The Deep Problems of Symmetry in Mathematical Physics and their Definitive Resolution

Abstract

Symmetry stands for a central notion in Mathematics with extremely important applications in Theoretical Physics. Mathematician Emmy Noether in 1915 famously axiomatized this notion which resoundingly made its way in fundamental physics in its chapters of Lagrangian mechanics and Hamiltonian mechanics. However there is only one dimension to symmetry in mathematics and mathematical physics: an invariance that is preserved under a particular transformation. Contrary to this approach, there exist many other dimensions to the artifact of symmetry, which, if correctly prospected, could have helped accurately answer the long-running question in mathematical physics concerning the very origin of physical symmetries and the reason for the physical mechanism of spontaneous symmetry-breaking. The ignorance of origin of observable symmetries as well as the existence of other categories of symmetry is accompanied by a misunderstanding of the very states of symmetry, which is reflected in an inexact assessment of the ultra-structures subtending mathematical symmetry. In this writing, we aim to to expose the caveats of symmetry in mathematics, specifically in Geometry, as viewed from the overarching Quanto-Geometric perspective, and to show their resolution with the application of special Differential Geometry principles emanating from the Quanto-Geometric framework.