Non-Hermitian $\cal PT$-symmetric quantum mechanics of relativistic particles with the restriction of mass
The modified Dirac equations for the massive particles with the replacement of the physical mass m with the help of the relation m → m1 + γ5m2 are investigated. It is shown that for a fermion theory with a γ5-mass term, the limiting of the mass specter by the value mmax = m1 /2m2 takes place. In this case the different regions of the unbroken PT symmetry may be expressed by means of the restriction of the physical mass m ≤ mmax. It should be noted that in the approach which was developed by C.Bender et al. for the PT -symmetric version of the massive Thirring model with γ5-mass term, the region of the unbroken PT -symmetry was found in the form m1 ≥ m2 [25]. However on the basis of the mass limitation m ≤ mmax we obtain that the domain m1 ≥ m2 consists of two different parametric sectors: i) 0 ≤ m2 ≤ m1/ √ 2 -this values of mass parameters m1,m2 correspond to the traditional particles for which in the limit mmax → ∞ the modified models are converting to the ordinary Dirac theory with the physical mass m; ii)m1/ √ 2 ≤ m2 ≤ m1 this is the case of the unusual particles for which equations of motion does not have a limit, when mmax → ∞. 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”. PACS numbers: 02.30.Jr, 03.65.-w, 03.65.Ge, 12.10.-g, 12.20.-m