Entanglement, Holography and Causal Diamonds

Abstract: We argue that the degrees of freedom in a d-dimensional CFT can be re-organized in an insightful way by studying observables on the moduli space of causal diamonds (or equivalently, the space of pairs of timelike separated points). This 2d-dimensional space naturally captures some of the fundamental nonlocality and causal structure inherent in the entanglement of CFT states. For any primary CFT operator, we construct an observable on this space, which is defined by smearing the associated one-point function over causal diamonds. Known examples of such quantities are the entanglement entropy of vacuum excitations and its higher spin generalizations. We show that in holographic CFTs, these observables are given by suitably defined integrals of dual bulk fields over the corresponding Ryu-Takayanagi minimal surfaces. Furthermore, we explain connections to the operator product expansion and the first law of entanglement entropy from this unifying point of view. We demonstrate that for small perturbations of the vacuum, our observables obey linear two-derivative equations of motion on the space of causal diamonds. In two dimensions, the latter is given by a product of two copies of a two-dimensional de Sitter space. For a class of universal states, we show that the entanglement entropy and its spin-three generalization obey nonlinear equations of motion with local interactions on this moduli space, which can be identified with Liouville and Toda equations, respectively. This suggests the possibility of extending the definition of our new observables beyond the linear level more generally and in such a way that they give rise to new dynamically interacting theories on the moduli space of causal diamonds. Various challenges one has to face in order to implement this idea are discussed.

• The paper introduces a novel 2d-dimensional moduli space of causal diamonds that reorganizes CFT observables to capture inherent entanglement and geometric structures.
• The paper defines new observables by smearing primary operator functions over causal diamonds, linking them to dual bulk fields through the Ryu-Takayanagi prescription.
• The paper demonstrates that these observables satisfy linear motion equations on the moduli space with nonlinear extensions resembling Liouville and Toda dynamics, offering fresh insights into quantum gravity.

Overview of the Paper: "Entanglement, Holography and Causal Diamonds"

The paper "Entanglement, Holography and Causal Diamonds" by Jan de Boer, Felix M. Haehl, Michal P. Heller, and Robert C. Myers explores a profound exploration of the connections between conformal field theories (CFTs), entanglement, and holography via the framework of causal diamonds. This conceptual framework proposes a novel organizational structure for the degrees of freedom in a $d$-dimensional CFT by examining them through observables defined over moduli spaces of causal diamonds. In essence, the authors propose that this perspective captures core aspects of nonlocality and causal relations inherent in the entanglement of CFT states.

Key Contributions

1. Moduli Space of Causal Diamonds: The authors introduce a $2d$-dimensional moduli space of causal diamonds as a fundamental organizational structure for observables in CFTs. This space provides a geometric and causal framework that reflects the inherent entanglement characteristics within holographic theories.
2. Observable Construction: For any primary operator in a CFT, the paper defines new observables by smearing the associated one-point function over the causal diamonds. This smearing enables the construction of a range of quantities that shed light on nonlocality and causal structure.
3. Holographic Interpretation: In the context of holographic CFTs, the observables tied to causal diamonds are computed as integrals of dual bulk fields over corresponding minimal surfaces. This provides a direct link to the geometry of the bulk via the Ryu-Takayanagi prescription, central to holographic entanglement entropy.
4. Equation of Motion on Causal Diamond Space: The paper demonstrates that under small perturbations of the vacuum, the defined observables satisfy linear equations of motion on the causal diamond moduli space. For two-dimensional CFTs, this moduli space can be productively viewed as two-dimensional de Sitter spaces.
5. Nonlinear Generalizations: For a class of universal states, the authors extend the framework to nonlinear regimes. They establish that the entanglement entropy and its higher spin analogs correspond to nonlinear equations akin to Liouville and Toda types on the moduli space.

Theoretical and Practical Implications

• The paper's framework offers insights into extending the organizational principles of CFTs, especially within holographic theories, through causal connections and nonlocal observables.
• Further, it hints towards potentially reorganizing existing theories into dynamical interacting models defined over this moduli space. This reorganization could open new avenues in modeling quantum gravity intricacies and deeper holographic insights.

Future Directions

The authors speculate on broadening these theories beyond linear regimes by leveraging the definitions of new observables. This forward-thinking approach suggests exploring non-linear dynamics on these moduli spaces could engender new dynamical theories. They also acknowledge various obstacles in realizing this potential, indicating this is a ripe area for further research and refinement.

In conclusion, by viewing entanglement and holography through the lens of causal diamond spaces, the paper contributes a fresh perspective to the description of CFTs. Its emphasis on geometric and causal aspects aligns with ongoing quests in theoretical physics to reconcile gravity and quantum mechanics under a unified framework. The proposed framework forms a robust foundation for further exploration into the depths of holographic principles and quantum entanglement in higher-dimensional and complex CFTs.