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International Conference on Mathematical Methods in Physics (ICMP)

Algebraic QFT and the area law for entropy of localized quantum matter

Bert Schroer

Published Apr 11, 2007• 0 citations
Publisher

Abstract

The algebraic approach to QFT, which for several decades has enriched QFT with structural theorems, has recently shown its utility in various constru ctions of actual interest. After some introductory remarkss which illustrate this point, the mai n content of the paper addresses the issue of algebraic holography. The example of lightfront holography is presented in some detail, and its use in studying thermal aspects of localization is ex plained. The vacuum polarization at the edge of the causal horizon of a wedge-localized algebra is identified as the cause of an area proportionality of localization-entropy of localized qua ntum matter. Finally the adaptation of the method to the analogous case of black holes with bifurcated Killing horizons is sketched.

DOI
10.22323/1.031.0027

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The algebraic approach to QFT, which for several decades has enriched QFT with structural theorems, has recently shown its utility in various constru ctions of actual interest. After some introductory remarkss which illustrate this point, the mai n content of the paper addresses the issue of algebraic holography. The example of lightfront holography is presented in some detail, and its use in studying thermal aspects of localization is ex plained. The vacuum polarization at the edge of the causal horizon of a wedge-localized algebra is identified as the cause of an area proportionality of localization-entropy of localized qua ntum matter. Finally the adaptation of the method to the analogous case of black holes with bifurcated Killing horizons is sketched.
The previously developed algebraic lightfront holography is used in conjunction with the tensor splitting of the chiral theory on the causal horizon. In this way a universal area law for the entanglement entropy of the vacuum relative to the split (tensor factorized) vacuum is obtained. The universality of the area law is a result of the kinematical structure of the properly defined lightfront degrees of freedom. We consider this entropy associated with causal horizon of the wedge algebra in Minkowski spacetime as an analog of the quantum Bekenstein black hole entropy similar to the way in which the Unruh temperature for the wedge algebra may be viewed as an analog in Minkowski spacetime of the Hawking thermal behavior. My more recent preprint hep-th/0202085 presents other aspects of the same problem.
The algebraic approach to QFT, which for several decades has enriched QFT with structural theorems, has recently shown its utility in various constructions of actual interest. In these lecture notes I explain how AQFT (in particular the modular theory of operator algebras) implies paradigmatic conceptual and mathematical changes while fully preserving the physical principles which underly QFT. As an illustration of actual interest I use holography on null-surfaces and the ensuing area law for entropy of localized matter in the vacuum state.
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
After a short presentation of KMS states and modular theory as the unifying description of thermalizing systems we propose the absence of transverse vacuum fluctuations in the holographic projections as the principle reason for an area behavior (the transverse area) of localization entropy as opposed to the volume dependence of ordinary heat bath entropy. Thermalization through causal localization is not a property of QM but results from the omnipresent vacuum polarization in QFT and does not require a Gibbs type ensemble avaraging (coupling to a heat bath).
Using the conformal equivalence of translational KMS states on chiral theories with dilational KMS states obtained from restricting the vacuum state to an interval (the chiral inversion of the Unruh-effect) it was shown in a previous publications that the diverging volume (length) factor of the thermodynamic limit corresponds to the logarithmic increase in the attenuation length of the localization-caused vacuum polarization cloud near the causal boundary. This is not a coincidence but rather a structural consequence of the fact that both operator algebras are of the same unique von Neumann type which is completely different from that met in quantum mechanical algebras. Together with the technique of holographic projection this leads to the universal area proportionality. The main aim in this paper is to present a derivation which is more in the spirit of recent work on entanglement entropy in condensed matter physics, especially to that of the replica trick as used by Cardy and collaborators. The essential new ingredient is the use of the split property which already has shown its constructive power in securing the existence of models of factorizing theories.
Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: the entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such ``area laws'' for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium the current status of area laws in these fields is reviewed. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation in quantum lattice models, and disordered systems, nonequilibrium situations, and topological entanglement entropies are discussed. These questions are considered in classical and quantum systems, in their ground and thermal states, for a variety of correlation measures. A significant proportion is devoted to the clear and quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. Matrix-product states, higher-dimensional analogs, and variational sets from entanglement renormalization are also discussed and the paper is concluded by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations of quantum states.