Quantum Extremality and Modular Witten Diagrams Calculate Entropy for Excited States with Amplitude Sources
Summarize this article with:
The behaviour of entropy in complex quantum systems remains a fundamental challenge in theoretical physics, and recent work by Abhirup Bhattacharya and Onkar Parrikar, both from the Tata Institute of Fundamental Research, sheds new light on this problem. They investigate entropy within excited states of holographic conformal field theories, exploring how subtle changes to these states impact the geometry of spacetime. Their calculations demonstrate a direct link between entropy and the shape of spacetime wedges, revealing how these wedges deform in response to external influences.
This research significantly advances our understanding of quantum gravity by explicitly connecting calculations on the quantum field theory side with geometric properties in the bulk, and it provides a novel method for calculating entropy using Witten diagrams that incorporates the effects of spacetime curvature and backreaction. A source term is introduced, activating a double-trace operator with a small amplitude, which initiates a cascade of effects on both the geometry of spacetime and the quantum state of matter within it. This process deforms the entanglement wedge, a region of spacetime connected by quantum entanglement, and this deformation becomes apparent in calculations of entanglement entropy, a measure of quantum connectedness. Scientists calculated these entropy changes to second order in the source amplitude, employing modular-flowed correlation functions and Witten diagrams to connect calculations in the quantum field theory to the geometry of the dual gravitational theory.
Boundary Field Theory Defines Bulk Gravity This work establishes a rigorous mathematical framework for understanding the deep connection between gravity, quantum field theory, and the holographic principle. Scientists aim to consistently compute quantities in a gravitational theory, such as energy, momentum, and entropy, using the tools of quantum field theory residing on the boundary of spacetime. They focus on systems not in thermal equilibrium and how to define conserved quantities within them, utilizing the modular Hamiltonian to describe entanglement structure. The research builds on the holographic principle and the AdS/CFT correspondence, which posits that gravity in certain spacetimes is equivalent to a quantum field theory on its boundary. Entanglement entropy and Rényi entropy, measures of quantum entanglement, are crucial for understanding the holographic principle and spacetime structure. Ultimately, scientists are building a mathematical toolkit for studying the connections between gravity, quantum mechanics, and information, providing a foundation for the holographic principle and a deeper understanding of the universe.
Entropy Deformation Reveals Bulk Geometry Changes Scientists have refined our understanding of entropy and symplectic structures within holographic conformal field theories, extending these concepts to semi-classical gravity. They calculated entropy for ball-shaped regions in excited states, preparing these states using a source that introduces a small amplitude into a double-trace operator. This source deforms both the bulk geometry and the state of matter fields, resulting in a corresponding deformation of the entanglement wedge, directly observable in entropy calculations. These calculations were performed perturbatively, reaching terms up to second order in the source amplitude, and utilized modular-flowed correlation functions and Witten diagrams. At first order, the established expression aligns with the Ryu-Takayanagi formula, confirming a fundamental relationship between entropy and geometry. At second order, results demonstrate a duality between boundary relative entropy and gravitational canonical energy, providing further validation of the holographic principle. Scientists generalized these results to semi-classical gravity, developing a framework for analyzing the gravitational symplectic form, which describes the geometry of spacetime, in the presence of quantum matter. They found that the gravitational symplectic form is not conserved on its own, but becomes conserved when combined with the Berry curvature of matter states, effectively cancelling contributions from the stress tensor. They defined a semi-classical symplectic 2-form, independent of the chosen reference frame, and demonstrated its consistency through a generalized HIW identity. Measurements confirm that the derived symplectic 2-form remains consistent even when restricted to localized areas of spacetime, providing a powerful tool for analyzing entropy and geometry. Holographic Entropy and Geometric Deformations Revealed Scientists have achieved a significant advance in understanding the relationship between gravity and quantum field theories through detailed calculations of entropy for excited states in holographic conformal field theories. Their work demonstrates how deformations of bulk geometry, arising from double-trace operators, directly influence entropy calculations, and successfully reproduces known results from the Ryu-Takayanagi formula, including the effects of backreaction and shape deformation. This was accomplished by explicitly calculating entropy perturbatively using modular-flowed correlation functions and employing Witten diagrams, a technique that connects calculations in the quantum field theory to the geometry of the dual gravitational theory.
The team’s calculations reveal a connection between the gravitational symplectic form in the bulk and contributions to entropy, specifically isolating and analyzing a graviton-exchange diagram. By rewriting this diagram, they were able to demonstrate its equivalence to the canonical energy term within the Ryu-Takayanagi formula, confirming the consistency between the field theory and gravitational descriptions. While focusing on a specific diagram, the researchers acknowledge that other diagrams likely contribute to the overall relative entropy, and suggest strategies, such as considering a large number of flavors, to systematically address these contributions. Future research will likely involve extending the analysis to include all relevant diagrams and exploring the implications of these findings for a more complete understanding of the holographic duality and the nature of quantum gravity. 👉 More information 🗞 Modular Witten Diagrams and Quantum Extremality 🧠 ArXiv: https://arxiv.org/abs/2512.11754 Tags: