In QFT, computations are done with plane-wave free solutions of the Dirac equations: if one were to consider full solutions of the Dirac equation in interaction with its own electrodynamic field, even without field quantization, what would one obtain? Does anybody know of full solutions, even at a classical level?
NOTE: the question is of purely mathematical interest, so I am not interested in reading that we commonly do not do that in standard computations, I would like to know what would happen if considering the problem with mathematical rigour.
exact solution of the full system (Dirac equation with gauge potential and Maxwell equations with spinor current).
The Schwinger model is the most well known example. And the Seiberg-Witten formalism provides examples in four and three spacetime dimensions: http://xxx.lanl.gov/abs/hep-th/9611118 and
Article Uniqueness of the Seiberg-Witten Effective Lagrangian
or http://www.its.caltech.edu/~matilde/swcosi.pdf
Thanks. Are there also models, always in (1+3)-dimensions, with no supersymmetry, that you know of?
No-because it's supersymmetry that expresses the fact that the quantum fluctuations are consistently described. However, with such examples it's possible to study how the superpartners might decouple-and this, of course, isn't completely known.
Let me add a simple observation. The Dirac equation in an external electromagnetic field is a system for which the initial value problem is easily treated by numerical methods. For fast and easy experimentation (1+1) dimensions are ideal, but the methods work in (1+3) equally well.
A small toy system can be found here:
”Relativistic Quantum Dynamics in 1D and the Klein Paradox” from The Wolfram Demonstrations Project. http://demonstrations.wolfram.com/RelativisticQuantumDynamicsIn1DAndTheKleinParadox/ Contributed by: Ulrich Mutze (published 2011-04-28)
In physics, a Dirac field doesn't interact with its own electromagnetic field. But it is a mathematical question, in this case it isn't this simple linear system to consider, for the reasons already given: run-away solutions, divergencies etc. and it is anyway a non linear system because of the interaction term.
Comm. Math. Phys. 112, Number 1 (1987), 21-49.
On global solutions of the Maxwell-Dirac equations
M. Flato, Jacques Simon, and Erik Taflin
Journal of Mathematical Physics 37, 4418 (1996);
Localized solutions of the Dirac–Maxwell equations
C.J. Radford.
Article Asymptotic completeness, global existence and the infrared p...
Moshe Flato, Jacques C.H. Simon, Erik Taflin
Is it possible that you missed the Volkov solutions? These are used routinely to study matter-field interaction in the strong e.m. field limit.
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