Hyperreal delta functions as a new general tool for modeling physical states with infinitely high densities

Abstract

This paper introduces the expanded real numbers as an ordered subring of the hyperreal number field that does not contain any infinitesimals, and defines the set of all integrable functions from the real numbers to the expanded real numbers. This allows to identify the Dirac delta with a special hyperreal-valued function of a real variable: the Dirac delta function thus defined is a general tool, applicable for the mathematical modeling of physical systems in which infinitely high densities occur.