Geometry, stochastic calculus and quantum fields in a non-commutative space-time

Abstract

The algebras of non-relativistic and of classical mechanics are unstable algebraic structures. Their deformation towards stable structures leads, respectively, to relativity and to quantum mechanics. Likewise, the combined relativistic quantum mechanics algebra is also unstable. Its stabilization requires the non-commutativity of the space-time coordinates and the existence of a fundamental length constant. The new relativistic quantum mechanics algebra has important consequences on the geometry of space-time, on quantum stochastic calculus and on the construction of quantum fields. Some of these effects are studied in this paper.