Time-travelling billiard-ball clocks: a quantum model

TL;DR

This paper introduces a quantum model of closed timelike curves using a billiard-ball analogy, comparing two quantum theories of CTCs, and finds distinct predictions for the system's state.

Contribution

It presents a novel quantum formulation of a billiard-ball CTC scenario and compares two leading quantum theories, revealing different outcomes.

Findings

D-CTCs produce a mixed state representing classical multiplicity.

P-CTCs predict an equal superposition of trajectories.

The model incorporates a vacuum state and a clock for operational distinction.

Abstract

General relativity predicts the existence of closed timelike curves (CTCs), along which an object could travel to its own past. A consequence of CTCs is the failure of determinism, even for classical systems: one initial condition can result in multiple evolutions. Here we introduce a new quantum formulation of a classic example, where a billiard ball can travel along two possible trajectories: one unperturbed and one, along a CTC, where it collides with its past self. Our model includes a vacuum state, allowing the ball to be present or absent on each trajectory, and a clock, which provides an operational way to distinguish the trajectories. We apply the two foremost quantum theories of CTCs to our model: Deutsch's model (D-CTCs) and postselected teleportation (P-CTCs). We find that D-CTCs reproduce the classical solution multiplicity in the form of a mixed state, while P-CTCs predict an equal superposition of the two trajectories, supporting a conjecture by Friedman et al. [Phys. Rev. D 42, 1915 (1990)].