The Logic of Bundles

Abstract

Since the work of Crown (J. Natur. Sci. Math. 15(1–2), 11–25 1975) in the 1970’s, it has been known that the projections of a finite-dimensional vector bundle E form an orthomodular poset (omp) $\mathcal {P}(E)$. This result lies in the intersection of a number of current topics, including the categorical quantum mechanics of Abramsky and Coecke (2004), and the approach via decompositions of Harding (Trans. Amer. Math. Soc. 348(5), 1839–1862 1996). Moreover, it provides a source of omps for the quantum logic program close to the Hilbert space setting, and admitting a version of tensor products, yet having important differences from the standard logics of Hilbert spaces. It is our purpose here to initiate a basic investigation of the quantum logic program in the vector bundle setting. This includes observations on the structure of the omps obtained as $\mathcal {P}(E)$ for a vector bundle E, methods to obtain states on these omps, and automorphisms of these omps. Key theorems of quantum logic in the Hilbert setting, such as Gleason’s theorem and Wigner’s theorem, provide natural and quite challenging problems in the vector bundle setting.