The minimum width in relativistic quantum mechanics

TL;DR

This paper refutes the long-held belief that relativistic quantum mechanics imposes a minimum localization width near the Compton wavelength, demonstrating that arbitrarily small widths are possible and addressing misconceptions about localization measures.

Contribution

It clarifies the correct relativistic position amplitude, shows Lorentz contraction allows smaller widths, and constructs amplitudes with widths below the supposed minimum, challenging prior assumptions.

Findings

Relativistic position amplitude becomes a delta function for localized states.

Lorentz contraction permits wavepacket widths to be arbitrarily small.

Constructed scalar amplitudes with widths smaller than the Compton wavelength.

Abstract

We challenge the widespread belief, originated by Newton and Wigner (Rev. Mod. Phys, 21, 400 (1949)) that the incorporation of special relativity into quantum mechanics implies that a massive particle cannot be localized within an arbitrarily small spatial extent, that there is a minimum width approximately equal to the Compton wavelength. Our argument is in four parts. First, the scalar function used by Newton and Wigner as a measure of localization is not a position probability amplitude. The correct relativistic position probability amplitude becomes a delta function for a state vector localized according to the criteria of Newton and Wigner. Second, the possibility of Lorentz contraction as observed from a boosted frame means that the wavepacket width in the boost direction can take arbitrarily small values. Third, we refer to the work of Almeida and Jabs (Am. J. Phys. 52, 921 (1984)) who show that the long time wavepacket spreading rate for relativistic position probability amplitudes is always less than the speed of light no matter how small the initial width of the wavepacket. Lastly, we show that it is a simple matter to construct scalar amplitudes with spatial widths smaller than the supposed minimum.