Why Be Normal?

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A normal state on a von Neumann algebra defines a countably additive probability measure over its projection lattice. The von Neumann algebras familiar from ordinary QM are algebras of all the bounded operators on a Hilbert space H, aka Type I factor von Neumann algebras. Their normal states are density operator states, and can be pure or mixed. In QFT and the thermodynamic limit of QSM, von Neumann algebras of more exotic types abound. Type III von Neumann algebras, for instance, have no pure normal states; the pure states they do have fail to be countably additive. I will catalog a number of temptations to accord physical significance to non-normal states, and then give some reasons to resist these temptations: (1) pure though they may be, non-normal states on non-Type I factor von Neumann algebras can't do the interpretive work we've come to expect from pure states on Type I factors; (2) our best accounts of state preparation don't work for the preparation of non-normal states; (3) there is a sense in which non-normal states fail to instantiate the laws of quantum mechanics.


A presentation at the Philosophy of Quantum Field Theory Workshop, which was held at The University of Western Ontario on April 24-26, 2009

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