Quantum mechanics is formulated as a geometric theory on a Hilbert manifold. Images of charts on the manifold are allowed to belong to arbitrary Hilbert spaces of functions including spaces of generalized functions. Tensor equations in this setting, also called functional tensor equations, describe families of functional equations on various Hilbert spaces of functions. The principle of functional relativity is introduced which states that quantum theory (QT) is indeed a functional tensor theory, i.e., it can be described by functional tensor (...) equations. The main equations of QT are shown to be compatible with the principle of functional relativity. By accepting the principle as a hypothesis, we then explain the origin of physical dimensions, provide a geometric interpretation of Planck’s constant, and find a simple model of the two-slit experiment and the process of measurement. (shrink)

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)

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 ...