Anti-de Sitter Space from Optimization of Path Integrals in Conformal Field Theories (1703.00456v2)

Abstract: We introduce a new optimization procedure for Euclidean path integrals which compute wave functionals in conformal field theories (CFTs). We optimize the background metric in the space on which the path integration is performed. Equivalently this is interpreted as a position-dependent UV cutoff. For two-dimensional CFT vacua, we find the optimized metric is given by that of a hyperbolic space and we interpret this as a continuous limit of the conjectured relation between tensor networks and Anti--de Sitter (AdS)/conformal field theory (CFT) correspondence. We confirm our procedure for excited states, the thermofield double state, the Sachdev-Ye-Kitaev model and discuss its extension to higher-dimensional CFTs. We also show that when applied to reduced density matrices, it reproduces entanglement wedges and holographic entanglement entropy. We suggest that our optimization prescription is analogous to the estimation of computational complexity.

• The paper introduces an optimization framework that alters the background metric in path integrals to produce hyperbolic geometries consistent with tensor network interpretations.
• It extends the method to various CFT states, including excited states and the SYK model, demonstrating clear links to entanglement wedges and holographic entropy.
• By minimizing the Liouville action, the research connects computational complexity in CFTs with emergent spacetime structures, suggesting pathways for higher-dimensional explorations.

Optimization of Path Integrals and Emergent Geometry in Conformal Field Theories

The paper "Anti--de Sitter Space from Optimization of Path Integrals in Conformal Field Theories" proposes a novel procedure to optimize Euclidean path integrals in Conformal Field Theories (CFTs), examining its connection to anti--de Sitter (AdS) spaces through the lens of AdS/CFT correspondence. The authors explore optimized background metrics for path integration, which they suggest can also be interpreted as position-dependent UV cutoffs, thus drawing a significant connection between tensor network formulations and holography.

Key Findings

1. Optimization Framework: The paper introduces an optimization process for wave functionals by altering the background metric utilized in path integrals. For two-dimensional CFT vacuum states, the optimized metric was identified as a hyperbolic space. This corroborates the tensor network interpretation of AdS/CFT, particularly for emergent spaces like the Multi-scale Entanglement Renormalization Ansatz (MERA).
2. Application to Various CFT States: The optimization method was extended beyond vacuum states to explore excited states, the thermofield double state, and even the Sachdev-Ye-Kitaev (SYK) model. A significant aspect investigated is the reduced density matrices, where this optimization yields entanglement wedges and holographic entanglement entropy, aligning well with established holographic principles.
3. Liouville Action Minimization: By minimizing the Liouville action, the paper highlights that a natural emergence of the hyperbolic space as an optimally efficient description is characterized by minimal action, analogous to proposing a measure of computational complexity within CFTs.
4. Extensions and Higher Dimensions: The authors speculate on extending the optimization approach to higher-dimensional spaces despite the challenges posed by metric constraints beyond simple Weyl scaling. Theoretical developments in this direction aim to corroborate the time slice geometries oft-discussed in AdS/CFT contexts.

Implications

The implications of this research range from practical procedures for enhancing simulation efficiency of quantum field theories to deep theoretical insights into the fabric of space-time geometry implied by fundamental theories like AdS/CFT. The work serves as a bridge connecting discrete, algorithmic descriptions of quantum geometry via tensor networks to continuous field-based descriptions, potentially influencing both computational approaches in physics and theoretical understandings of holography.

Future Developments

The paper sets the stage for several pathways in theoretical and computational physics. Future endeavors may include:

• Refinement of the optimization techniques in higher dimensions and time-dependent cases.
• Investigations into the geometric duals implicated by more general field theories and the extension to non-conformal frameworks.
• Exploring time-evolution within this framework, providing potential insights into dynamical holography and quantum information flow.

In summary, the paper offers substantial advancements in the understanding of the relationship between CFT optimizations and emergent space-time geometries, advocating for structured methods that build upon the conceptual interface between quantum mechanics and general relativity. It challenges researchers to explore the limits and applications of these foundational principles across various domains in modern theoretical physics.