Vacuum polarization of a complex automorphic scalar field in two-dimensional spacetime with closed null geodesics and the time-machine problem

The quantum authomorphic scalar field in two-dimensional space-time with closed null geodesics (the Cauchy horizon) is considered. The stress energy tensor $\langle T_{\mu\nu}\rangle^{\mathrm{ren}}$ is obtained. It is shown that the value of $\langle T_{\mu\nu}\rangle^{\mathrm{ren}}$ is regular at the Cauchy horizon for specific authomorphic parameter.