Bell's theorem for trajectories

TL;DR

This paper formulates a Bell's inequality for classical trajectories, proposes a way to test its quantum violation over time, and concludes that quantum trajectories cannot be predetermined in local-realistic theories.

Contribution

It introduces a Bell's inequality for trajectories, a novel no-go theorem, and demonstrates its violation by quantum mechanics over extended time intervals.

Findings

Quantum mechanics violates the trajectory Bell's inequality.

Violations persist over entire time intervals, not just instantaneously.

Trajectories cannot be predetermined within local-realistic theories.

Abstract

In classical theory, the trajectory of a particle is entirely predetermined by the complete set of initial conditions via dynamical laws. Based on this, we formulate a no-go theorem for the dynamics of classical particles, i.e., a Bell's inequality for trajectories, and discuss its possible violation in a quantum scenario. A trajectory, however, is not an outcome of a quantum measurement, in the sense that there is no observable associated with it, and thus there is no "direct" experimental test of the Bell's inequality for trajectories. Nevertheless, we show how to overcome this problem by considering a special case of our generic inequality that can be experimentally tested point-by-point in time. Such inequality is indeed violated by quantum mechanics, and the violation persists during an entire interval of time and not just at a particular singular instant. We interpret the violation to imply that trajectories (or at least pieces thereof) cannot exist predetermined, within a local-realistic theory.