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

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. 1 The instability of relativistic quantum mechanics and a fundamental length Physical models and theories are mere approximations to Nature and the physical constants can never be known with absolute precision. Therefore, if a fine tuning of the parameters is needed to reproduce some particular phenomenon, it is probable that the model is basically unsound and that its other predictions are unreliable. A wider range of validity is expected for theories that do not change in a qualitative manner for a small change of parameters. Such theories are called stable or rigid. A mathematical structure is said to be stable (or rigid) for a class of deformations if any deformation in this class leads to an equivalent (isomorphic) structure. The idea of stability of structures provides a guiding principle to test either the validity or the need for generalization of a physical theory. Namely, if the mathematical structure of a given theory turns out to be unstable, one might just as well deform it, until one falls into a stable one, which has a good chance of being a generalization of wider validity.