# Enhancement of superluminal weak values under Lorentz boost

## TL;DR

This paper investigates how Lorentz boosts affect the probability of superluminal weak values in relativistic quantum theories, revealing different enhancement mechanisms for Klein-Gordon and Dirac waves.

## Contribution

It numerically analyzes the impact of Lorentz boosts on superluminal weak values in Klein-Gordon and Dirac theories, highlighting distinct enhancement mechanisms.

## Key findings

- Lorentz boost increases superluminal probability fraction.
- Klein-Gordon waves show enhancement due to asymmetry in velocity distribution.
- Dirac waves exhibit enhancement regardless of initial velocity distribution symmetry.

## Abstract

We study the local group velocity defined as the weak value of the velocity operator in the (1+1) dimensional Klein-Gordon as well as Dirac theory. It was shown by Berry [ J. Phys. A 45, 185308 (2012)] that when the pre- and post-selected states for evaluating the weak value are chosen at random from an ensemble of available states, the local group velocity has a universal probability distribution which can have both subluminal and superluminal components. In this work, we numerically explore the role of Lorentz boost and its impact on the superluminal fraction of the total probability distribution. We show that the dependence (enhancement) of the superluminal fraction on Lorentz boost of the total probability distribution differs both qualitatively and quantitatively for the Klein-Gordon waves and Dirac waves. For the Klein-Gordan waves, the asymmetry in the distribution of group velocities around the zero velocity point in the laboratory frame is entirely responsible for the observation of relative enhancement in the boosted frame. On the other hand, for the Dirac waves, we observe an enhancement irrespective of whether the laboratory frame velocity distribution is symmetric or not.