Real Functions for Physics

TL;DR

This paper introduces a new classification of real functions and distributions called 'combed functions', which are sufficient for physics and simplify mathematical handling of generalized functions.

Contribution

It proposes a classification based on low pass-filter behavior, including diverse objects like analytic functions and delta functions, focusing on those relevant for physics.

Findings

Combed functions include real analytic functions, delta functions, and derivatives.

The classification simplifies the mathematical treatment of generalized functions.

Combed functions are sufficient for classical and quantum physics applications.

Abstract

A new classification of real functions and other related real objects defined within a compact interval is proposed. The scope of the classification includes normal real functions and distributions in the sense of Schwartz, referred to jointly as "generalized functions". This classification is defined in terms of the behavior of these generalized functions under the action of a linear low pass-filter, which can be understood as an integral operator acting in the space of generalized functions. The classification criterion defines a class of generalized functions which we will name "combed functions", leaving out a complementary class of "ragged functions". While the classification as combed functions leaves out many pathological objects, it includes in the same footing such diverse objects as real analytic functions, the Dirac delta "function", and its derivatives of arbitrarily high orders, as well as many others in between these two extremes. We argue that the set of combed functions is sufficient for all the needs of physics, as tools for the description of nature. This includes the whole of classical physics and all the observable quantities in quantum mechanics and quantum field theory. The focusing of attention on this smaller set of generalized functions greatly simplifies the mathematical arguments needed to deal with them.