Billiard-ball paradox for a quantum wave packet

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