Symmetries in Exact Bohrification

Abstract

The ‘Bohrification” program in the foundations of quantum mechanics implements Bohr’s doctrine of classical concepts through an interplay between commutative and non-commutative operator algebras. Following a brief conceptual and mathematical review of this program, we focus on one half of it, called “exact” Bohrification, where a (typically noncommutative) unital -algebra A is studied through its commutative unital -subalgebras organized into a poset This poset turns out to be a rich invariant of A (Hamhalter in J Math Anal Appl 383:391–399, 2011, [19], Hamhalter in J Math Anal Appl 422:1103-1115, 2015, [20], Landsman in Bohrification: From classical concepts to commutative algebras. Chicago, Chicago University Press [34]). To set the stage, we first give a general review of symmetries in elementary quantum mechanics (i.e., on Hilbert space) as well as in algebraic quantum theory, incorporating as a new kid in town. We then give a detailed proof of a deep result due to Hamhalter (J Math Anal Appl 383:391–399, 2011, [19]), according to which determines A as a Jordan algebra (at least for a large class of -algebras). As a corollary, we prove a new Wigner-type theorem to the effect that order isomorphisms of are (anti) unitarily implemented. We also show how is related to the orthomodular poset of projections in A. These results indicate that is a serious player in -algebras and quantum theory.