# Could this model have soliton solutions?

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We consider a theory described by the Lagrangian,

$$\mathcal{L}=i\bar{\Psi}\gamma^\mu\partial_\mu\Psi-m\bar{\Psi}\Psi+\frac{1}{2}g(\bar{\Psi}\Psi)^2$$

The corresponding field equations are, $$(i\gamma^\mu\partial_\mu-m+g\bar{\Psi}\Psi)\Psi=0$$

Could this model have soliton solutions? Without the last term, it is just a Dirac field (if $g=0$), but it has to be included. This is similar to the Thirring model. I was looking for this field in books and papers but I haven't found it. If you know about it could you give me any reference?

This post imported from StackExchange Physics at 2014-06-25 21:06 (UCT), posted by SE-user Anthonny
I guess you are trying to make a Fermionic mexican hat. Please say so--- because either sign of m in the action gives a positive mass for the Fermion.

This post imported from StackExchange Physics at 2014-06-25 21:06 (UCT), posted by SE-user Ron Maimon
I guess m is positive as in dirac equation. What is the problem?

This post imported from StackExchange Physics at 2014-06-25 21:06 (UCT), posted by SE-user Anthonny
Actually Im trying to know if some kind of soliton (or at least a solitary wave) is possible in this model.

This post imported from StackExchange Physics at 2014-06-25 21:06 (UCT), posted by SE-user Anthonny
@Anthony: this model is Fermionic. Solitons are coherent superpositions bosonic excitations. But the model conserves a U(1) charge which counts the Fermions, so that you can make a Fermi sea with a large numbers of fermions, and perhaps get a superconducting condensate, which can then have solitons. But I don't think this is what you meant. Perhaps you can say exactly what kind of soliton you are after? If you want a classical solution of the form $\psi(x)$, it's not going to work, because $\psi$ is Fermi.

This post imported from StackExchange Physics at 2014-06-25 21:06 (UCT), posted by SE-user Ron Maimon
I'm not sure if it is very important, but I'd want to know why you say that a classical solution is not going to work. I wonder if this model is known, as I say before I I haven't found it in any reference and I would like to know if you have any.

This post imported from StackExchange Physics at 2014-06-25 21:06 (UCT), posted by SE-user Anthonny
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