Must Physics Be Constrained by the Archimedean Axiom ? Relativity and Quanta with Scalars given by Reduced Power Algebras
Résumé
It is shown that, unknown to nearly everyone, modern theoretical Physics is significantly {\it constrained} by the tacit acceptance of the ancient Archimedean Axiom imposed upon Geometry by Euclid more than two millennia ago, an axiom which does not seem to have any modern physical motivation. By freeing oneself of this axiom a large variety of scalar fields and algebras other, and larger than the usual fields $\mathbb{R}$ and $\mathbb{C}$ of real, respectively complex numbers becomes available for mathematical modelling in theoretical Physics. This paper shows the validity of such modelling in Special Relativity and Quantum Mechanics, namely, in the case of the Lorentz transformations, the Heisenberg Uncertainty and the No Cloning property. The advantages in using such alternative scalars, specifically, given by {\it reduced power algebras} or {\it ultrapower fields}, are multiple. Among them, one can in a simple and direct way eliminate the so called ''infinities in physics". More generally, one can introduce many levels of precision in theoretical Physics. This is much unlike the present situation when, with the use of $\mathbb{R}$ as the only basic scalar field, there can exist only one single level of precision. Also, one can establish a {\it Second Relativity Principle} in which the covariance of the equations of Physics and of basic physical phenomena and properties is considered not only with respect to changes of reference frames, but also with changes of algebras or fields of scalars. In this regard, this paper shows that the Lorentz transformations, the Heisenberg Uncertainty and the No Cloning property are indeed covariant with respect to a large variety of scalars given by reduced power algebras, or in particular, ultrapower fields.
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