The paper summarizes, generalizes and reveals the physical content of a recently proposed framework that unifies the standard formalisms of special relativity and quantum mechanics. The framework is based on Hilbert spaces H of functions of four space-time variables x,t, furnished with an additional indefinite inner product invariant under Poincaré transformations. The indefinite metric is responsible for breaking the symmetry between space and time variables and for selecting a family of Hilbert subspaces that are preserved under Galileo transformations. Within these (...) subspaces the usual quantum mechanics with Shrödinger evolution and t as the evolution parameter is derived. Simultaneously, the Minkowski space-time is embedded into H as a set of point-localized states, Poincaré transformations obtain unique extensions to H and the embedding commutes with Poincaré transformations. Furthermore, the framework accommodates arbitrary pseudo-Riemannian space-times furnished with the action of the diffeomorphism group. (shrink)

An appropriate kind of curved Hilbert space is developed in such a manner that it admits operators of $\mathcal{C}$ - and $\mathfrak{D}$ -differentiation, which are the analogues of the familiar covariant and D-differentiation available in a manifold. These tools are then employed to shed light on the space-time structure of Quantum Mechanics, from the points of view of the Feynman ‘path integral’ and of canonical quantisation. (The latter contains, as a special case, quantisation in arbitrary curvilinear coordinates when space is (...) flat.) The influence of curvature is emphasised throughout, with an illustration provided by the Aharonov-Bohm effect. (shrink)

To clarify and illuminate the place of probability in science Ellery Eells and James H. Fetzer have brought together some of the most distinguished philosophers ...