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
L = ψ ¯ ( i ∂ / − m ) ψ + g 2 ( ψ ¯ ψ ) 2 {\displaystyle {\mathcal {L}}={\overline {\psi }}\left(i\partial \!\!\!/-m\right)\psi +{\frac {g}{2}}\left({\overline {\psi }}\psi \right)^{2}}
where g {\displaystyle g} is the coupling constant,
∂ / = ∑ μ = 0 3 γ μ ∂ ∂ x μ {\displaystyle \partial \!\!\!/=\sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}} in the Feynman slash notations, ψ ¯ = ψ ∗ γ 0 {\displaystyle {\overline {\psi }}=\psi ^{*}\gamma ^{0}} . Here γ μ {\displaystyle \gamma ^{\mu }} , 0 ≤ μ ≤ 3 {\displaystyle 0\leq \mu \leq 3} , are Dirac gamma matrices. The corresponding equation can be written as
i ∂ ∂ t ψ = − i ∑ j = 1 3 α j ∂ ∂ x j ψ + m β ψ − g ( ψ ¯ ψ ) β ψ {\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 } , where α j {\displaystyle \alpha ^{j}} , 1 ≤ j ≤ 3 {\displaystyle 1\leq j\leq 3} , and β {\displaystyle \beta } are the Dirac matrices. In one dimension, this model is known as the massive Gross–Neveu model.
Generalizations A commonly considered generalization is
L = ψ ¯ ( i ∂ / − m ) ψ + g ( ψ ¯ ψ ) k + 1 k + 1 {\displaystyle {\mathcal {L}}={\overline {\psi }}\left(i\partial \!\!\!/-m\right)\psi +g{\frac {\left({\overline {\psi }}\psi \right)^{k+1}}{k+1}}}
with k > 0 {\displaystyle k>0} , or even
L = ψ ¯ ( i ∂ / − m ) ψ + F ( ψ ¯ ψ ) {\displaystyle {\mathcal {L}}={\overline {\psi }}\left(i\partial \!\!\!/-m\right)\psi +F\left({\overline {\psi }}\psi \right)} , where F {\displaystyle F} 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 k = 1 {\displaystyle k=1} and in one dimension only, and non-renormalizable for higher values of k {\displaystyle k} and in higher dimensions.
Solitary wave solutions The Soler model admits solitary wave solutions of the form
… excerpt ends here. Continue reading the full article.
