QUANTUM THEORY, HYPERBOLIC GEOMETRY AND RELATIVITY

Abstract

We construct a Lorentz-invariant quantum theory on velocity hyperboloids, with Schrödinger theory as its Euclidean analogue and with the property that the scattering operator is Poincaré invariant. This allows us to introduce the classical space-time concept for a macroscopic description of some properties of the microscopic scattering operator. This gives a completely novel approach to the relation between microscopic quantum theory and classical macroscopic sapce-time concepts. Hyperbolic integral geometry will be developed and used extensively in the construction of the theory, which might be called hyperbolic quantum theory. We stress that this is a radically new kind of relativistic quantum theory, where the term, relativistic, has a new meaning, dictated by the quantum theory. Special relativity was extracted from Maxwell's classical theory and should be properly adapted to quantum theory.