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Looking at the Dirac equation of the form

$$\Big(i\gamma^{\mu}(\partial_{\mu}-iA_{\mu})-m\Big)\psi=0$$

There is a simple solution to this equation, which is

$$\psi=\exp\Big(i\int^xA_{\mu}dx^{\mu}\Big)\psi_0$$

Where $\psi_0$ solves the Dirac equation in the absence of the gauge field.

My question is, are there any other solutions to this equation? Presumably there are, or else it appears the gauge field has almost negligible effect on the dynamics of the solution.

Is there a nice way of categorizing these different types of solutions?

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The solution you wrote down is only valid if the integral doesn’t depend on the specific path you take. This is equivalent to saying that the exterior derivative of A is zero, which is equivalent to saying that A describes zero field.

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  • $\begingroup$ Why? Is this to avoid multivalued-ness of the wavefunction? $\endgroup$
    – fewfew4
    May 22, 2020 at 23:29
  • $\begingroup$ Nevermind, I think I understand. Thank you! $\endgroup$
    – fewfew4
    May 22, 2020 at 23:44
  • $\begingroup$ @Jahan - If the gauge field $A_{\mu}$ is a matrix field, say, a $3 \times 3$ matrix field $A_{\mu}^{a} T_{3}^{a}$, where $T_{3} = \frac{1}{2}\lambda^{a}$ with $\lambda^{a}$ being Gell-Mann matrices, will the solution be in the form $\psi = exp \left[ i \int^{x} \left( A_{\mu}^{a} T_{3}^{a}\right) dx^{\mu}\right] \psi_{0}$? $\endgroup$
    – Shen
    May 20, 2022 at 8:06

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