[Paper Review] Modular Hamiltonians for Euclidean Path Integral States

Srivatsan Balakrishnan, Onkar Parrikar|arXiv (Cornell University)|Jan 31, 2020

Black Holes and Theoretical Physics53 references4 citations

TL;DR

This paper derives a manifestly Lorentzian formula for the modular Hamiltonian of half-space/Rindler regions in relativistic quantum field theories, applicable to excited states created by sources in the Euclidean path integral. It shows that the modular Hamiltonian can be expressed as a perturbative series in source strength, with all-order resummation possible for null deformations, yielding exact agreement with known vacuum modular Hamiltonians for null cuts of the Rindler horizon.

ABSTRACT

We study half-space/Rindler modular Hamiltonians for excited states created by turning on sources for local operators in the Euclidean path integral in relativistic quantum field theories. We derive a simple, manifestly Lorentzian formula for the modular Hamiltonian to all orders in perturbation theory in the sources. We apply this formula to the case of shape-deformed half spaces in the vacuum state, and obtain the corresponding modular Hamiltonian to all orders in the shape deformation in terms of products of half-sided null energy operators, i.e., stress tensor components integrated along the future and past Rindler horizons. In the special case where the shape deformation is purely null, our perturbation series can be resummed, and agrees precisely with the known formula for vacuum modular Hamiltonians for null cuts of the Rindler horizon. Finally, we study some universal properties of modular flow (corresponding to Euclidean path integral states) of local operators inside correlation functions in conformal field theories. In particular, we show how the flow becomes the local boost in the limit where the operator being flowed approaches the entanglement cut.

Motivation & Objective

• To develop a general, manifestly Lorentzian expression for the modular Hamiltonian in half-space regions of relativistic quantum field theories beyond the vacuum state.
• To extend perturbative methods for modular Hamiltonians to excited states created by local sources in the Euclidean path integral formalism.
• To derive the modular Hamiltonian for shape-deformed half-spaces in the vacuum, expressing it in terms of products of half-sided null energy operators.
• To show that the perturbation series for null deformations can be exactly re-summed, recovering the known vacuum modular Hamiltonian for null cuts.
• To investigate universal properties of modular flow for local operators in conformal field theories, particularly in the limit approaching the entanglement cut.

Proposed method

• Perturbative expansion of the modular Hamiltonian in the source strength λ, treating the excited state as a deformation of the vacuum via a source coupled to a local operator in the Euclidean path integral.
• Derivation of a closed-form expression for the modular Hamiltonian to all orders in perturbation theory, given by equation (3), involving time-ordered products of modular-flowed operators.
• Use of modular flow to define the time-ordered operators O(s,Y) via conjugation with the vacuum modular Hamiltonian K, as in equation (6).
• Application of the replica trick and analytic continuation to Euclidean path integral states, enabling Lorentzian formulation of the modular Hamiltonian.
• Use of step functions and topological sorting of ordered variables to reorganize and simplify the perturbative series, particularly for the T-term and P-term in the summation.
• Identification of a cancellation mechanism between terms in the perturbative series by mapping permutations to binary strings, leading to exact cancellation and a simplified final expression.

Experimental results

Research questions

• RQ1 Can a manifestly Lorentzian formula for the modular Hamiltonian be derived for excited states in relativistic QFTs created via sources in the Euclidean path integral?
• RQ2 How does the modular Hamiltonian for shape-deformed half-spaces in the vacuum state depend on the deformation profile, and can it be expressed in terms of stress tensor components?
• RQ3 Is the perturbative series for the modular Hamiltonian in the source expansion exactly re-summedable in special cases, such as purely null deformations?
• RQ4 What universal behavior emerges in the modular flow of local operators in conformal field theories as they approach the entanglement cut?
• RQ5 How do the analytic structures of the modular flow and the resulting Hamiltonian relate to the geometry of the entanglement wedge in holographic theories?

Key findings

• The modular Hamiltonian for Euclidean path integral states is derived to all orders in perturbation theory as a manifestly Lorentzian expression, given by equation (3), involving time-ordered products of modular-flowed operators.
• For shape-deformed half-spaces in the vacuum, the modular Hamiltonian is expressed as a sum of products of half-sided null energy operators, corresponding to integrals of stress tensor components along future and past Rindler horizons.
• In the case of purely null deformations, the perturbation series is exactly re-summed, and the result matches the known formula for the vacuum modular Hamiltonian of null cuts of the Rindler horizon.
• The modular flow of local operators in conformal field theories approaches a local boost symmetry in the limit where the operator approaches the entanglement cut, revealing a universal geometric behavior.
• A cancellation mechanism is identified between terms in the perturbative series, achieved by mapping permutations of ordered variables to binary strings, leading to simplification and exact results.
• The derivation establishes a direct link between the structure of the modular Hamiltonian and the underlying causal and geometric structure of the entanglement region, particularly through the role of the Rindler horizon and modular flow.