Paper detail

Conformal transformations and doubling of the particle states

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 $q_μ$ ($q'_μ=q_μ+h_μ$, $q'_μ=Λ^ν_μq_ν$, $q'_μ=λq_μ$ and $q'_μ=-M^2q_μ/q^2$) and the corresponding rotations on the 6D cone $κ_Aκ^A=0$ $(A=μ;5,6\equiv 0,1,2,3;5,6)$ with $q_μ=M\ κ_μ/(κ_{5}+κ_{6})$ and the scale parameter $M$. The 4D reduction of the 6D fields on the cone $κ_Aκ^A=0$ require the intermediate 5D projection of the fields which are placed into two 5D hyperboloids $q_μq^μ+ q_5^2= M^2$ and $q_μq^μ- q_5^2=- M^2$ in order to cover the whole domain $(-\infty,\infty)$ of $q^2\equiv q_μq^μ$ with $(q_5^2\ge 0$. The resulting 5D and 4D fields $φ(x,x_5=0)=Φ(x)$ in the coordinate space consist of two parts $φ=φ_1+φ_2$ and $Φ=Φ_1+Φ_2$, where the Fourier conjugate of $φ_1(x,x_5)$ and $φ_2(x,x_5)$ are defined on the hyperboloids $q_μq^μ+ q_5^2= M^2$ and $q_μq^μ- q_5^2=- M^2$ respectively. The present relationship between the 6D, 5D and 4D fields require two kinds of 5D fields $φ_{\pm}=φ_1\pmφ_2$ and their 4D reductions $φ_{\pm}(x_5=0)=Φ_{\pm}=Φ_1\pmΦ_2$ with the same quantum numbers and with the different masses and the source operators. This doubling of the 4D fields $Φ_{\pm}=Φ_1\pm Φ_2$ is in agreement with the observed mass splitting of the electron and muon, $π$ and $π(1300)$-mesons, N and N(1440)-nucleons etc [1].