Quantum coherence and closed timelike curves

Various calculations of the S matrix have shown that it seems to be nonunitary for interacting fields when there are closed timelike curves. It is argued that this is because there is loss of quantum coherence caused by the fact that part of the quantum state circulates on the closed timelike curves and is not measured at infinity. A prescription is given for calculating the superscattering matrix $ on spacetimes whose parameters can be analytically continued to obtain a Euclidean metric. It is illustrated by a discussion of a spacetime in which two disks in flat space are indentified. If the disks have an imaginary time separation, this corresponds to a heat bath. An external field interacting with the heat bath will lose quantum coherence. One can then analytically continue to an almost real separation of the disks. This will give closed timelike curves but one will still get loss of quantum coherence. A comparison is made with the work of authors who find a nonunitary S matrix. It is shown that this is because the $ does not factor into an S matrix and its adjoint when the spacetime does not have the property of asymptotic completeness.