Physics:Soler model

Short description: Type of 3+1 dimensional quantum field theory

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

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

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

The corresponding equation can be written as

[math]\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 }[/math],

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

Generalizations

A commonly considered generalization is

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

with [math]\displaystyle{ k\gt 0 }[/math], or even

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

where [math]\displaystyle{ F }[/math] 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 [math]\displaystyle{ k=1 }[/math] and in one dimension only, and non-renormalizable for higher values of [math]\displaystyle{ k }[/math] and in higher dimensions.

Solitary wave solutions

The Soler model admits solitary wave solutions of the form [math]\displaystyle{ \phi(x)e^{-i\omega t}, }[/math] where [math]\displaystyle{ \phi }[/math] is localized (becomes small when [math]\displaystyle{ x }[/math] is large) and [math]\displaystyle{ \omega }[/math] 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 [math]\displaystyle{ (\bar\psi\psi)^2=J_\mu J^\mu }[/math], with [math]\displaystyle{ \bar\psi\psi=\psi^*\sigma_3\psi }[/math] the relativistic scalar and [math]\displaystyle{ J^\mu=(\psi^*\psi,\psi^*\sigma_1\psi,\psi^*\sigma_2\psi) }[/math] the charge-current density. The relation follows from the identity [math]\displaystyle{ (\psi^*\sigma_1\psi)^2+(\psi^*\sigma_2\psi)^2+(\psi^*\sigma_3\psi)^2 =(\psi^*\psi)^2 }[/math], for any [math]\displaystyle{ \psi\in\Complex^2 }[/math].

See also

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