# Conformal transformations and doubling of the particle states

Research paper by **A. I. Machavariani**

Indexed on: **09 Jan '14**Published on: **09 Jan '14**Published in: **Mathematical Physics**

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#### 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].