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 $\gamma ^{\mu }$, $0\leq \mu \leq 3$, 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 $\alpha ^{j}$, $1\leq j\leq 3$, 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.

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 $\phi (x)e^{-i\omega t},$ 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 $\psi \in \mathbb {C} ^{2}$.

See also

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