Modular Hamiltonians on the null plane and the Markov property of the vacuum state

By H. Casini, Eduardo Testé, G. Torroba. 2017

151 citations recorded by the source corpus.

We compute the modular Hamiltonians of regions having the future horizon lying on a null plane. For a CFT this is equivalent to regions with a boundary of arbitrary shape lying on the null cone. These Hamiltonians have a local expression on the horizon formed by integrals of the stress tensor. We prove this result in two different ways, and show that the modular Hamiltonians of these regions form an infinite dimensional Lie algebra. The corresponding group of unitary transformations moves the fields on the null surface locally along the null generators with arbitrary null line dependent velocities, but act non-locally outside the null plane. We regain this result in greater generality using more abstract tools on the algebraic quantum field theory. Finally, we show that modular Hamiltonians on the null surface satisfy a Markov property that leads to the saturation of the strong sub-additive inequality for the entropies and to the strong super-additivity of the relative entropy.

Subfields: Applied Mathematics, Astronomy and Astrophysics, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/da89aece4bee6e237230d937c06db8c4ef7ab926

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Citations in the corpus

Light-ray operators in conformal field theory (2018, background): “The ANEC has a growing list of interesting applications in conformal field theory [41, 43, 44, 99–102].”
Entanglement entropy: Holography and renormalization group (2018, background): “…2016; Karthik and Narayanan, 2016a,b).1 The same sort of argument with the quantum inequality of entanglement has been extended more recently (Casini et al., 2017a; Lashkari, 2017), and yielded monotonic functions along RG flows in higher dimensions, which provides an alternative proof of…”
A general proof of the quantum null energy condition (2017, methodology): “ are not local in the sense that they do not generate local ows. With the result (2.12) in hand one can then just go and calculate the algebra when Aand Bare both deformed null cuts (see AppendixAand [34]): K0fX Bg;K 0fX Ag = i(K0fX Ag K 0fX Bg) = iP fX B X Ag (2.15) such that: e 0iK fX B gseiK 0fX ”