It is currently believed that when expressed in terms of classical fields fundamental physics involves only scalar fields, Yang-Mills potentials and spinors, as well as the gravitational field. The actions which determine the dynamics of these fields can be written in terms of exterior derivatives, covariant derivatives and Dirac operators, as well as the space-time metric. In Chapter 5 we shall see that in a rather large class of algebras a generalization of the Dirac operator can be introduced. It would seem then that field theory could be studied on a large class of noncommutative geometries. If the associative algebra which underlies the geometry is too abstract or too different from an ordinary algebra of functions on space-time, it is impossible to interpret its elements in terms of known classical observables but in principle every associative algebra A can be used to define a noncommutative geometry. In Section 6.1 we shall show in fact how one can construct a differential calculus over arbitrary A. This construction contains however absolutely no information about the structure of A and even for finite algebras the corresponding algebra of forms is of infinite dimension.

The purpose of the present chapter is to introduce a large class of associative algebras of infinite dimension which can be used in constructing noncommutative geometries which might be of interest in physics. In the first section we make a few general remarks about formal algebras and we present some examples which have been used in the construction of noncommutative geometries.