Unitarity of quantum theory and closed timelike curves

Interacting quantum fields on spacetimes containing regions of closed timelike curves (CTC’s) are subject to a nonunitary evolution X. Recently, a prescription has been proposed that restores unitarity of the evolution by modifying the inner product on the final Hilbert space. We give a rigorous description of this proposal and note an operational problem which arises when one considers the composition of two or more nonunitary evolutions. We propose an alternative methd by which unitarity of the evolution may be regained by extending X to a unitary evolution on a larger (possibly indefinite) inner product space. The proposal removes the ambiguity noted by Jacobson in assigning expectation values to observables localized in regions spacelike separated form the CTC region. We comment on the physical significance of the possible indefiniteness of the inner product introduced in our proposal.