Vacuum polarization in a one-dimensional effective quantum-electrodynamics model

TL;DR

The paper develops and analyzes a one-dimensional effective QED model of a hydrogen-like atom with a delta-function nucleus to study vacuum polarization without explicit photons. It derives the exact 1D Dirac Hamiltonian, computes the vacuum-polarization density via momentum-space Green functions, and decomposes the first-order QED correction to the bound-state energy into direct, exchange, and Breit-type terms, noting the Dirac-delta contribution. A central contribution is demonstrating and addressing the slow convergence of vacuum-polarization densities in a finite plane-wave basis, proposing a momentum-space regularization that isolates the regular VP part and recovers the correct delta contribution. The results show that, in this 1D setting, the total VP density does not vanish and can lead to a net positive QED shift, offering insights into how similar calculations might be handled in the 3D effective QED theory for atoms and molecules. The methods and regularization scheme provide a path toward practical finite-basis implementations of vacuum-polarization effects in more realistic, higher-dimensional effective QED models.

Abstract

With the aim of progressing toward a practical implementation of an effective quantum-electrodynamics (QED) theory of atoms and molecules, which includes the effects of vacuum polarization through the creation of virtual electron-positron pairs but without the explicit photon degrees of freedom, we study a one-dimensional effective QED model of the hydrogen-like atom with delta-potential interactions. This model resembles the three-dimensional effective QED theory with Coulomb interactions while being substantially simpler. We provide some mathematical details about the definition of this model, calculate the vacuum-polarization density, and the Lamb-type shift of the bound-state energy, correcting and extending results of previous works. We also study the approximation of the model in a finite plane-wave basis, and in particular we discuss the basis convergence of the bound-state energy and eigenfunction, of the vacuum-polarization density, and of the Lamb-type shift of the bound-state energy. We highlight the difficulty of converging the vacuum-polarization density in a finite basis and we propose a way to improve it. The present work could give hints on how to perform similar calculations for the three-dimensional effective QED theory of atoms and molecules.

Figures (8)

• Figure 1: The Uehling and total vacuum-polarization densities $n^{\text{vp},(1)}(x)$ [Eq. (\ref{['nvp1x']})] and $n^{\text{vp}}(x)$ [Eq. (\ref{['nvpx']})] for $m = c = Z = 1$. The vertical line represents a Dirac-delta function.
• Figure 2: First-order QED correction to the bound-state energy [Eq. (\ref{['Evp1b']})] for $m=Z=1$ as a function of the inverse speed of light $1/c$, using either the (a) Uehling or (b) total vacuum-polarization local density matrix [Eqs. (\ref{['nvpUxmat']}) and (\ref{['n1vpx']})]. The total correction, as well as the direct-Coulomb-type (DC) [Eq. \ref{['Evp1bDC']}], exchange-Coulomb-type (XC) [Eq. \ref{['Evp1bXC']}], and exchange-Breit-type (XB) [Eq. \ref{['Evp1bXB']}] contributions are shown.
• Figure 3: Convergence of the bound-state energy of the 1D hydrogen-like Dirac model with a plane-wave basis as a function of (a) the IR cutoff parameter $L$ and (b) the UV cutoff parameter $\Lambda$ for $m=c=Z=1$. The exact value in the limits $L\to \infty$ and $\Lambda \to \infty$ is $\varepsilon_\text{b} = 0.6$.
• Figure 4: Convergence of the (a) large component and (b) small component of the bound-state eigenfunction of the 1D hydrogen-like Dirac model with a plane-wave basis as a function of the UV cutoff parameter $\Lambda$ for an IR cutoff parameter $L=10$ and $m=c=Z=1$. The exact eigenfunction [Eq. (\ref{['psi0tilde']})] corresponds to the limits $L\to \infty$ and $\Lambda \to \infty$ .
• Figure 5: (a) Uehling and (b) total vacuum-polarization densities for $m = c = Z = 1$, calculated exactly [Eqs. (\ref{['nvp1x']}) and (\ref{['nvpx']})] and with a plane-wave basis for a IR cutoff parameter $L=10$ and a UV cutoff parameter $\Lambda = 50$ [from Eq. (\ref{['nvpxbasis']})].