Gleasons sats

Abstract

This paper aims to present Gleason’s theorem and a full proof, by the most elementary methods of analysis possible. Gleason’s theorem is an important theorem in the mathematical foundations of quantum mechanics. It characterizes measures on closed subspaces of separable Hilbert spaces of dimension at least 3. The theorem can be formulated in terms of so-called frame functions. It states that all bounded frame functions, on the specified 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 geometric arguments on the unit sphere, and methods relating to convergence of sequences. It is then 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 own additions and clarifications in order to make it more accessible and correct. A lemma of single-variable analysis has been expanded, an oversight in the proof of the geometric lemma 5 (Piron) has been fixed and an erroneous topological argument has led to the much rewritten proposition 2 about extremal values of frame functions. The motivation for the sufficiency of the proof in R3 for higher-dimensional Hilbert spaces follows the ideas of the original proof by Andrew M. Gleason.