Non-Archimedean Mathematics and the formalism of Quantum Mechanics

TL;DR

This paper introduces a non-Archimedean mathematical framework using Euclidean numbers and ultrafunctions, aiming to provide a novel formalism for quantum mechanics that extends traditional methods.

Contribution

It develops a new non-Archimedean approach with Euclidean numbers and ultrafunctions, applying it to the formalism of quantum mechanics for potential advancements.

Findings

Introduction of Euclidean numbers E and {\Lambda}-limits

Definition of ultrafunctions as generalized functions

Application to quantum mechanics formalism

Abstract

This paper is divided in four parts. In the introduction, we discuss the program and the motivations of this paper. In section 2, we introduce the non-Archimedean field of Euclidean numbers E and we present a summary of the theory of {\Lambda}-limits which can be considered as a different approach to nonstandard methods. In the third part (section 3), we define axiomatically the space of ultrafunctions which are a kind of generalized function based on the field of Euclidean numbers E. Finally, we describe an application of the previus theory to the formalism of classical Quantum Mecanics.