a $d+2$ dimensional gravitational spacetime corresponds to a quantum state in an associated Hilbert space $\mathcal{H}_{\Sigma}$. This correspondence has profound implications for how spacetime geometry is conceptualized. Notably, the surface $\Sigma$ is not required to coincide with a physical boundary, as traditionally prescribed by holographic principles. Furthermore, quantum states tied to topologically trivial surfaces are pure states, whereas those linked to surfaces with non-trivial topology manifest as mixed states.

The paper also proposes that entanglement entropy, a fundamental measure of quantum correlations, can be understood as proportional to the area of the extremal surface in spacetime, establishing a direct linkage to the Bekenstein-Hawking entropy. The formulation for entanglement entropy here is expanded beyond the traditional boundary-centric holography to apply to any point in the bulk spacetime, provided certain geometric conditions of convexity are met.

Numerical Results and Theoretical Implications

Central to their analysis is the extension of this correspondence to account for entanglement entropy and the Fisher information metric. The authors utilize the Cramer-Rao bound from quantum estimation theory to elucidate that the quantum fluctuations of the radial coordinate in AdS space diminish significantly in the large $N$ limit, aligning with the emergence of classical spacetime as understood in large $N$ field theories.

By employing tensor networks as a heuristic, the paper also explores the implications for effective entropy and the effective dimensionality of quantum states associated with these surfaces. This approach has profound implications—potentially offering a more seamless integration of inertia and gravitational effects in quantum field theoretic frameworks.

Future Directions and Speculations

The authors hint at the potential of their framework to offer insights into longstanding challenges in cosmological and theoretical physics, such as de Sitter space holography and the internal compact dimensions typically encountered in string theory. More specifically, the work proposes resolving complexities unique to dS spaces, where boundaries traditionally serve as spacelike rather than timelike structures.

Given its radical departure from traditional boundaries, this work could catalyze developments in string theory's engagement with cosmological models and inspire novel computational methods for evaluating quantum gravity theories where traditional holographic foundations, such as boundaries, may not naturally exist.

In summary, this paper makes a substantive contribution to the evolving narrative of holographic theory by significantly relaxing boundary constraints, potentially broadening the application of holographic ideas to a wider variety of cosmological scenarios. This nascent direction deserves further exploration, particularly in instances where gravitational duals to specific quantum states can provide fresh insights into the nature of spacetime itself.