Covariance and Quantum Logic

Considering the fundamental role symmetry plays throughout physics, it is remarkable how little attention has been paid to it in the quantum-logical literature. In this paper, we discuss G-test spaces—that is, test spaces hosting an action by a group G—and their logics. The focus is on G-test spaces having strong homogeneity properties. After establishing some general results and exhibiting various specimens (some of them exotic), we show that a sufficiently symmetric G-test space having an invariant, separating set of states with affine dimension n, is always representable in terms of a real Hilbert space of dimension n+1, in such a way that orthogonal outcomes are represented by orthogonal unit vectors.