This paper provides an explanation of the type classification of von Neumann algebras, which has made many appearances in the recent work on entanglement in quantum field theory and quantum gravity. The goal is to bridge a gap in the literature between resources that are too technical for the non-expert reader, and resources that seek to explain the broad intuition of the theory without giving precise definitions. Reading this paper will provide you with: (i) an argument for why “factors” are the fundamental von Neumann algebras that one needs to study; (ii) an intuitive explanation of the type classification of factors in terms of renormalization schemes that turn unnormalizable positive operators into “effective density matrices”; (iii) a mathematical explanation of the different types of renormalization schemes in terms of the allowed traces on a factor; (iv) an intuitive characterization of type I and type II factors in terms of their “standard forms”; and (v) a list of some interesting connections between type classification and modular theory, including the argument for why type III 1 factors are believed to be the relevant ones in quantum field theory. None of the material is new, but the pedagogy is different from other sources the author has read; it is most similar in spirit to the recent work on gravity and the crossed product by Chandrasekaran, Longo, Penington, and Witten.
This paper provides an explanation of the type classification of von Neumann algebras, which has made many appearances in the recent work on entanglement in quantum field theory and quantum gravity. The goal is to bridge a gap in the literature between resources that are too technical for the non-expert reader, and resources that seek to explain the broad intuition of the theory without giving precise definitions. Reading this paper will provide you with: (i) an argument for why “factors” are the fundamental von Neumann algebras that one needs to study; (ii) an intuitive explanation of the type classification of factors in terms of renormalization schemes that turn unnormalizable positive operators into “effective density matrices”; (iii) a mathematical explanation of the different types of renormalization schemes in terms of the allowed traces on a factor; (iv) an intuitive characterization of type I and type II factors in terms of their “standard forms”; and (v) a list of some interesting connections between type classification and modular theory, including the argument for why type III 1 factors are believed to be the relevant ones in quantum field theory. None of the material is new, but the pedagogy is different from other sources the author has read; it is most similar in spirit to the recent work on gravity and the crossed product by Chandrasekaran, Longo, Penington, and Witten.
An early result of algebraic quantum field theory is that the algebra of any subregion in a QFT is a von Neumann factor of type III _{1} 1 , in which entropy cannot be well-defined because such algebras do not admit a trace or density states. However, associated to the algebra is a modular group of automorphisms characterizing the local dynamics of degrees of freedom in the region, and the crossed product of the algebra with its modular group yields a type II _∞ ∞ factor, in which traces and hence von Neumann entropy can be well-defined. In this work, we generalize recent constructions of the crossed product algebra for the TFD to, in principle, arbitrary spacetime regions in arbitrary QFTs, formally paving the way to the study of entanglement entropy without UV divergences. In contrast to previous works, we emphasize that this construction is independent of gravity. In this sense, the crossed product construction represents a refinement of Haag’s assignment of nets of observable algebras to spacetime regions by providing a natural construction of a type II factor. We present several concrete examples: a QFT in Rindler space, a CFT in an open ball of Minkowski space, and arbitrary boundary subregions in AdS/CFT. In the holographic setting, we provide a novel argument for why the bulk dual must be the entanglement wedge, and discuss the distinction arising from boundary modular flow between causal and entanglement wedges for excited states and disjoint regions.
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
One of von Neumann’s motivations for developing the theory of operator algebras and his and Murray’s 1936 classification of factors was the question of possible decompositions of quantum systems into independent parts. For quantum systems with a finite number of degrees of freedom the simplest possibility, i.e., factors of type I in the terminology of Murray and von Neumann, are perfectly adequate. In relativistic quantum field theory (RQFT), on the other hand, factors of type III occur naturally. The same holds true in quantum statistical mechanics of infinite systems. In this brief review some physical consequences of the type III property of the von Neumann algebras corresponding to localized observables in RQFT and their difference from the type I case will be discussed. The cumulative effort of many people over more than 30 years has established a remarkable uniqueness result: The local algebras in RQFT are generically isomorphic to the unique, hyperfinite type III1 factor in Connes ’ classification of 1973. Specific theories are characterized by the net structure of the collection of these isomorphic algebras for different space-time regions, i.e., the way they are embedded into each other. John von Neumann was the father of the Hilbert space formulation of quantum mechanics [1] that has been the basis of almost all mathematically rigorous investigations of the theory to this day. We start by recalling the main concepts and explaining some notations.
In quantum systems with infinitely many degrees of freedom, states can be infinitely entangled across a pair of subsystems. But are there different forms of infinite entanglement? To answer this, we provide a rigorous framework for studying information-theoretic properties in infinite systems, formulated in terms of von Neumann algebras. In this framework, we show that there are operational tasks that can distinguish between different forms of infinite entanglement. We find that the type classification of von Neumann algebras (types I, II, III, and their respective subtypes) is in one-to-one correspondence with operational entanglement properties. For instance, the Connes classification of type III factors corresponds to the system’s capability at the operational task of “embezzling” entanglement. Our findings promote the type classification from algebraic bookkeeping to a classification of infinite quantum systems based on their operational entanglement properties.
In an algebraic understanding of holographic gravity, a major question is: how can a boundary theory with Type-III von Neumann algebras of local observables (where density operators and entropy are ill-defined) be equivalent to a bulk theory with Type-I von Neumann algebras of local observables (where density operators and entropy are well-defined)? Replica wormholes and problems like the factorization problem suggest that instead of full equivalence, there is an approximation. When approximate equality breaks down, corrections to bulk theory predictions are required, just as in replica wormholes. Quantum measurements are then described as operations that can only be described as identity operations in the bulk theory but are non-trivial in the boundary theory. The breakdown of approximate equality between the boundary theory and the bulk theory generates illusions of wavefunction collapse, when there actually is no wavefunction collapse in the boundary theory with Type-III von Neumann algebras.
This short document presents no new results. It is a collection of related ideas that may lead to a way of quantizing gravity. To achieve this it may be relevant to use my recent new approach to quantum theory, which is based on quantum events. Given that most standard quantum texts do not define a mathematical object which corresponds to a physical event, the definition of a quantum event is motivated and discussed in detail. This in turn leads to the idea that the mathematical object that contains enough quantum events for describing a quantum system is a von Neumann algebra. The next idea is that the kinematics of special relativity (as embodied in flat Minkowski space) should be quantized as a first step in the program of the quantization of general relativity, that is, quantizing Einstein's equation. It must be emphasized that the events of special relativity are viewed as part of the kinematics, rather than of the dynamics of that theory. The same holds for general relativity. Similarly, I am viewing quantum events as part of the kinematics of quantum theory, even though some find this to be counterintuitive. An Appendix shows how ``collapse'' is easily understood in this new approach to quantum theory.