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Dirac Wave Functions of Positive Energy with Arbitrarily Small Position Uncertainty

Ilmar Bürck, Roderich Tumulka

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 $σ_x$; 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.)

Dirac Wave Functions of Positive Energy with Arbitrarily Small Position Uncertainty

Abstract

We consider wave functions in the Hilbert space of a single Dirac particle, specifically from the positive-energy subspace of the free Dirac Hamiltonian. Over the decades, various authors conjectured that for wave functions from , there is a positive lower bound to the position uncertainty ; in other words, that such states cannot be arbitrarily narrow in . 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.)
Paper Structure (12 sections, 3 theorems, 60 equations, 3 figures)

This paper contains 12 sections, 3 theorems, 60 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Claims and Arguments for a Positive Minimal Size
  3. Pair Production
  4. Width of $P_+\delta^3$
  5. Convolution Argument
  6. Review of the Bracken-Melloy Construction
  7. Proofs
  8. Derivation of the Kernel
  9. Proof of Theorem \ref{['thm:1']}
  10. Proof of Theorem \ref{['thm:2']}
  11. Proof of Remark \ref{['rem:converse']}
  12. Proof of Lemma \ref{['lem:1']}

Key Result

Theorem 1

For $\psi\in\mathcal{H}_+$ with $\|\psi\|=1$, $\sigma_\psi$ can be arbitrarily close to 0. In other words, for every $\varepsilon>0$ there exists $\psi\in\mathcal{H}_+$ with $\|\psi\|=1$ and $\sigma_\psi < \varepsilon$. In other words, there is a sequence $\psi_n\in\mathcal{H}_+$ with $\|\psi_n\|=1$

Figures (3)

  • Figure 1: Plots of the functions $K_0(|x|)/|x|$ and $K_1(|x|)/|x|$ that show up as coefficients in \ref{['eqn: Kernel']}
  • Figure 2: $\sigma_{\psi_n}$ for $n = 1,\dots,19$, illustrating that $\sigma_{\psi_n}\to 0$ as $n\to\infty$
  • Figure 3: $4\pi r^2|\psi_n(r)|^2$ for $n$ = 1, 2, 5, 10 with $r$ in units of the Compton wave length

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Lemma 1