  Past studies of the billiard-ball paradox, a problem involving an object that
travels back in time along a closed timelike curve (CTC), typically concern
themselves with entirely classical histories, whereby any trajectorial effects
associated with quantum mechanics cannot manifest. Here we develop a quantum
version of the paradox, wherein a (semiclassical) wave packet evolves through a
region containing a wormhole time machine. This is accomplished by mapping all
relevant paths on to a quantum circuit, in which the distinction of the various
paths is facilitated by representing the billiard particle with a clock state.
For this model, we find that Deutsch's prescription (D-CTCs) provides
self-consistent solutions in the form of a mixed state composed of terms which
represent every possible configuration of the particle's evolution through the
circuit. In the equivalent circuit picture (ECP), this reduces to a binomial
distribution in the number of loops of time machine. The postselected
teleportation prescription (P-CTCs) on the other hand predicts a pure-state
solution in which the loop counts have binomial coefficient weights. We then
discuss the model in the continuum limit, with a particular focus on the
various methods one may employ in order to guarantee convergence in the average
number of clock evolutions. Specifically, for D-CTCs, we find that it is
necessary to regularise the theory's parameters, while P-CTCs alternatively
require more contrived modification.
