6 M ay 2 00 5 What Scalars Should We Use ?

There are compelling historical and mathematical reasons why we ended up, among others in Physics, with using the scalars given by the real numbers in R, or the complex numbers in C. Recently, however, infinitely many easy to construct and use other algebras of scalars have quite naturally emerged in a number of branches of Applied Mathematics. These algebras of scalars can deal with the long disturbing difficulties encountered in Physics, related to such phenomena as " infinities in Physics " , " re-normalization " , the " Feynman path integral " , and so on. Specifically, as soon as one is dealing with scalars in algebras which-unlike R and Care no longer Archimedean, one can deal with a large variety of " infinite " quantities and do so within the usual rules and with the usual operations of algebra. Here we present typical constructions of these recently emerged algebras of scalars, most of them non-Archimedean. Algebraic Note In order to avoid possible undesirable overlap between terminology used in mathematics, and on the other hand, in physics, we recall here the following. In mathematics the notion of field means a ring in which every non-zero element has an inverse with respect to multiplication. Typical fields are given by the set Q of rational numbers, the set R of real numbers, the set C of complex numbers, as well as the set H of Hamil-ton's quaternions.