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 where
is the coupling constant,
in the Feynman slash notations,
ψ ¯
ψ
, are Dirac gamma matrices.
The corresponding equation can be written as where
are the Dirac matrices.
In one dimension, this model is known as the massive Gross–Neveu model.
[3][4] A commonly considered generalization is with
is a smooth function.
Besides the unitary symmetry U(1), in dimensions 1, 2, and 3 the equation has SU(1,1) global internal symmetry.
[5] 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.
The Soler model admits solitary wave solutions of the form
is localized (becomes small when
is a real number.
[6] In spatial dimension 2, the Soler model coincides with the massive Thirring model, due to the relation
ψ ¯
ψ
ψ ¯
ψ =
ψ
ψ
the relativistic scalar and
the charge-current density.
The relation follows from the identity