Subfactors, bimodules, and quantum mechanics

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The project involves continued work on subfactor and planar algebra theory and a new investigation of the relation between the Connes tensor product of bimodules over von Neumann algebras and very strongly intertwined quantum systems. The planar operad can be used to axiomatise a large class of hyperfinite subfactors and we intend to exploit this new point of view to better understand existing examples and discover new ones, as well as exploring planar algebras beyond the positivity condition required for subfactors. We say that two quantum systems are very strongly intertwined if there is an algebra of "common observables" which means that certain of one system automatically yield measurement of the other system. We would then expect the Hilbert space for the joint system to be the Connes tensor product of the individual Hilbert space, taken over the von Neumann algebra of common observables. We shall look for such systems and see if this kind of intertwining has observable consequences.The project is a continuing investigation of the mathematical structure of quantum mechanics-the study of the universe on a very small scale. The states of a system ("wave functions") are defined by a Hilbert space and operators on that Hilbert Space represent measurements. A von Neumann algebra is a collection of operators with certain physically relevant closure properties. "Factors" are von Neumann algebras with no operators commuting with all others in the algebra. The algebra of all observables localized in a region of space-time is a factor. Subfactors occur in interesting ways when considering the causal geometry of space-time. The project focuses on subfactors and a related way of combining two quantum systems called the Connes tensor product which is capable of identifying a von Neumann algebra of observables on one system with such an algebra on the other. There are potential applications of these ideas to quantum computing, especially through the approach of Michael Freedman.