Are there any nonlinear variations of Dirac's Equation analogous to the Nonlinear Schrodinger Equation, that have been studied and published in any mainstream journals or books?

Perhaps such a thing as a NDE would apply to non-quantum situations the way NSE applies to optics or hydrodynamics, but where relativistic speeds and spin matter. It might apply to extrapolations to quantum theory. But I don't recall seeing anything like that in the literature.


As well as the Nonlinear Dirac Equation Wikipedia Page spoken of in the comments, see the Maxwell-Dirac equation in e.g.:

as well as the works of the late Hilary Booth of the Australian National University.

Basically, this nonlinear system is pretty much what its name says, i.e an intriguing and uncluttered "first quantized" or "baby" formulation of QED (appealing especially to non quantum field theorists like me):

$$\gamma^\mu\left(i \partial_\mu - q A_\mu\right) \psi + V \psi - \psi = 0$$

$$\partial_\nu F^{\nu\,\mu} = q\,\bar{\psi} \gamma^\mu \psi$$

$$F_{\mu\,\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$$

with the Lorenz Gauge

$$\partial_\mu A^\mu = 0$$.

i.e. intuitively, the current source for the Maxwell equations is $q$ times the electron's probability current density (here $\bar{\psi}$ stands for charge conjugated $\psi$). Thus, a first quantized electron field is nonlinearly coupled to a first quantized photon field. This nonlinear system can be solved exactly for the hydrogen atom potential $V$ and in other situations and these solutions (particularly the hydrogen atom ones) can indeed model the Lamb shift and spontaneous emission. The series solution (given in the Barut and Kraus paper) comes out to something very like the standard QED perturbation terms and indeed one must step in and "renormalize" this solution too.

I've had quite a bit of experience with nonlinear Schrödinger equations to model optical solitons in a former life so I understand the Maxwell-Dirac equation is likely a little different from the kind of thing you were looking for, but I hope you find this interesting nonetheless.


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