On nonlocality, lattices and internal symmetries

Department of Applied Mathematics & Theoretical Physics, University of Cambridge, Cambridge CB3 9EW, UK.

Received: 14 July 1997
Accepted: 12 September 1997

Abstract

We study functional analytic aspects of two types of correction terms to the Heisenberg algebra. One type of correction terms is known to induce a finite lower bound to the resolution of distances, a short-distance cut-off which is motivated from string theory and quantum gravity. It implies the existence of families of lattices of position eigenvalues which form representations of certain unitary groups. Due to the finite these lattices cannot be resolved on the given geometry. Within the framework, degrees of freedom that correspond to structure smaller than the resolvable (Planck) scale then turn into “internal” degrees of freedom with these unitary groups as symmetries. The other type of correction terms differs by a crucial sign and its basics are here considered for the first time.