  Recently, there has been much interest in the evolution of quantum particles
on closed time-like curves (CTCs). However, such models typically assume
point-like particles with only two degrees of freedom - a very questionable
assumption given the relativistic setting of the problem. We show that it is
possible to generalise the Deutsch model of CTCs to fields using the equivalent
circuit formalism. We give examples for coherent, squeezed and single-photon
states interacting with the CTC via a beamsplitter. The model is then
generalised further to account for the smooth transition to normal quantum
mechanics as the CTC becomes much smaller than the size of the modes
interacting on it. In this limit, we find that the system behaves like a
standard quantum mechanical feedback loop.
