Conformal transformations and doubling of the particle states

Abstract

The 6D and 5D representations of the four-dimensional (4D) interacted fields and the corresponding equations of motion are obtained using equivalence of the conformal transformations of the four-momentum $q_{\mu}$ ($q'_{\mu}=q_{\mu}+h_{\mu}$, $q'_{\mu}=\Lambda^{\nu}_{\mu}q_{\nu}$, $q'_{\mu}=\lambda q_{\mu}$ and $q'_{\mu}=-M^2q_{\mu}/q^2$) and the corresponding rotations on the 6D cone $\kappa_A\kappa^A=0$ $(A=\mu;5,6\equiv 0,1,2,3;5,6)$ with $q_{\mu}=M\ \kappa_{\mu}/(\kappa_{5}+\kappa_{6})$ and the scale parameter $M$. The 4D reduction of the 6D fields on the cone $\kappa_A\kappa^A=0$ require the intermediate 5D projection of the fields which are placed into two 5D hyperboloids $q_{\mu}q^{\mu}+ q_5^2= M^2$ and $q_{\mu}q^{\mu}- q_5^2=- M^2$ in order to cover the whole domain $(-\infty,\infty)$ of $q^2\equiv q_{\mu}q^{\mu}$ with $(q_5^2\ge 0$. The resulting 5D and 4D fields $\varphi(x,x_5=0)=\Phi(x)$ in the coordinate space consist of two parts $\varphi=\varphi_1+\varphi_2$ and $\Phi=\Phi_1+\Phi_2$, where the Fourier conjugate of $\varphi_1(x,x_5)$ and $\varphi_2(x,x_5)$ are defined on the hyperboloids $q_{\mu}q^{\mu}+ q_5^2= M^2$ and $q_{\mu}q^{\mu}- q_5^2=- M^2$ respectively. The present relationship between the 6D, 5D and 4D fields require two kinds of 5D fields $\varphi_{\pm}=\varphi_1\pm\varphi_2$ and their 4D reductions $\varphi_{\pm}(x_5=0)=\Phi_{\pm}=\Phi_1\pm\Phi_2$ with the same quantum numbers and with the different masses and the source operators. This doubling of the 4D fields $\Phi_{\pm}=\Phi_1\pm \Phi_2$ is in agreement with the observed mass splitting of the electron and muon, $\pi$ and $\pi(1300)$-mesons, N and N(1440)-nucleons etc [1].