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 formula_2 is the coupling constant, formula_3 in the Feynman slash notations, formula_4. Here formula_5, formula_6, are Dirac gamma matrices. The corresponding equation can be written as where formula_8, formula_9, and formula_10 are the Dirac matrices. In one dimension, this  model is known as the massive Gross-Neveu model. Generalizations. A commonly considered generalization is with formula_12, or even where formula_14 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 formula_15 and in one dimension only, and non-renormalizable for higher values of formula_16 and in higher dimensions. Solitary wave solutions. The Soler model admits solitary wave solutions of the form formula_17 where formula_18 is localized (becomes small when formula_19 is large) and  formula_20 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 formula_21, with formula_22 the relativistic scalar and formula_23 the charge-current density. The relation follows from the identity formula_24, for any formula_25.