  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.
