# numerical formulation of Dirac equation plus electromagnetic field

I have the following equations describing the electron field in a (classic) electromagnetic field:

$$c\left(\alpha _i\right.{\cdot (P - q(A + A_b) + \beta mc) \psi = E \psi }$$

where $A_b$ is the background field and $A$ is the one generated by the local Dirac field

I presume that the equation for the electromagnetic field $A$ generated by the electron would be:

$$\nabla_{\mu}\nabla^{\mu} A_{\nu} = \frac{\psi \gamma^{\nu} \psi}{\epsilon_0}$$

Question: Is there a way to numerically solve these systems of equations to find eigenstates of the system?

Side Show Question: Are these eigenstates physically meaningful? do i still need to apply second quantization procedure in order to know which eigenstates are physically meaningful (i.e: stable) and which are not?

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I guess these are tough questions, as the system is nonlinear. I can only give some references. In some of them, some numerical solutions of this system were found: Phys. Rev. A 60, 4291–4300 (1999) (also arXiv:physics/0001038 ), http://maths-old.anu.edu.au/research.publications/proceedings/039/CMAproc39-booth.pdf and references there. For what it's worth, in my work arXiv:1111.4630 it is shown how to eliminate spinor field from the system (but complex electromagnetic potentials are introduced, which produce the same electromagnetic field, so their imaginary part is defined by one common function, so you have just 5 unknown real functions).

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