Dirac Wave Functions of Positive Energy with Arbitrarily Small Position Uncertainty

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.)