Billiard-ball paradox for a quantum wave packet

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.

arXiv comment18 pages, 9 figures

References in corpus (19)

• The warp drive: hyper-fast travel within general relativity
• Closed timelike curves via post-selection: theory and experimental demonstration
• The quantum mechanics of time travel through post-selected teleportation
• Quantum computation with indefinite causal structures
• Path Integrals, Density Matrices, and Information Flow with Closed Timelike Curves
• Information Flow of quantum states interacting with closed timelike curves
• Relativistic quantum information and time machines
• Closed Timelike Curves and "Effective" Superluminal Travel with Naked Line Singularities
• Billiard in the space with a time machine
• Treating Time Travel Quantum Mechanically
• Reversible time travel with freedom of choice
• Reversible dynamics with closed time-like curves and freedom of choice
• Space-time qubits
• A time machine for free fall into the past
• Spinning up a Time Machine
• Time-travelling billiard-ball clocks: a quantum model
• Three-Dimensional Billiards with Time Machine
• Time travel without paradoxes: Ring resonator as a universal paradigm for looped quantum evolutions
• Reply to comment on "Information Flow of quantum states interacting with closed timelike curves"