Heisenberg Uncertainty in Reduced Power Algebras

Abstract

The Heisenberg uncertainty relation is known to be obtainable by a purely mathematical argument. Based on that fact, here it is shown that the Heisenberg uncertainty relation remains valid when Quantum Mechanics is re-formulated within far wider frameworks of {\it scalars}, namely, within one or the other of the infinitely many {\it reduced power algebras} which can replace the usual real numbers $\mathbb{R}$, or complex numbers $\mathbb{C}$. A major advantage of such a re-formulation is, among others, the disappearance of the well known and hard to deal with problem of the so called "infinities in Physics". The use of reduced power algebras also opens up a foundational question about the role, and in fact, about the very meaning and existence, of fundamental constants in Physics, such as Planck's constant $h$. A role, meaning, and existence which may, or on the contrary, may not be so objective as to be independent of the scalars used, be they the usual real numbers $\mathbb{R}$, complex numbers $\mathbb{C}$, or scalars given by any of the infinitely many reduced power algebras, algebras which can so easily be constructed and used.