What scalars should we use ?

Abstract

There are compelling historical and mathematical reasons why we ended up, among others in Physics, with using the scalars given by the real or the complex numbers. 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 the reals R and complex numbers C - are 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.

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• Two Essays on the Archimedean versus Non-Archimedean Debate
• Where Infinitesimals Come From ..
• Two Theories of Special Relativity ?
• Deficient Mathematical Models of Quantum Theory
• Local Functions : Algebras, Ideals, and Reduced Power Algebras
• No-Cloning in Reduced Power Algebras