Failure of unitarity for interacting fields on spacetimes with closed timelike curves

The scattering of free quantum fields is well defined on a class of asymptotically flat spacetimes with closed timelike curves (CTC's), and, at least on these spacetimes, the $S$ matrix is unitary as well. For interacting fields, however, the preceding paper has obtained a set of unitarity relations that must be satisfied by the Feynman propagator if the scattering is to be unitary to each order in perturbation theory. In a globally hyperbolic spacetime, the causal form of the propagator guarantees that the relations are satisfied, but for spacetimes with CTC's, the form of the propagator is altered, and we show that the unitarity relations are not satisfied for interacting fields. We consider the $\lambda {\phi }^4$ theory in detail, but the results appear to hold for a wide class of fields. Although a conventional interpretation of quantum mechanics leads to inconsistency, a path-integral interpretation appears to allow a consistent assignment of probabilities to histories.