| dc.description.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. | sv |
