Abstract
The Copenhagen Interpretation describes individual systems, using the same Hilbert space formalism as does the statistical ensemble interpretation (SQM). This leads to the well-known paradoxes surrounding the Measurement Problem. We extend this common mathematical structure to encompass certain natural bundles with connections over the Hilbert sphere S. This permits a consistent extension of the statistical interpretation to interacting individual systems, thereby resolving these paradoxes.
Suppose V is a physical system in interaction with another system W. The state vector of V+W has a set of polar decompositions with a vector q of complex coefficients. These are parameterized by the right toroid T of amplitudes q, and comprise a singular toroidal bundle over S, which comprises the enlarged state space of V+W. We prove that each T has a unique natural convex partition yielding the correct SQM probabilities. In the extended theory V and W synchronously assume pure spectral states according to which member of the partition contains q. The apparent indeterminism of SQM is thus attributable to the effectively random distribution of initial phases.