Modular Hamiltonians for Deformed Half-Spaces and the Averaged Null Energy Condition (1605.08072v1)

Abstract: We study modular Hamiltonians corresponding to the vacuum state for deformed half-spaces in relativistic quantum field theories on $\mathbb{R}^{1,d-1}$. We show that in addition to the usual boost generator, there is a contribution to the modular Hamiltonian at first order in the shape deformation, proportional to the integral of the null components of the stress tensor along the Rindler horizon. We use this fact along with monotonicity of relative entropy to prove the averaged null energy condition in Minkowski space-time. This subsequently gives a new proof of the Hofman-Maldacena bounds on the parameters appearing in CFT three-point functions. Our main technical advance involves adapting newly developed perturbative methods for calculating entanglement entropy to the problem at hand. These methods were recently used to prove certain results on the shape dependence of entanglement in CFTs and here we generalize these results to excited states and real time dynamics. We also discuss the AdS/CFT counterpart of this result, making connection with the recently proposed gravitational dual for modular Hamiltonians in holographic theories.

• The paper introduces a novel framework for modular Hamiltonians in deformed half-spaces, revealing an extra stress tensor term that supports the ANEC in QFT.
• It employs advanced perturbative techniques to extend entanglement entropy analyses from stationary to dynamic and deformed state configurations.
• The findings bridge quantum field theory and holographic duality, reinforcing bounds on CFT three-point functions and fundamental quantum energy constraints.

Overview of Modular Hamiltonians and Quantum Field Theories

The paper "Modular Hamiltonians for Deformed Half-Spaces and the ANEC in QFT" tackles the intricate subject of modular Hamiltonians in relativistic quantum field theories (QFTs), with a focus on deformed half-spaces. This work probes the theoretical landscaping of entanglement entropy and modular Hamiltonians, engaging with the interplay between these concepts within the framework of QFTs and conformal field theories (CFTs).

Modular Hamiltonians in QFTs

The paper embarks on the exploration of modular Hamiltonians, specifically pertaining to vacuum states in deformed half-spaces. A principal claim made therein is the existence of a supplementary term to the modular Hamiltonian when considering shape deformations from the original half-space, characterized by the integral over the null components of the stress tensor along the Rindler horizon. This is instrumental in reaffirming the averaged null energy condition (ANEC), particularly in flat spacetime configurations, leading to a refreshed proof of the Hofman-Maldacena bounds on parameters present in CFT three-point functions.

Methodological Advances

A key technical advancement demonstrated in the paper encompasses the adoption and adaptation of perturbative techniques for evaluating entanglement entropy, previously established for stationary configurations in CFTs, to examine these dynamics in deformed geometries and dynamic states. The utility of this methodological approach is exemplified in its application to not only vacuum states but also to excited states, facilitating a comprehensive analysis that transcends beyond stationary assumptions.

Implications and Theoretical Significance

One of the cardinal implications of this research involves establishing a direct linkage between the entanglement structure and fundamental quantum constraints. The positive September strides in proving the ANEC employing relativistic QFTs accentuates the robust foundation of entropic inequalities in governing and constraining physical phenomena within these fields. While the standard focusing theorems, topological censorship, and other results have historically leaned on averaged energy conditions in classical general relativity, this work expands the understanding by contextualizing these conditions within quantum frameworks.

Holographic Dual and AdS/CFT

A remarkable facet of the paper is the discussion on the Anti-de Sitter/Conformal Field Theory (AdS/CFT) correspondence to bridge these quantum findings with gravitational paradigms. The insights gained from modular Hamiltonians for deformed half-spaces significantly interact with holographic principles, where gravitational dualities offer a macroscopic lens for these modular constructs. The authors make a persuasive case for these dualities by aligning their findings with the Jafferis-Lewkowycz-Maldacena-Suh (JLMS) proposal on holographic duals for modular Hamiltonians.

Future Directions

While the research makes significant strides towards understanding the role of modular Hamiltonians in quantum field theories and the critical validation of the ANEC, it also opens doors to further questions and explorations. Potential investigations could explore other spacetime configurations, more complex entangling surfaces, or even extend these methodologies to tackle broader geometric deformations within both quantum and gravitational contexts.

In summary, this paper delivers a scholarly scaffold that enriches the dialogue between quantum field theories and the broad vistas of theoretical physics, reinforcing the quintessential role of entanglement and modular Hamiltonians in unmasking the nuanced architecture of the universe at both quantum and cosmic scales. The thoughtful examination of modular Hamiltonians not only challenges pre-existing narratives but also solidifies key quantum necessities such as the averaged null energy condition, weaving them into the multilayered tapestry of theoretical physics.