  Interacting quantum fields on spacetimes containing regions of closed
timelike curves (CTCs) are subject to a non-unitary evolution $X$. Recently, a
prescription has been proposed, which 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 non-unitary evolutions. We propose
an alternative method 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 localised in regions spacelike
separated from the CTC region. We comment on the physical significance of the
possible indefiniteness of the inner product introduced in our proposal.
