Analyticity and the Unruh effect: a study of local modular flow (2403.18937v4)

Abstract: The Unruh effect can be formulated as the statement that the Minkowski vacuum in a Rindler wedge has a boost as its modular flow. In recent years, other examples of states with geometrically local modular flow have played important roles in understanding energy and entropy in quantum field theory and quantum gravity. Here I initiate a general study of the settings in which geometric modular flow can arise, showing (i) that any geometric modular flow must be a conformal symmetry of the background spacetime, and (ii) that in a well behaved class of "weakly analytic" states, geometric modular flow must be future-directed. I further argue that if a geometric transformation is conformal but not isometric, then it can only be realized as modular flow in a conformal field theory. Finally, I discuss a few settings in which converse results can be shown -- i.e., settings in which a state can be constructed whose modular flow reproduces a given vector field.

• The paper demonstrates that any geometric modular flow must conform to spacetime symmetries through a conformal Killing field.
• The study reveals that in weakly analytic states, modular flow is constrained to be future-directed, affecting non-equilibrium quantum dynamics.
• The research establishes that modular flow’s conformal nature, distinct from isometry, bridges theoretical insights in quantum field theory and gravity.

Analyticity and Geometric Modular Flow: Exploring the Unruh Effect

The paper by Jonathan Sorce presents an investigation into the conditions under which geometric modular flow arises, with a specific focus on its relationship to the Unruh effect within the field of theoretical physics. The research advances our understanding of energy, entropy, and modular flow in quantum field theory and quantum gravity settings by delineating precise constraints that govern when modular flows can be considered geometric.

Key Contributions

The paper begins by revisiting the Unruh effect, which posits that for an observer in constant acceleration through the Minkowski vacuum, the vacuum state acts as a thermal bath. This phenomenon is well-understood in the context of Rindler wedges, where the Minkowski vacuum's modular flow corresponds to a Lorentz boost.

The principal contributions of the paper are as follows:

1. Conformal Symmetric Conditions: It is demonstrated that any geometric modular flow must act as a conformal symmetry of the underlying spacetime. This is fundamentally governed by microcausality and implies that the generator of the flow must be a conformal Killing vector field. The result effectively extends the conditions necessary for the presence of geometric modular flow to any conformal spacetime.
2. Directionality in Weakly Analytic States: The paper asserts that within a well-behaved class of states denoted as "weakly analytic", modular flows must be directed towards the future. This finding is particularly profound as it impacts the nature of thermodynamic behavior in non-equilibrium states like those exhibited in the Unruh effect. Weakly analytic states encourage analyticity assumptions about the support of states on large momentum, reframing how we consider non-equilibrium thermodynamic states and modular Hamiltonians.
3. Conformality vs. Isometry: Another critical argument presented is that if the geometric transformation by modular flow is conformal but not isometric, it can only manifest within a conformal field theory. This bridges the conditions of modular flow to broader theoretical contexts, hence contributing to our theoretical understanding of the structures where modular flow manifests.

Theoretical and Practical Implications

The findings in the paper have potential implications both theoretically and practically:

• Theoretical Implications: From a theoretical standpoint, the results clarify longstanding conceptual issues in quantum field theories related to the applicability of modular theory. By demanding that modular flow be aligned with conformal transformations, the research aligns a branch of quantum field theory with the geometric structure of spacetime, reinforcing its intuitions from relativity and providing a more robust framework for future explorations in holography and quantum gravity.
• Practical Implications: Practically, this investigation may guide new pathways in the examination of black hole thermodynamics and radiation properties. Additionally, understanding the conformal requirements of modular flow can help in constraining conditions for quantum experiments that aim to simulate or better understand properties related to the Unruh effect.

Future Developments

The research opens several future directions. A fascinating area of exploration is refining the conditions under which modular flow exists without relying on assumptions of analyticity, potentially expanding our understanding beyond weakly analytic states. Moreover, extending this theory in practical, non-conformal field theories might yield new quantum gravity insights, particularly within the context of asymptotically anti-de Sitter spaces or other non-standard spacetime geometries.

In summary, this paper extends the discourse on modular flow and its geometric analogs within quantum field theories, presenting well-defined constraints essential for both theoretical rigor and potential empirical exploration.