Orthomorphisms of Projections

Abstract

The standard formalism of quantum mechanical system consists of three basic components: the algebra of observables, the set of states on this algebra (which describes the probability structure in question), and the group of automorphisms of the algebra (which expresses the time development of the system). It is the ambition of the logico-algebraic approach to quantum mechanics, as it was articulated by Mackey [224], to recover all these aspects from the structure of propositions, i.e. from the quantum logic of Jro’ ections. The observables can be reconstructed from projections by making use of the spectral resolution. As seen before, linear states are in a one-to-one correspondence with finitely-additive measures on projections (the Generalized Gleason Theorem discussed in Chapter 5). The aim of the present chapter is to complete this program by showing that also the dynamics of the system is uniquely determined by the automorphisms and homomorphisms of the projection lattices. In the following Section 8.1 we shall apply the Generalized Gleason Theorem to prove that orthomorphisms between projection lattices of von Neumann algebras extend uniquely to Jordan homomorphisms. The Dye Theorem on automorphisms of projection lattices as well as the celebrated Wigner Unitary-Antiunitary Theorem will then be derived as special consequences of this result.