Validity of one-dimensional QED for a system with spatial symmetry

We examine the accuracy of an intrinsically one-dimensional (1D) quantum electrodynamics to predict accurately the forces and charges of a three-dimensional (3D) system that has a high degree of symmetry and therefore depends effectively only on a single coordinate. As a test case we analyze two charged capacitor plates that are infinitely extended along two coordinate directions. Using the lowest-order fine-structure correction to the photon propagator we compute the vacuum's induced charge polarization density and show that the force between the charged plates is increased. Although a one-dimensional theory cannot take the transverse character of the virtual (force-mediating) photons into account, nevertheless it predicts, in lowest order of the fine-structure constant, the Coulomb force law between the plates correctly. However, the quantum correction to the classical result is slightly different between the 1D and 3D theories with the polarization charge density induced from the vacuum underestimated by the 1D approach.

Figure 1

Sketch of the infinitely extended plane-parallel capacitor plates with width $2w$ each and edge-to-edge spacing $d$. The parameter $q$ is the (two-dimensional) charge density (charge per unit area in the $x_1-x_2$ direction). We also sketch the vacuum's induced polarization charges.

Figure 2

Numerical value of the fine-structure constant ${\alpha }^{\left(1\text{D}\right)}$ (in units of $\alpha /{\lambda }_C^2)$ for one-dimensional QED as a function of the width $2w$ of the plate, based on the requirement that the total induced polarization charge from the vacuum on the plate should be identical for the 1D and 3D QED.

Figure 3

Ratio between the induced polarized charge $Q_{\mathrm{pol}}$ and the initial charge $Q_0$ as a function of the width of the plate $w$. The solid line is obtained from the 3D QED approach and the dashed line is the result of the 1D calculation. The unperturbed density of the plate was $q=1.602\times {10}^{-4}\text{C}/{\mathrm{m}}^2$.

Figure 4

Polarization charge density ${\rho }_{\mathrm{pol}}\left(z\right)$ measured in $\text{C}/{\text{m}}^2$ around the positive charged plate according to the three-dimensional (solid line) and the one-dimensional theory (dashed line) as a function of the position $z$. The unperturbed density of the plate was $q=1.602\times {10}^{-4}\text{C}/{\mathrm{m}}^2$ and the plate has a width of $2w=1.213\times {10}^{-11}$ m.

Figure 5

Correction to the force per unit area (in $\text{N}/{\text{m}}^2)$ between the two oppositely charged capacitor plates according to the three-dimensional (solid line) and the one-dimensional theory (dashed line) as a function of the distance $d$ between them. The unperturbed density of each plate is $q=\pm 1.6\times {10}^{-4}\text{C}/{\mathrm{m}}^2$ and each plate has a width of $2w=1.213\times {10}^{-11}\text{m}$. For comparison, the classical force is $1450\text{N}/{\mathrm{m}}^2$.

Figure 6

Leading-order diagram (a) and one-loop diagrams (b)–(j) that contribute to the electric potential between charge carriers in the two plates. In the long-range limit (low-energy transfer limit) we consider, the potential is dominated by diagrams (a) and (h). The wavy lines represent photons and the straight lines represent electrons.

Figure 7

Modified photon propagator at one loop. It is the modified photon propagator that will contribute to the long-range electric potential. The wavy line with the shaded circle on the left represents the combined effects of the tree-level photon propagator shown as the first term on the right and the one-loop correction to the photon propagator shown as the second term on the right.