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arXiv:2607.20678

Uniqueness of null-local modular flow

Gautam Satishchandran, Jonathan Sorce

Published Jul 22, 2026• 0 citations
arXivHTMLPDF

Abstract

In arXiv:2306.01837, it was conjectured that one can engineer a large class of quantum field theory states for which the modular flow on a spacelike slice is "instantaneously local." Here we show that on null slices, such flows are highly constrained; ultraviolet universality essentially requires null-local modular flow to be unique. Concretely, we study massless Klein-Gordon theory in Minkowski spacetime, and construct, for any sufficiently regular future-directed vector field on the past null boundary of a causal diamond, a state that has this vector field as its instantaneous modular flow. We then show by explicit computation that no two distinct states in this class can be realized in the same local Hilbert space. Using a more abstract argument, we also show that in the vacuum sector of the theory, the only null-local modular flow in a causal diamond is provided by the vacuum state itself. We also comment on the construction of null-local modular flow for massive scalars, free Maxwell fields, free gravitons, and in curved backgrounds.

DOI
10.48550/arxiv.2607.20678

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In arXiv:2306.01837, it was conjectured that one can engineer a large class of quantum field theory states for which the modular flow on a spacelike slice is "instantaneously local." Here we show that on null slices, such flows are highly constrained; ultraviolet universality essentially requires null-local modular flow to be unique. Concretely, we study massless Klein-Gordon theory in Minkowski spacetime, and construct, for any sufficiently regular future-directed vector field on the past null boundary of a causal diamond, a state that has this vector field as its instantaneous modular flow. We then show by explicit computation that no two distinct states in this class can be realized in the same local Hilbert space. Using a more abstract argument, we also show that in the vacuum sector of the theory, the only null-local modular flow in a causal diamond is provided by the vacuum state itself. We also comment on the construction of null-local modular flow for massive scalars, free Maxwell fields, free gravitons, and in curved backgrounds.
A bstract Modular flows probe important aspects of the entanglement structures, especially those of QFTs, in a dynamical framework. Despite the expected non-local nature in the general cases, the majority of explicitly understood examples feature local space-time trajectories under modular flows. In this work, we study a particular class of non-local modular flows. They are associated with the relativistic vacuum state and sub-regions whose boundaries lie on a planar null-surface. They satisfy a remarkable algebraic property known as the half-sided modular inclusion, and as a result the modular Hamiltonians are exactly known in terms of the stress tensor operators. To be explicit, we focus on the simplest QFT of a massive or massless free scalar in 1 + 2 dimensions. We obtain explicit expressions for the generators. They can be separated into a sum of local and non-local terms showing certain universal pattern. The preservation of von Neumann algebra under modular flow works in a subtle way for the non-local terms. We derive a differential-integral equation for the finite modular flow, which can be analyzed in perturbation theory of small distance deviating from the entanglement boundary, and re-summation can be performed in appropriate limits. Comparison with the general expectation of modular flows in such limits are discussed.
A bstract In two-dimensional conformal field theories (CFT) in Minkowski spacetime, we study the spacetime distance between two events along two distinct modular trajectories. When the spatial line is bipartite by a single interval, we consider both the ground state and the state at finite different temperatures for the left and right moving excitations. For the free massless Dirac field in the ground state, the bipartition of the line given by the union of two disjoint intervals is also investigated. The modular flows corresponding to connected subsystems preserve relativistic causality. Locality along the modular flows of some fields is explored by evaluating their (anti-)commutators. In particular, the bilocal nature of the modular Hamiltonian of two disjoint intervals for the massless Dirac field provide multiple trajectories leading to Dirac delta contributions in the (anti-)commutators even when the initial points belong to different intervals, thus being spacelike separated.
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.
We consider the algebras generated by observables in quantum field theory localized in regions in the null plane. For a scalar free field theory, we show that the one-particle structure can be decomposed into a continuous direct integral of lightlike fibres and the modular operator decomposes accordingly. This implies that a certain form of QNEC is valid in free fields involving the causal completions of half-spaces on the null plane (null cuts). We also compute the relative entropy of null cut algebras with respect to the vacuum and some coherent states.
We compute the vacuum local modular Hamiltonian associated with a space ball region in the free scalar massive Quantum Field Theory. We give an explicit expression on the one particle Hilbert space in terms of the massive Legendre differential operator and the Green integral for the Helmholtz operator. The quadratic form of the massive modular Hamiltonian is expressed in terms of an integral of the energy density with parabolic distribution and of a Yukawa potential, that here appears intrinsically. We then get the formula for the local entropy of a Klein-Gordon wave packet. This gives the vacuum relative entropy of a coherent state on the double cone von Neumann algebras associated with the free scalar QFT. Among other points, we have a criterion for a Bogoliubov automorphism to be locally weakly inner and we provide the passivity characterisation of the modular Hamiltonian within the standard subspace set up.
A bstract We prove a conjectured lower bound on 〈 T__ ( x )〉 ψ in any state ψ of a CFT on Minkowski space, dubbed the Quantum Null Energy Condition (QNEC). The bound is given by the second order shape deformation, in the null direction, of the geometric entanglement entropy of an entangling cut passing through x . Our proof involves a combination of the two independent methods that were used recently to prove the weaker Averaged Null Energy Condition (ANEC). In particular the properties of modular Hamiltonians under shape deformations for the state ψ play an important role, as do causality considerations. We study the two point function of a “probe” operator O \mathcal{O} O O in the state ψ and use a lightcone limit to evaluate this correlator. Instead of causality in time we consider causality on modular time for the modular evolved probe operators, which we constrain using Tomita-Takesaki theory as well as certain generalizations pertaining to the theory of modular inclusions. The QNEC follows from very similar considerations to the derivation of the chaos bound and the causality sum rule. We use a kind of defect Operator Product Expansion to apply the replica trick to these modular flow computations, and the displacement operator plays an important role. We argue that the proof extends to more general relativistic QFT with an interacting UV fixed point and also prove a higher spin version of the QNEC. Our approach was inspired by the AdS/CFT proof of the QNEC which follows from properties of the Ryu-Takayanagi (RT) surface near the boundary of AdS, combined with the requirement of entanglement wedge nesting. Our methods were, as such, designed as a precise probe of the RT surface close to the boundary of a putative gravitational/stringy dual of any QFT with an interacting UV fixed point.