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, . 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.[3][4]
A commonly considered generalization is
with , or even
where 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 where is localized (becomes small when is large) and is a real number.[6]
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 .[7]