Relativistic PT-symmetric fermionic theories in 1+1 and 3+1 dimensions

Relativistic PT-symmetric fermionic interacting systems are studied in 1+1 and 3+1 dimensions. The objective is to include non-Hermitian PT-symmetric interaction terms that give {\it real} spectra. Such interacting systems could describe new physics. The simplest non-Hermitian Lagrangian density is $L=L_0+L_{int}=\bar\psi(i\not\partial-m)\psi-g\bar\psi\gamma^5\psi$. The associated relativistic Dirac equation is PT invariant in 1+1 dimensions and the associated Hamiltonian commutes with PT. However, the dispersion relation $p^2=m^2-g^2$ shows that the PT symmetry is broken in the chiral limit $m\to0$. For interactions $L_{int}=-g(\bar\psi\gamma^5\psi)^N$ with N=2,3, if the associated Dirac equation is PT invariant, the dispersion relation gives complex energies as $m\to0$. Other models are studied in which x-dependent PT-symmetric potentials such as $ix^3$, $-x^4$, $i\kappa/x$, Hulth\'en, or periodic potentials are coupled to $\psi$ and the classical trajectories plane are examined. Some combinations of these potentials give a real spectrum. In 3+1 dimensions, the simplest system $L=L_0+L_{int}=\bar\psi(i\not\partial-m)\psi-g\bar\psi\gamma^5\psi$ resembles the 1+1-dimensional case but the Dirac equation is not PT invariant because $T^2=-1$. This explains the appearance of complex eigenvalues as $m\to0$. Other Lorentz-invariant 2-point and 4-point interactions give non-Hermitian PT-symmetric terms in the Dirac equation. Only the axial vector and tensor Lagrangian interactions $L_{int}=-i\bar\psi\tilde B_\mu\gamma^5\gamma^\mu\psi$ and $L_{int}=-i\bar\psi T_{\mu\nu}\sigma^{\mu\nu}\psi$ fulfil both requirements of PT invariance of the associated Dirac equation and non-Hermiticity. Both models give complex spectra as $m\to0$. The effect on the spectrum of the additional constraint of selfadjointness of the Hamiltonian with respect to the PT inner product is investigated.