  A ring resonator involves a scattering process where a part of the output is
fed again into the input. The same formal structure is encountered in the
problem of time travel in a neighborhood of a closed timelike curve (CTC). We
know how to describe quantum optics of ring resonators, and the resulting
description agrees with experiment. We can apply the same formal strategy to
any looped quantum evolution, in particular to the time travel. The argument is
in its essence a topological one and thus does not refer to any concrete
geometry. It is shown that the resulting paradigm automatically removes logical
inconsistencies associated with chronology protection, provided all
input-output relations are given by unitary maps. Examples of elementary loops
and a two-loop time machine illustrate the construction. In order to apply the
formalism to quantum computation one has to describe multi-qubit systems
interacting via CTC-based quantum gates. This is achieved by second
quantization of loops. An example of a multiparticle system, with oscillators
interacting via a time machine, is explicitly calculated. However, the
resulting treatment of CTCs is not equivalent to the one proposed by Deutsch in
his classic paper.
