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 hypothesized 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 $x$. Using a sequence of wave functions introduced by Bracken and Melloy, we show that this hypothesis is false. (In fact, they already stated that it is false, but their proof that their sequence is a counter-example had a gap.)