The solution to the Hardy's paradox

TL;DR

This paper resolves Hardy's paradox by analyzing the experiment with a complex parameter, showing the paradox is only apparent at a specific value and explaining it through quantum uncertainty principles.

Contribution

It introduces a complex parameter to analyze Hardy's experiment, demonstrating the paradox's dependence on this parameter and providing a new interpretation based on quantum uncertainty.

Findings

The paradox disappears for most parameter values.

The paradox only appears at a specific parameter value ($=1$).

Particles can cross the intersection point at different times due to energy-time uncertainty.

Abstract

By using both, the weak-value formulation as well as the standard probabilistic approach, we analyze the Hardy's experiment introducing a complex and dimensionless parameter ($\epsilon$) which eliminates the assumption of complete annihilation when both, the electron and the positron departing from a common origin, cross the intersection point $P$. We then find that the paradox does not exist for all the possible values taken by the parameter. The apparent paradox only appears when $\epsilon=1$; however, even in this case we can interpret this result as a natural consequence of the fact that the particles can cross the point $P$, but at different times due to a natural consequence of the energy-time uncertainty principle.