State-Dependent Bulk-Boundary Maps and Black Hole Complementarity (1310.6335v2)

Abstract: We provide a simple and explicit construction of local bulk operators that describe the interior of a black hole in the AdS/CFT correspondence. The existence of these operators is predicated on the assumption that the mapping of CFT operators to local bulk operators depends on the state of the CFT. We show that our construction leads to an exactly local effective field theory in the bulk. Barring the fact that their charge and energy can be measured at infinity, we show that the commutator of local operators inside and outside the black hole vanishes exactly, when evaluated within correlation functions of the CFT. Our construction leads to a natural resolution of the strong subadditivity paradox of Mathur and Almheiri et al. Furthermore, we show how, using these operators, it is possible to reconcile small corrections to effective field theory correlators with the unitarity of black hole evaporation. We address and resolve all other arguments, advanced in arxiv:1304.6483 and arxiv:1307.4706, in favour of structure at the black hole horizon. We extend our construction to states that are near equilibrium, and thereby also address the "frozen vacuum" objections of arxiv:1308.3697. Finally, we explore an intriguing link between our construction of interior operators and Tomita-Takesaki theory.

• The paper introduces a state-dependent mapping framework that translates CFT operators into local bulk operators for black hole interiors.
• It demonstrates that this mapping preserves local commutativity and resolves paradoxes such as the strong subadditivity and firewall issues.
• The approach leverages Tomita-Takesaki theory to extend the framework to near-equilibrium states, offering new insights for quantum gravity research.

State-Dependent Bulk-Boundary Maps and Black Hole Complementarity

This paper presents a detailed examination of the state-dependent mapping between boundary and bulk operators in the context of the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence. It aims to address fundamental issues related to the quantum description of black hole interiors, specifically the information paradox and the nature of the black hole horizon. The authors, Papadodimas and Raju, provide a framework that enables the construction of local bulk operators capable of describing the interior of a black hole, contingent on the state-dependence of the bulk-boundary operator mapping. This perspective is instrumental in resolving several paradoxes associated with black hole physics.

Key Contributions and Methodologies

1. State-Dependence and Complementarity: The central hypothesis of the paper is the state-dependent nature of the bulk-boundary operator mapping. The authors propose that the transformation of Conformal Field Theory (CFT) operators into local bulk operators is inherently dependent on the state of the CFT. This approach does not contravene the principles of quantum mechanics, as the operators in question merely represent a state-dependent physical interpretation of standard quantum operators.
2. Construction of Interior Operators: Papadodimas and Raju demonstrate an explicit method for constructing local bulk effective field theories in the black hole interior. Their construction ensures that the commutators of local operators, when evaluated within CFT correlation functions, vanish exactly, except for their charge and energy measured at infinity, thus maintaining locality in the bulk.
3. Resolution of Paradoxes: Utilizing the state-dependent mapping, the authors resolve the strong subadditivity paradox, which has been a pivotal challenge in understanding black hole complementarity. Their methodology shows that the operators inside and outside the black hole, although secretly acting on the same degrees of freedom, maintain a default commutativity essential for resolving paradoxes associated with firewall proposals.
4. Tomita-Takesaki Theory: A significant theoretical underpinning of their framework is the linkage to Tomita-Takesaki theory of von Neumann algebras. This connection enriches their approach with formalism that provides modular isomorphisms between algebras, offering insights into the algebraic structure of quantum field theories.
5. Handling Equilibrium and Near-Equilibrium States: The paper expands the construction to states near equilibrium, which addresses objections of a "frozen vacuum" at the black hole horizon by showing how interior operators can adapt to incorporate non-equilibrium dynamics, preserving the unitarity of black hole evaporation.

Implications and Future Directions

The framework developed in this paper suggests a reconciliatory path forward for longstanding theoretical challenges in black hole physics, particularly concerning the smoothness of black hole horizons and the nature of quantum information degradation through evaporation.

Practical Implications: This construction informs our understanding of how holographic correspondences can resolve classical paradoxes of general relativity with quantum mechanical principles, providing an innovative approach to formulating quantum gravity theories.

Theoretical Implications: The work challenges the assumption of fixed, universal mappings in quantum field theory, advocating instead for a nuanced, state-dependent approach reflective of the underlying quantum state. The connection to Tomita-Takesaki theory signifies potential avenues for applying algebraic quantum field theory concepts more broadly in holographic and gravitational contexts.

Speculative Developments: Future research might explore the full implications of state-dependent operators beyond AdS/CFT, potentially impacting other holographic principles or fields that deal with entanglement and quantum phase transitions. Moreover, the further mathematical formalization of quantum geometries using state dependence could lead to new formulations in quantum gravity.

Overall, Papadodimas and Raju propose a compelling construction of black hole interiors within the AdS/CFT paradigm, marking significant strides in addressing the philosophical and technical barriers presented by quantum gravity, and paving the way for further exploration and refinement of complementarity principles in holographic contexts.