Billiard-ball paradox for a quantum wave packet

• URL: http://arxiv.org/abs/2205.05399v2
• Date: Tue, 16 Aug 2022 10:58:29 GMT
• Title: Billiard-ball paradox for a quantum wave packet
• Authors: Lachlan G. Bishop, Timothy C. Ralph, Fabio Costa
• Abstract summary: billiard-ball paradox is a problem involving an object that travels back in time along a closed timelike curve. We develop a quantum version of the paradox, wherein a (semiclassical) wave packet evolves through a region containing a wormhole time machine. We discuss the model in the continuum limit, with a particular focus on the various methods one may employ in order to guarantee convergence.
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• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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