Quantum Complex Scalar Field in the Two-Dimensional Spacetime with   Closed Timelike Curves and a Time-Machine Problem
  It is considered the quantum complex scalar field which obeys the authomorphic condition in the two-dimensional spacetime with closed timelike curves and the chronology horizon. The renormalized stress-energy tensor is obtained. It is shown that the value of the stress-energy tensor is regular at the chronology horizon for specific authomorphic parameters. Thus the particular example of field configuration is given for which the Hawking's chronology protection conjecture is violated. 
Sergey Sushkov