In this chapter we shall consider the idea of Kaluza and Klein that space-time has more than 4 dimensions. We shall slightly reformulate this and suggest that what appears to be a point will, at a sufficiently small length scale, be seen to possess an algebraic structure which can be described by a noncommutative geometry. It would seem that the particles which appear in nature do so in finite multiplets and there is no indication that this is due to a lack of the accelerator energy which would be necessary to excite an as yet unseen infinite multiplet structure. We shall take this fact as motivation for considering as extra structure the finite noncommutative geometries based on matrix algebras which were studied in Chapter 3. From a practical point of view in any case a finite structure will always be a sufficient if not the most esthetic description of the known particle spectrum.

In the first section we shall show that to a certain extent spin can be interpreted as an additional noncommutative structure to space-time. Since this will involve choosing a unitary representation of the Lorentz group, the structure will necessarily be an infinite one. In the next section we shall examine the physical theories which would appear if one were to study ordinary electrodynamics in a space-time with an additional algebraic structure. We shall see that these contain some but not all of the properties one might require of an extension of the standard model of the electroweak interactions. In the last section we shall examine the theory of gravity in the same extended geometry. This yields for the appropriate algebraic structure one of the truncated versions of ordinary Kaluza-Klein theory.