Chronology Protection and Quantized Fields: Complex Automorphic Scalar Field in Misner Space

The renormalized stress-energy tensor $⟨T_{\mu \nu }⟩$ of the quantized complex massless scalar field which obeys the automorphic condition in Misner space is obtained. It is shown that there exists the special value of the automorphic parameter for which $⟨T_{\mu \nu }⟩$ is regular on the chronology horizon and, so, can not act as a protector of chronology through a back reaction on a spacetime metric. However, it is shown that, at the same time, the value of field square $⟨{\varphi }^2⟩$, which characterizes the quantum field fluctuations, is divergent on the chronology horizon. The assumption is suggested that the infinitely growing quantum field fluctuations, which appear if a (self)interaction of the scalar field is taken into account, would prevent the chronology horizon formation.