Generalization of the continuous symmetry measure: the symmetry of vectors, matrices, operators and functions

Abstract

In this paper we generalize a method for evaluating the continuous symmetry measure, which is a quantitative estimate of the degree of symmetry of a given object. The generalization makes it possible to calculate the degree of symmetry content for any mathematical entity that is part of metric spaces such as vectors, matrices, operators and functions. Furthermore, by this new approach one can calculate the symmetry-content values for any compact symmetry groups either finite or infinite. An advantage of the new methodology is the ability to investigate analytically problems of symmetry changes. Examples of symmetry evaluation calculations are provided, including mixing of ideal gases, evaluation of the symmetry content of a Hamiltonian operator, the 2p_z orbital of the hydrogen atom, and more.