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 $C^*$-algebra A is studied through its commutative unital $C^*$-subalgebras $C\subseteq A$, organized into a poset $\mathscr {C}(A)$. 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 $\mathscr {C}(A)$ 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 $\mathscr {C}(A)$ determines A as a Jordan algebra (at least for a large class of $C^*$-algebras). As a corollary, we prove a new Wigner-type theorem to the effect that order isomorphisms of $\mathscr {C}(B(H))$ are (anti) unitarily implemented. We also show how $\mathscr {C}(A)$ is related to the orthomodular poset ${\mathscr {P}}(A)$ of projections in A. These results indicate that $\mathscr {C}(A)$ is a serious player in $C^*$-algebras and quantum theory.