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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?

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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. –  Ron Maimon Oct 24 '11 at 19:03
    
I guess m is positive as in dirac equation. What is the problem? –  Anthonny Oct 24 '11 at 19:19
    
Oh--- ok--- I was wrong. I thought you were trying to make solitons like those that occur in the bosonic form of this action (which doesn't work). –  Ron Maimon Oct 24 '11 at 19:25
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@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. –  Ron Maimon Oct 24 '11 at 23:00
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@Anthony: Fermionic fields don't have classical solutions. This model is heavily studied--- it's the Gross Neveu model. –  Ron Maimon Oct 25 '11 at 15:10
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