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PT-Symmetric Pseudo-Hermitian Relativistic Quantum Mechanics With a Maximal Mass

The quantum-field model described by non-Hermitian, but a ${\cal PT}$-symmetric Hamiltonian is considered. It is shown by the algebraic way that the limiting of the physical mass value $m \leq m_{max}= {m_1}^2/2m_2$ takes place for the case of a fermion field with a $γ_5$-dependent mass term ($m\rightarrow m_1 +γ_5 m_2 $). In the regions of unbroken $\cal PT$ symmetry the Hamiltonian $H$ has another symmetry represented by a linear operator $ \cal C$. We exactly construct this operator by using a non-perturbative method. In terms of $ \cal C$ operator we calculate a time-independent inner product with a positive-defined norm. As a consequence of finiteness mass spectrum we have the $\cal PT$-symmetric Hamiltonian in the areas $(m\leq m_{max})$, but beyond this limits $\cal PT$-symmetry is broken. Thus, we obtain that the basic results of the fermion field model with a $γ_5$-dependent mass term is equivalent to the Model with a Maximal Mass which for decades has been developed by V.Kadyshevsky and his colleagues. In their numerous papers the condition of finiteness of elementary particle mass spectrum was introduced in a purely geometric way, just as the velocity of light is a maximal velocity in the special relativity. The adequate geometrical realization of the limiting mass hypothesis is added up to the choice of (anti) de Sitter momentum space of the constant curvature.