Which symmetry group for elementary particles with an electric charge today and in the past?

Abstract

In this paper, we revisit the Kaluza-Klein theory from the perspective of the classification of elementary particles based on the coadjoint orbit method. The key conjecture is to consider the electric charge as an extra momentum on an equal footing with the mass and the linear momentum. We study the momentum map of the corresponding symmetry group ${\hat{G}}_1$ which conserves the hyperbolic metric. We show that the electric charge is not an invariant, i.e. it depends on the reference frame, which is in contradiction with the experimental observations. In other words, it is not the symmetry group of the Universe today as we know it. To avert this paradox, we scale the fifth coordinate and consider the limit in which the cylinder radius ω vanishes. For the corresponding group ${\hat{G}}_0$, the charge is an invariant and therefore independent of the frame of reference and the observer. On this basis, we propose a cosmological scenario in which the elementary particles of the early Universe are classified from the momenta of the group ${\hat{G}}_1$, with the three former dimensions inflating quickly while the fifth one shrinks, leading to a 4D era in which as today the particles are characterized by the momenta of the group ${\hat{G}}_0$. By this mechanism, the elementary particles can acquire electric charge as a by-product of the $4+1$ symmetry breaking of the Universe. This work opens the way for the geometric quantization of charged elementary particles. We construct the corresponding ${\hat{G}}_0$-connections as a pullback connection over the space-time. Requiring that the linear 5-momentum is parallel-transported, we recover the conservation of the charge and the equation of motion with the Lorentz force. We revisit the variational relativity and obtain the field equations for both the gravitation and electromagnetic interactions with coupling terms which are negligible in the Newtonian approximation, allowing recovery of the Maxwell equations.