Department Colloquium

Department Colloquium

Department of Mathematics
Princeton University

April 26st, 2000

On the Quantum Mechanics of Individual Systems

Taking standard quantum mechanics (SQM) as a statistical theory, we extend the standard Hilbert space formulation to a mathematical model of the individuals which comprise the statistical ensembles of SQM. The model of two interacting systems is a singular toroidal bundle over the unit sphere in the Hilbert space of the composite system, together with a natural connection which permits the Schrodinger evolution in the sphere to be lifted to the bundle.
The main mathematical innovation required is the construction of convex periodic tilings of Euclidian spaces (which is new even in 3 dimensions). These tilings descend to partitions of the toroidal fibers. The states of the subsystems are determined by which tile contains the lifted evolution. The toroidal tilings are the unique functorial convex partitions consistent with SQM. This is joint work with Simon Kochen.