Infinite and finite Gleason's theorems and the logic of indeterminacy

Abstract

In the first half of the paper I prove Gleason's lemma: Every non-negative frame function on the set of rays in ℛ3 is continuous. This is the central and most difficult part of Gleason's theorem. The proof is a reconstruction of Gleason's idea in terms of orthogonality graphs. The result is a demonstration that this theorem is actually combinatorial in nature. It depends only on a finite graph structure. In the second half of the paper I use the graph construction to obtain results about probability distributions (non-negative frame functions with weight one) on finite sets of rays. For example, given any two distinct nonorthogonal rays a and b, I construct a finite set of rays Γ that contains them, and has the following property: No probability distribution on Γ assigns both a and b a truth value (probability zero or one) unless they are both false. Thus the principle of indeterminacy turns into a theorem of propositional quantum logic (or partial Boolean algebras). © 1998 American Institute of Physics.