Non-Hermitian $\cal PT$-symmetric quantum mechanics of relativistic particles with the restriction of mass

Abstract

The modified Dirac equations for the massive particles with the replacement of the physical mass $m$ with the help of the relation $m\rightarrow m_1+ \gamma_5 m_2$ are investigated. It is shown that for a fermion theory with a $\gamma_5$-mass term, the limiting of the mass specter by the value $ m_{max}= {m_1}^2/2m_2$ takes place. In this case the different regions of the unbroken $\cal PT$ symmetry may be expressed by means of the restriction of the physical mass $m\leq m_{max}$. It should be noted that in the approach which was developed by C.Bender et al. for the $\cal PT$-symmetric version of the massive Thirring model with $\gamma_5$-mass term, the region of the unbroken $\cal PT$-symmetry was found in the form $m_1\geq m_2$ \cite{ft12}. However on the basis of the mass limitation $m\leq m_{max}$ we obtain that the domain $m_1\geq m_2$ consists of two different parametric sectors: i) $0\leq m_2 \leq m_1/\sqrt{2}$ -this values of mass parameters $m_1,m_2$ correspond to the traditional particles for which in the limit $m_{max}\rightarrow \infty$ the modified models are converting to the ordinary Dirac theory with the physical mass $m$; ii)$m_1/\sqrt{2}\leq m_2 \leq m_1$ - this is the case of the unusual particles for which equations of motion does not have a limit, when $m_{max}\rightarrow \infty$. The presence of this possibility lets hope for that in Nature indeed there are some "exotic fermion fields". As a matter of fact the formulated criterions may be used as a major test in the process of the division of considered models into ordinary and exotic fermion theories. It is tempting to think that the quanta of the exotic fermion field have a relation to the structure of the "dark matter".