The Taming of Closed Time-like Curves

We formulate QFT on a $R^{1,d}/Z_2$ orbifold, in a manner which is invariant under the $Z_2$ time and space reversal. This is a background with closed time-like curves. It is also relevant for the elliptic interpretation of de Sitter space. We calculate the one-loop vacuum expectation value of the stress tensor in the invariant QFT, and show that it does not diverge at the boundary of the region of closed time-like curves. Rather, the only divergence is at the initial time slice of the orbifold, analogous to a spacelike Big-Bang singularity. We then calculate the one-loop graviton tadpole in bosonic string theory, and show that the answer is the same as if the target space would be just the Minkowski space $R^{1,d}$, suggesting that the tadpole vanishes for the superstring. Finally, we argue that it is possible to define local S-matrices, even if the spacetime is globally time-nonorientable.