Dirac Wave Functions of Positive Energy with Arbitrarily Small Position Uncertainty

TL;DR

This paper demonstrates that for positive-energy Dirac wave functions, there is no positive lower bound to their position uncertainty, countering longstanding conjectures and clarifying the nature of localization in relativistic quantum mechanics.

Contribution

The authors provide a rigorous proof that positive-energy Dirac wave functions can have arbitrarily small position uncertainty, resolving a long-standing conjecture with a gap in previous proofs.

Findings

Positive-energy Dirac states can be arbitrarily localized

Counter-example to the conjecture of a positive lower bound

Clarification of localization limits in relativistic quantum mechanics

Abstract

We consider wave functions in the Hilbert space $\mathcal{H}=L^2(\mathbb{R}^3,\mathbb{C}^4)$ of a single Dirac particle, specifically from the positive-energy subspace $\mathcal{H}_+$ of the free Dirac Hamiltonian. Over the decades, various authors conjectured that for wave functions from $\mathcal{H}_+$, there is a positive lower bound to the position uncertainty $\sigma_x$; in other words, that such states cannot be arbitrarily narrow in $x$. Building on work by Bracken and Melloy, we show that this conjecture is false. (In fact, they already stated that this conjecture is false and already had a counter-example, but their proof that it is a counter-example had a gap.)