# Covariant four-potential in the Dirac equation in QED

• A
• Shen712
In summary, the Dirac equation for an electron is given by (1) or (2), where ##A_{\mu}## is the covariant four-potential of the electrodynamic field generated by the electron itself, and ##B_{\mu}## is the external field imposed by an external source. The components of ##A_{\mu}## are given by equations (3) and (4) where ##\rho## and ##\mathbf{j}## are the charge density and current density respectively. The term ##e\gamma^{\mu} A_{\mu}\psi## in the Dirac equation represents the interaction of the electron with its own electromagnetic field. This interaction is described by gauge theory

#### Shen712

TL;DR Summary
In the Dirac equation for an electron, the covariant four-potential is generated by the electron itself. How can this four-potential interact with the electron itself?
Under the entry "Quantum electrodynamics" in Wikipedia, the Dirac equation for an electron is given by

$$i\gamma^{\mu}\partial_{\mu}\psi - e\gamma^{\mu}\left( A_{\mu} + B_{\mu} \right) \psi - m\psi = 0 ,\tag 1$$

or

$$i\gamma^{\mu}\partial_{\mu}\psi - m\psi = e\gamma^{\mu}\left( A_{\mu} + B_{\mu} \right) \psi .\tag 2$$

where ##A_{\mu}## is the covariant four-potential of the electrodynamic field generated by the electron itself, and ##B_{\mu}## is the external field imposed by external source.

Under another entry "Electromagnetic four-potential" in Wikipedia, the components of the covariant four-potential ##A_{\mu} = (\phi, \mathbf{A})## are given by

$$\phi \left( \mathbf{r}, t\right) = \frac{1}{4\pi\epsilon_{0}} \int d^{3}x' \frac{\rho \left( \mathbf{r'}, t_{r}\right)}{|\mathbf{r} - \mathbf{r'}|} ,\tag 3$$

$$\mathbf{A} \left( \mathbf{r}, t\right) = \frac{\mu_{0}}{4\pi} \int d^{3}x' \frac{j\left( \mathbf{r'}, t_{r}\right)}{|\mathbf{r} - \mathbf{r'}|} ,\tag 4$$

where ##\rho\left( \mathbf{r}, t\right)## and ##\mathbf{j}\left( \mathbf{r}, t\right)## are charge density and current density respectively, and

$$t_{r} = t - \frac{|\mathbf{r} - \mathbf{r'}|}{c} \tag 5$$

is the retarded time.

When I try to solve the Dirac equation (2), I have problem dealing with the first term on the right-hand side, ##e\gamma^{\mu} A_{\mu} \psi##. The four-potential ##A_{\mu} = (\phi, \mathbf{A})## is generated by the electron itself, how can the electron interact with ##A_{\mu}##? Specifically, in this case, the charge density ##\rho## and current density ##\mathbf{j}## belong to the electron itself, and ##\mathbf{r} = \mathbf{r'}##. How can I calculate the term ##e\gamma^{\mu} A_{\mu} \psi##?

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• topsquark
I don't know, why they introduce the complication with the external field ##B_{\mu}## in the Wikipedia article. Let's first deal with standard QED, which describes a closed system consisting of electrons, positrons, and the electromagnetic field. The electrons and positrons are described by the quantized Dirac field, and the electromagnetic field is quantized too.

First of all as any relativistic QFT QED describes a system with a non-fixed number of particles, i.e., it describes electrons and positrons interacting through the electromagnetic field but indeed also on the interaction of a single electron with its own electromagnetic field. As is well known these higher-order "radiative corrections" lead to divergent integrals, which have to be resolved in perturbation theory by renormalization, but after this problem is resolved QED leads to astonishingly accurate predictions of the associated phenomena like the Lamb shift of hydrogen levels or the anomalous magnetic moment of the electron.

Note that the same problem with the "radiation reaction" exists also in the classical case. The only problem is that it's so much more severe than in the QFT case that it cannot even be resolved at all orders of perturbation theory. The best one can come up with in the classical domain is the Landau-Lifshitz approximation of the Lorentz-Abraham-Dirac equation.

For a very good treatment, see

K. Lechner, Classical Electrodynamics, Springer International
Publishing AG, Cham (2018),
https://doi.org/10.1007/978-3-319-91809-9

C. Nakhleh, The Lorentz-Dirac and Landau-Lifshitz
equations from the perspective of modern renormalization
theory, Am. J. Phys 81, 180 (2013),
https://dx.doi.org/10.1119/1.4773292
https://arxiv.org/abs/1207.1745

• malawi_glenn