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

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

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

The corresponding equation can be written as

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

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

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

with $k>0$, or even

$ℒ=\bar{\psi }(i\partial /-m)\psi +F\left(\bar{\psi }\psi \right)$,

where $F$ is a smooth function.

Features

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 $\varphi (x)e^{-i\omega t},$ where $\varphi$ 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 $(\overline{\psi }\psi {)}^2=J_{\mu }{J}^{\mu }$, with $\overline{\psi }\psi ={\psi }^{*}{\sigma }_3\psi$ 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 $\psi \in {ℂ}^2$.

See also

• Dirac equation
• Gross–Neveu model
• Nonlinear Dirac equation
• Thirring model