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I've read that if you have a Hamiltonian for the Dirac Equation, you can add a potential term to it simply by adjusting the momentum operator so that $p^\mu \rightarrow p^\mu-A^\mu$, where $A^\mu$ is the relevant potential. But how do you calculate $A^\mu$? For example, what would $A^\mu$ be for an electron in an electromagnetic field given by the tensor $F^{\alpha\beta}$?

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Use the classical vector potential (the one you get from maxwell equations in potential form), but the correct tools for that situation should be quantum field theory –  Angel Joaniquet Tukiainen May 29 '13 at 20:56

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The field tensor can be derived from the vector potential like so:

$$F^{\mu \nu}=\partial^\mu A^\nu-\partial^{\nu}A^\mu $$

If $F$ is simple enough, you can usually construct an appropriate $A$ without too much difficulty. Otherwise you're stuck inverting this with a bunch of indefinite integrals.

Note that $A$ is not uniquely determined by this relation. If $A^\mu$ is a valid vector potential, then for any analytic function $\phi$ $$A'^\mu=A^\mu + \partial^\mu\phi $$ will give equivalent results.

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