Gleason Theorem

Abstract

In the Hilbert space model of quantum mechanics, the states of a physical system are typically identified with probability measures on the lattice P(H) of all orthogonal projections in a Hilbert space, H, which embodies the “logic” of the quantum system. A natural example of a probability measure on projections is the restriction of a normed positive linear functional on the von Neumann algebra B(H) of all bounded operators acting on H. Whether or not all probability measures are of this very form was one of the basic questions of mathematical foundations of quantum mechanics. This question has led to many deep mathematical results known today as Gleason type theorems. The aim of this chapter is to provide a proof of Gleason Theorem on linear extension of bounded completely additive measure on a Hilbert space projection lattice and its ramifications in the light of recent results of quantum measure theory. The contents is organized as follows. In the first section we show how one can reduce the problem to the real three-dimensional Hilbert space. In the second step we prove Gleason Theorem for the real three-dimensional Hilbert space by using the concept of the frame function. Finally, in the last section we show that the boundedness of measure can be relaxed in Gleason Theorem whenever the corresponding Hilbert space has infinite dimension, which is a rather deep result by Dorofeev and Sherstnev. We conclude with historical remarks and some links with recent investigations.