The Correct Formulation of Gleason’s Theorem in Quaternionic Hilbert Spaces

Abstract

Quantum Theories can be formulated in real, complex or quaternionic Hilbert spaces as established in Solér's theorem. Quantum states are here pictured in terms of σ-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 hypothes 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 [Va68] 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.

Quantum theory in quaternionic Hilbert space: How Poincaré symmetry reduces the theory to the standard complex one

“…where H is a real, complex or quaternionic Hilbert space. The set of trace-class operators turns out to be a closed two-sided * -ideal of B(H) (the unital real C * -algebra of bounded operators A : H → H) in the three considered cases [24]. This correspondence between µ and T µ exists for the three cases as demonstrated by the celebrated Gleason's theorem valid for R and C [10].…”

“…Above P T µ P is explicitely selfadjoint and thecyclic property of the trace together with P P = P proves that tr(P T µ P ) = tr(P T µ ) in the complex and real cases, finding the standard statement of Gleason theorem in those cases. Cyclicity of the trace does not hold in the quanternionic case [24]. An alternative, equivalent, and much more effective approach [24] is to state Gleason's identity in the three cases as…”

“…Cyclicity of the trace does not hold in the quanternionic case [24]. An alternative, equivalent, and much more effective approach [24] is to state Gleason's identity in the three cases as…”

“…In fact, it turns out that the real part of the trace is basis independent in quaternionic Hilbert space (and also in the remaining two cases). Finally (1) is equivalent to (2) because Re (tr(P T µ )) = tr(P T µ P ) by elementary properties of the trace operation in the three considered cases (see [24] for details). Gleason's result is valid (but the correspondence of measures and positive unit-trace thrace-class operators ceases to be one-to-one) for separable complex Hilbert spaces when L 1 (H) L(H) and L 1 (H) is the projector lattice of a von Neumann algebra whose canonical decomposition into definite-type von Neumann algebras does not contain type-I 2 algebras [6].…”

An operational construction of the sum of two non-commuting observables in quantum theory and related constructions

“…Several rigorous results exist in the literature about this subject. However, to the best of our knowledge, almost all are concentrated on two physically well-motivated approaches: the generalization of Feynman's functional calculus for T -ordered products of operators and extensions of Weyl calculus-see [13,15,16,28] for exhaustive reviews on these approaches-with several original contributions from outstanding authors [3,23,26]. The mathematical technology, the types of functional spaces and the interactions with some other mathematical objects like the Feynman-Kac integral share some similarities with the content of in this work-c.f.…”