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Quantum states as measures on the spectral presheaf

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States of a physical system—classical or quantum—can be seen as normed positive linear functionals (i.e., integrals) on the algebra of physical quantities. In classical physics, where the algebra is abelian, states areequivalently described by measures on the state space. A quantum system has a nonabelian algebra of physical quantities, and there is no obvious notion of a state space. We will show how quantum states can also be understood as measures, but now defined on a certain presheaf, called the spectral presheaf, which takes the role of the state space of the quantum system. The spectral presheaf is a central object in the topos approach to quantum theory. It will be shown how the spectral presheaf is related to, but different from the Gel’fand spectrum of a certain topos-internal operator algebra.

This talk is part of the Category Theory Seminar series.

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