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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
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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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But how do we know in general, just by looking at (any) Lagrangian, whether or not it will have soliton solutions? (sorry I joined late ... 2.5 yrs late ...) –  The Dark Side Jun 23 '14 at 7:36

1 Answer 1

The answer to this question is no assuming dimensions higher than 1+1. This can be seen observing that the equation of motion for the fermionic field is just the limit of mass going to infinity of a scalar field coupled to a fermionic field. This can be seen in the following way. Consider the Lagrangian $$ L = \frac{1}{2}(\partial\phi)^2-\frac{1}{2}m^2\phi^2+\bar\psi(i\gamma\cdot\partial-g\phi)\psi. $$ The equations of motion are easily obtained to be $$ \partial^2\phi+m^2\phi=g\bar\psi\psi\qquad (i\gamma\cdot\partial-g\phi)\psi=0. $$ The equation of the scalar field can be immediately integrated to give $$ \phi=g\int d^Dx\Delta(x-y)\bar\psi(y)\psi(y) $$ with the propagator that of a free particle. This propagator, in the limit of a very large mass of the scalar field, is just proportional to $\delta^D(x-y)$. This observation is crucial for the following. So, we are left with the equation $$ (i\gamma\cdot\partial-\kappa\bar\psi\psi)\psi=0 $$ where $\kappa$ is a constant that depends on the parameters $m$ and $g$ of the Lagrangian we started from. In this way we have recovered the equation proposed by the OP but we have just proved that this is the large mass limit of a scalar field coupled to a fermion field.

Now, we do quantum field theory on the initial Lagrangian and write down the partition function as $$ Z[j,\bar\eta,\eta]=\int[d\phi][d\bar\psi][d\psi]e^{i\int d^Dx\left[\frac{1}{2}(\partial\phi)^2-\frac{1}{2}m^2\phi^2+\bar\psi(i\gamma\cdot\partial-g\phi)\psi\right]} e^{i\int d^Dx\left[j\phi+\bar\eta\psi-\bar\psi\eta\right]}. $$ The fermion part can be immediately integrated generating a potential to the scalar field in the form $$ V(\phi)=-i{\rm tr}\ {\rm ln}\left[i\gamma\cdot\partial-g\phi\right]_{(x,x)}. $$ This can be evaluated by a loop expansion as $$ V(\phi)=-g_1\phi^3-g_2\phi^4+\ldots. $$ Now, independently on the value of the mass of the scalar field and the presence of a finite v.e.v., using Derrick theorem we know that stationary localized solutions to a nonlinear wave equation or nonlinear Klein-Gordon equation in dimensions three and higher are unstable. This concludes the proof that the model equation yielded by the OP does not have soliton solutions in dimensions three and higher. In dimensions 1+1 these can exist.

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