Soler model

The soler model is a quantum field theory model of Dirac fermions interacting via four fermion interactions in 3 spatial and 1 time dimension. It was introduced in 1938 by Dmitri Ivanenko and re-introduced and investigated in 1970 by Mario Soler as a toy model of self-interacting electron.

This model is described by the Lagrangian density


 * $$\mathcal{L}=\overline{\psi} \left(i\partial\!\!\!/-m \right) \psi + \frac{g}{2}\left(\overline{\psi} \psi\right)^2$$

where $$g$$ is the coupling constant, $$\partial\!\!\!/=\sum_{\mu=0}^3\gamma^\mu\frac{\partial}{\partial x^\mu}$$ in the Feynman slash notations, $$\overline{\psi}=\psi^*\gamma^0$$. Here $$\gamma^\mu$$, $$0\le\mu\le 3$$, are Dirac gamma matrices.

The corresponding equation can be written as


 * $$i\frac{\partial}{\partial t}\psi=-i\sum_{j=1}^{3}\alpha^j\frac{\partial}{\partial x^j}\psi+m\beta\psi-g(\overline{\psi} \psi)\beta\psi$$,

where $$\alpha^j$$, $$1\le j\le 3$$, and $$\beta$$ are the Dirac matrices. In one dimension, this model is known as the massive Gross–Neveu model.

Generalizations
A commonly considered generalization is


 * $$\mathcal{L}=\overline{\psi} \left(i\partial\!\!\!/-m \right) \psi + g\frac{\left(\overline{\psi} \psi\right)^{k+1}}{k+1}$$

with $$k>0$$, or even


 * $$\mathcal{L}=\overline{\psi} \left(i\partial\!\!\!/-m \right) \psi + F\left(\overline{\psi} \psi\right)$$,

where $$F$$ is a smooth function.

Internal symmetry
Besides the unitary symmetry U(1), in dimensions 1, 2, and 3 the equation has SU(1,1) global internal symmetry.

Renormalizability
The Soler model is renormalizable by the power counting for $$k=1$$ and in one dimension only, and non-renormalizable for higher values of $$k$$ and in higher dimensions.

Solitary wave solutions
The Soler model admits solitary wave solutions of the form $$\phi(x)e^{-i\omega t},$$ where $$\phi$$ is localized (becomes small when $$x$$ is large) and $$\omega$$ is a real number.

Reduction to the massive Thirring model
In spatial dimension 2, the Soler model coincides with the massive Thirring model, due to the relation $$ (\bar\psi\psi)^2=J_\mu J^\mu$$, with $$\bar\psi\psi=\psi^*\sigma_3\psi$$ the relativistic scalar and $$J^\mu=(\psi^*\psi,\psi^*\sigma_1\psi,\psi^*\sigma_2\psi)$$ the charge-current density. The relation follows from the identity $$ (\psi^*\sigma_1\psi)^2+(\psi^*\sigma_2\psi)^2+(\psi^*\sigma_3\psi)^2 =(\psi^*\psi)^2$$, for any $$\psi\in\Complex^2$$.