The role of type III factors in quantum field theory
Jakob YngvasonOne 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 III, 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.
One of the key subtleties about trying to study quantum information in a field theory is that you can’t formally decompose the Hilbert space into a tensor product of spatially local subsystems. The reasons are technical, and rarely explained well. This paper is an exception, giving an excellent introduction to the key ideas, in a manner accessible to a quantum (non-field) information theorist. (See related work by Yngvason this blogpost by Tobias Osborne and my previous discussion re: Reeh-Schielder theorem.)
A historically important but little known debate regarding the necessity and meaning of macroscopic superpositions, in particular those containing different gravitational fields, is reviewed and discussed from a modern perspective.
This paper is Dieter Zeh’s in-line commentary on what might be Feynman’s most explicit exposition of his interpretation of quantum mechanics:
As far as I know, Feynman never participated in the published debate about interpretational problems, such as quantum measurements. So I was surprised when I recently discovered a little known report about a conference regarding the role of gravity and the need for its quantization, held at the University of North Carolina
in Chapel Hill in 1957, since it led at some point to a discussion of the measurement problem and of the question about the existence and meaning of macroscopic superpositions. This session was dominated by Feynman’s presentation of a version of Schrodinger’s cat, in which the cat with its states of being dead or alive is replaced by a macroscopic massive ball being centered at two different positions with their distinguishable gravitational fields. I found this part of the report so remarkable for historical reasons that I am here quoting it in detail for the purpose of discussing and commenting it from a modern point of view….The discussion to be quoted below certainly deserves to become better known and discussed because of the influence it seems to have had on several later developments.
Towards a Formulation of Quantum Theory as a Causally Neutral Theory of Bayesian Inference
M. S. Leifer, R. W. SpekkensQuantum theory can be viewed as a generalization of classical probability theory, but the analogy as it has been developed so far is not complete. Whereas the manner in which inferences are made in classical probability theory is independent of the causal relation that holds between the conditioned variable and the conditioning variable, in the conventional quantum formalism, there is a significant difference between how one treats experiments involving two systems at a single time and those involving a single system at two times. In this article, we develop the formalism of quantum conditional states, which provides a unified description of these two sorts of experiment. In addition, concepts that are distinct in the conventional formalism become unified: channels, sets of states, and positive operator valued measures are all seen to be instances of conditional states; the action of a channel on a state, ensemble averaging, the Born rule, the composition of channels, and nonselective state-update rules are all seen to be instances of belief propagation. Using a quantum generalization of Bayes' theorem and the associated notion of Bayesian conditioning, we also show that the remote steering of quantum states can be described within our formalism as a mere updating of beliefs about one system given new information about another, and retrodictive inferences can be expressed using the same belief propagation rule as is used for predictive inferences. Finally, we show that previous arguments for interpreting the projection postulate as a quantum generalization of Bayesian conditioning are based on a misleading analogy and that it is best understood as a combination of belief propagation (corresponding to the nonselective state-update map) and conditioning on the measurement outcome.
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