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arXiv:2011.14622

Localized type I algebras and their entropy in quantum field theories

Heide Narnhofer

Published Nov 30, 2020• 0 citations
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Abstract

It is shown that under the assumption of the nuclearity condition for local regions the resulting Doplicher-Longo-algebra between two double cones which due to nuclearity is type I allows to estimate its entropy by nuclearity bounds.
DOI
10.48550/arxiv.2011.14622

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This essay was completed as partial fulfillment of the requirements forthe MASt in Applied Mathematics degree (Part III of the Mathemat-ical Tripos) carried out at the DAMPT, Cambridge. After reviewingthe essentials of C*-algebras and von Neumann algebras, we present theMurray–von Neumann classification of factors into types I, II and III.We then introduce the algebraic approach to Quantum Field Theory(QFT) and discuss existing arguments for the association of type IIIalgebras to local regions of spacetime. We explain why, as a conse-quence, the entanglement entropy of a region is not well defined, andhow relative entropy offers an alternative, reviewing the fundamentalsof entanglement entropy in QFT. Finally, we see how including grav-ity turns the algebra of observables outside a black hole into one oftype II∞, where a well-defined notion of entropy can be recovered. This work should be regarded as introductory notes to operator al-gebras with a particular emphasis on their applications in theoreticalphysics. They are intended to synthesize the mathematical content fora physics-oriented audience, using worked-out examples throughout toset a foundation and gradually build up to some direct examples of theusage of the framework. They are aimed at advanced undergraduatestudents or early postgraduate students, and only some basic knowl-edge on quantum mechanics, QFT, group theory and linear algebra areassumed
We consider the problem of defining the entanglement entropy for chiral nets in the framework of algebraic quantum field theory.Considering a Möbius covariant local net with the split property, we give a sensible definition for the entropy H E I of a state restricted to a local algebra of an open connected non-dense interval I ⊂ S 1 , with a given "conformal energy cutoff E".Considering the vacuum state restricted to any such interval, we prove that the latter is finite, and we give some upper bound estimates in terms of the dimensions of eigenspaces of the conformal Hamiltonian.This thesis is based on a joint work with Yoh Tanimoto.
In the framework of Algebraic Quantum Field Theory, several operator algebraic notions of entanglement entropy can be associated to a couple of causally disjoint and distant spacetime regions SA\mathcal{S}_ASA​ and SB\mathcal{S}_BSB​. In this work, we show that the mutual information is finite in any local QFT verifying a modular ppp-nuclearity condition for some 0<p<10 < p <10<p<1. A similar result is proved for another recently studied entanglement measure. Furthermore, if we assume conformal covariance then by comparison with other entanglement measures we can state that the mutual information satisfies lower bounds of area law type when the distance between SA\mathcal{S}_ASA​ and SB\mathcal{S}_BSB​ approaches to zero. As an application, in 1+11+11+1-dimensional integrable models with factorizing S-matrices, we study the asymptotic behaviour of different entanglement measures as the distance between two causally disjoint wedges diverges.