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

TL;DR

This paper presents a novel approach using hyperreal delta functions within expanded real numbers to model physical states with infinite densities, providing a new mathematical tool for such applications.

Contribution

It introduces expanded real numbers without infinitesimals and defines hyperreal-valued delta functions for modeling infinitely dense physical states.

Findings

Hyperreal delta functions can represent infinite densities.

The framework extends classical delta functions to hyperreal contexts.

Applicable to modeling physical systems with infinite 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.