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These fields scale as, $\tilde\phi (\psi)=\Omega^\xi\,\phi(\psi)$ and the metric tensor as $\tilde{g}_{\mu\nu}=\Omega^2\,g_{\mu\nu}$, where $\Omega$ and $\xi$ are called the conformal factor and wight, respectively. The conformal mass in Klein-Gordon mass ($\tilde m$) is related to original scalar mass, ($m$) by $\tilde{m}=\Omega^{-1}\,m$\,. Moreover, the Klein-Gordon equation in the conformal's frame reduces to the quantum Telegraph equation of a particle whose mass is given by, $M=(\xi+1)\,m$, in Minkowski's frame. The conformal wave equation in 2 dimensions with $\xi=1$ yields the quantum Telegraph equation with a mass. We have found that the conformal wave equation in 2 dimensions yields the Dirac equation for $\xi=\pm\,i$ in flat space. In 4 dimensions the mass of the conformal spinor field scales as $\tilde{m}=\Omega^{-2}m$. The spinor charge ($q$) is influenced by the conformal transformation and becomes $Q_c=q\xi/(\xi+\frac{3}{2})\,$. The conformal factor for a spinor field is found to be equal to the phase factor of the spinor field. Moreover, the conformal transformation preserves the probability of the spinor particle. There exists a certain conformal transformation that transforms the Klein-Gordon equation into the Dirac equation. An Aharonov- Bohm-like effect is found to occur due to a conformal transformation of the spinor field. Breaking of conformal invariance is found to give rise to a mass of the particle that is tantamount to the Higgs mechanism.
