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
where is the coupling constant,
in the Feynman slash notations,.
Here,, are Dirac gamma matrices.
The corresponding equation can be written as
where,,
and are the Dirac matrices.
In one dimension,
this model is known as the massive Gross-Neveu model.

Generalizations

A commonly considered generalization is
with, or even
where is a smooth function.

Features

Internal symmetry

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

Renormalizability

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

Solitary wave solutions

The Soler model admits solitary wave solutions
of the form
where is localized
and 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
with
the relativistic scalar
and
the charge-current density.
The relation follows from the identity
for any.