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 ^[1] and re-introduced and investigated in 1970 by Mario Soler^[2] as a toy model of self-interacting electron.

This model is described by the Lagrangian density

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

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

The corresponding equation can be written as

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

where ${\displaystyle \alpha ^{j}}$, ${\displaystyle 1\leq j\leq 3}$, and ${\displaystyle \beta }$ are the Dirac matrices. In one dimension, this model is known as the massive Gross–Neveu model.^[3][4]

Generalizations

A commonly considered generalization is

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

with ${\displaystyle k>0}$, or even

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

where ${\displaystyle F}$ 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.^[5]

Renormalizability

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

Solitary wave solutions

The Soler model admits solitary wave solutions of the form ${\displaystyle \phi (x)e^{-i\omega t},}$ where ${\displaystyle \phi }$ is localized (becomes small when ${\displaystyle x}$ is large) and ${\displaystyle \omega }$ is a real number.^[6]

Reduction to the massive Thirring model

In spatial dimension 2, the Soler model coincides with the massive Thirring model, due to the relation ${\displaystyle ({\bar {\psi }}\psi )^{2}=J_{\mu }J^{\mu }}$, with ${\displaystyle {\bar {\psi }}\psi =\psi ^{*}\sigma _{3}\psi }$ the relativistic scalar and ${\displaystyle J^{\mu }=(\psi ^{*}\psi ,\psi ^{*}\sigma _{1}\psi ,\psi ^{*}\sigma _{2}\psi )}$ the charge-current density. The relation follows from the identity ${\displaystyle (\psi ^{*}\sigma _{1}\psi )^{2}+(\psi ^{*}\sigma _{2}\psi )^{2}+(\psi ^{*}\sigma _{3}\psi )^{2}=(\psi ^{*}\psi )^{2}}$, for any ${\displaystyle \psi \in \mathbb {C} ^{2}}$.^[7]

See also

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