Dynamical Symmetries, Super-coherent States and Noncommutative Structures: Categorical and Geometrical Quantization Analysis

Abstract

The relation between fundamental spacetime structures and dynamical symmetries are treated beyond the geometrical and topological viewpoint. To this end analyze, taking into account the concept of categories and quasi hamiltonian structures, a recent research (Cirilo-Lombardo and Arbuzov in Int J Geom Methods Mod Phys 15(01):1850005, 2017) where one linear and one quadratic in curvature models were constructed and where a dynamical breaking of the \(\textit{SO}(4,2)\) group symmetry arises. We explain there how and why coherent states of the Klauder-Perelomov type are defined for both cases taking into account the coset geometry and some hints on the possibility to extend they to the categorical (functorial) status are given. The new spontaneous compactification mechanism that was defined in the subspace invariant under the stability subgroup is commented in the context of future developments as the main tool for the treatment of the internal symmetries, as the electroweak in the Standard Model (SM). The physical implications of the symmetry rupture as the introduction of a noncommutative structure in the context of non-linear realizations and direct gauging are analyzed and briefly discussed in this new theoretical framework.