Abstract
Quantum Theories can be formulated in real, complex or quaternionic Hilbert spaces as established in Soler's theorem. Quantum states are here pictured in terms of sigma-additive probability measures over the non-Boolean lattice of orthogonal projectors of the considered Hilbert space. Gleason's theorem proves that, if the Hilbert space is either real or complex and some technical hypotheses are ...
Abstract
Quantum Theories can be formulated in real, complex or quaternionic Hilbert spaces as established in Soler's theorem. Quantum states are here pictured in terms of sigma-additive probability measures over the non-Boolean lattice of orthogonal projectors of the considered Hilbert space. Gleason's theorem proves that, if the Hilbert space is either real or complex and some technical hypotheses are true, then these measures are one-to-one with standard density matrices used by physicists recovering and motivating the familiar notion of state. The extension of this result to quaternionic Hilbert spaces was obtained by Varadarajan in 1968. Unfortunately, the formulation of this extension (Varadarajan in Geometry of quantum theory, Van Nostrand Reinhold Inc., Washington, 1968) is partially mathematically incorrect due to some peculiarities of the notion of trace in quaternionic Hilbert spaces. A minor issue also affects Varadarajan's statement for real Hilbert space formulation. This paper is devoted to present Gleason-Varadarajan's theorem into a technically correct and physically meaningful form valid for the three types of Hilbert spaces. In particular, we prove that only the real part of the trace enters the formalism of Quantum Theories (also dealing with unbounded observables and symmetries) and it can be safely used to formulate and prove a common statement of Gleason's theorem.
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