Gleasons sats

Detta är en Kandidat-uppsats från Göteborgs universitet/Institutionen för matematiska vetenskaper

Sammanfattning: This paper aims to present Gleason’s theorem and a full proof, by the most elementarymethods of analysis possible. Gleason’s theorem is an important theorem in the mathematicalfoundations of quantum mechanics. It characterizes measures on closed subspacesof separable Hilbert spaces of dimension at least 3. The theorem can be formulated interms of so-called frame functions. It states that all bounded frame functions, on thespecified Hilbert spaces, must have the form hAx, xi, for some self-adjoint operator A.The theorem is proved by first proving the statement in R3, through mostly geometricarguments on the unit sphere, and methods relating to convergence of sequences. It isthen shown that this implies the theorem in general Hilbert spaces of higher dimension.The bulk of our proof follows the ideas of Cooke, Keane and Moran [2] with some ownadditions and clarifications in order to make it more accessible and correct. A lemma ofsingle-variable analysis has been expanded, an oversight in the proof of the geometriclemma 5 (Piron) has been fixed and an erroneous topological argument has led to themuch rewritten proposition 2 about extremal values of frame functions. The motivationfor the sufficiency of the proof in R3 for higher-dimensional Hilbert spaces follows theideas of the original proof by Andrew M. Gleason.

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