Gerbes, uncertainty and quantization

Abstract

The explanation of the photoelectric effect by Einstein and Maxwell's field theory of electromagnetism have motivated De Broglie to make the hypothesis that matter exhibits both waves and particles like-properties. These representations of matter are enlightened by string theory which represents particles with stringlike entities. Mathematically, string theory can be formulated with a gauge theory on loop spaces which is equivalent to the differential geometry of gerbes. In this paper, we show that the descent theory of Giraud and Grothendieck can enable to describe the wave-like properties of the matter with the quantization of a theory of particles. The keypoint is to use the fact that the state space is defined by a projective bundle over the parametrizing manifold which induces a gerbe which represents the geometry obstruction to lift this bundle to a vector bundle. This is equivalent to saying that a phase is determined up to a complex number of module $1$. When this uncertainty occurs, we cannot locate precisely the position of a particle, and the smallest dimensional quantity that can be described is a $1$ dimensional manifold; this leads to the concept of wave properties of the matter and string theory.