Wave packets in QFT: leading order width corrections to decay rates and clock behaviour under Lorentz boosts

TL;DR

This paper investigates how localized wave packets in quantum field theory affect decay rates and clock behavior under Lorentz boosts, revealing corrections to standard assumptions and implications for relativistic quantum clocks.

Contribution

It provides the first calculation of decay rate corrections for Gaussian wave packets and analyzes their non-ideal Lorentz boost behavior as quantum clocks.

Findings

Decay rate correction of order a^2/M^2 for wave packets

Lorentz boost induces a correction of order a^2v^2/M^2 in decay rates

Wave packet clocks do not perfectly follow Lorentz transformations but remain consistent with relativity

Abstract

Decay rates in quantum field theory (QFT) are typically calculated assuming the particles are represented by momentum eigenstates (i.e. plane waves). However, strictly speaking, localized free particles should be represented by wave packets. This yields width corrections to the decay rate and to the clock behaviour under Lorentz boosts. We calculate the decay rate of a particle of mass $M$ modeled as a Gaussian wavepacket of width $a$ and centered at zero momentum. We find the decay rate to be $\Gamma_0 \big[1- \frac{3 a^2}{4 M^2} +\mathcal{O}\big(\tfrac{a^4}{M^4}\big)\big]$ where $\Gamma_0$ is the decay rate of the particle at rest treated as a plane wave. The leading correction is then of order $\tfrac{a^2}{M^2}$. We then perform a Lorentz boost of velocity $v$ on the above Gaussian and find that its decay rate does not decrease \textit{exactly} by the Lorentz factor $\sqrt{1-v^2}$. There is a correction of order $\tfrac{a^2v^2}{M^2}$. Therefore, the decaying wave packet does not act exactly like a typical clock under Lorentz boosts and we refer to it is a "WP clock" (wave packet clock). A WP clock does not move with a single velocity relative to an observer but has a spread in velocities (more specifically, a spread in momenta). Nonetheless, it is best viewed as a single clock as the wave packet represents a one-particle state in QFT. WP clocks do not violate Lorentz symmetry and are not based on new physics: they are a consequence of the combined requirements of special relativity, quantum mechanics and \textit{localized} free particles.