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 ^[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

${\displaystyle {\mathcal {L}}={\overline {\psi }}\left(i\partial \!\!\!/-m\right)\psi +{\frac {g}{2}}\left({\overline {\psi }}\psi \right)^{2}}${\displaystyle {\mathcal {L}}={\overline {\psi }}\left(i\partial \!\!\!/-m\right)\psi +{\frac {g}{2}}\left({\overline {\psi }}\psi \right)^{2}}

where ${\displaystyle g}${\displaystyle g} is the coupling constant, ${\displaystyle \partial \!\!\!/=\sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}}${\displaystyle \partial \!\!\!/=\sum _{\mu =0}^{3}\gamma ^{\mu }{\frac {\partial }{\partial x^{\mu }}}} in the Feynman slash notations, ${\displaystyle {\overline {\psi }}=\psi ^{*}\gamma ^{0}}${\displaystyle {\overline {\psi }}=\psi ^{*}\gamma ^{0}}. Here ${\displaystyle \gamma ^{\mu }}${\displaystyle \gamma ^{\mu }}, ${\displaystyle 0\leq \mu \leq 3}${\displaystyle 0\leq \mu \leq 3}, are Dirac gamma matrices.

The corresponding equation can be written as

${\displaystyle i{\frac {\partial }{\partial t}}\psi =-i\sum _{j=1}^{3}\alpha ^{j}{\frac {\partial }{\partial x^{j}}}\psi +m\beta \psi -g({\overline {\psi }}\psi )\beta \psi }${\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 ${\displaystyle \alpha ^{j}}${\displaystyle \alpha ^{j}}, ${\displaystyle 1\leq j\leq 3}${\displaystyle 1\leq j\leq 3}, and ${\displaystyle \beta }${\displaystyle \beta } are the Dirac matrices. In one dimension, this model is known as the massive Gross–Neveu model.^{[3] [4]}

A commonly considered generalization is

${\displaystyle {\mathcal {L}}={\overline {\psi }}\left(i\partial \!\!\!/-m\right)\psi +g{\frac {\left({\overline {\psi }}\psi \right)^{k+1}}{k+1}}}${\displaystyle {\mathcal {L}}={\overline {\psi }}\left(i\partial \!\!\!/-m\right)\psi +g{\frac {\left({\overline {\psi }}\psi \right)^{k+1}}{k+1}}}

with ${\displaystyle k>0}${\displaystyle k>0}, or even

${\displaystyle {\mathcal {L}}={\overline {\psi }}\left(i\partial \!\!\!/-m\right)\psi +F\left({\overline {\psi }}\psi \right)}${\displaystyle {\mathcal {L}}={\overline {\psi }}\left(i\partial \!\!\!/-m\right)\psi +F\left({\overline {\psi }}\psi \right)},

where ${\displaystyle F}${\displaystyle F} 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 ${\displaystyle k=1}${\displaystyle k=1} and in one dimension only, and non-renormalizable for higher values of ${\displaystyle k}${\displaystyle k} and in higher dimensions.

The Soler model admits solitary wave solutions of the form ${\displaystyle \phi (x)e^{-i\omega t},}${\displaystyle \phi (x)e^{-i\omega t},} where ${\displaystyle \phi }${\displaystyle \phi } is localized (becomes small when ${\displaystyle x}${\displaystyle x} is large) and ${\displaystyle \omega }${\displaystyle \omega } is a real number.^[6]

In spatial dimension 2, the Soler model coincides with the massive Thirring model, due to the relation ${\displaystyle ({\bar {\psi }}\psi )^{2}=J_{\mu }J^{\mu }}${\displaystyle ({\bar {\psi }}\psi )^{2}=J_{\mu }J^{\mu }}, with ${\displaystyle {\bar {\psi }}\psi =\psi ^{*}\sigma _{3}\psi }${\displaystyle {\bar {\psi }}\psi =\psi ^{*}\sigma _{3}\psi } the relativistic scalar and ${\displaystyle J^{\mu }=(\psi ^{*}\psi ,\psi ^{*}\sigma _{1}\psi ,\psi ^{*}\sigma _{2}\psi )}${\displaystyle J^{\mu }=(\psi ^{*}\psi ,\psi ^{*}\sigma _{1}\psi ,\psi ^{*}\sigma _{2}\psi )} the charge-current density. The relation follows from the identity ${\displaystyle (\psi ^{*}\sigma _{1}\psi )^{2}+(\psi ^{*}\sigma _{2}\psi )^{2}+(\psi ^{*}\sigma _{3}\psi )^{2}=(\psi ^{*}\psi )^{2}}${\displaystyle (\psi ^{*}\sigma _{1}\psi )^{2}+(\psi ^{*}\sigma _{2}\psi )^{2}+(\psi ^{*}\sigma _{3}\psi )^{2}=(\psi ^{*}\psi )^{2}}, for any ${\displaystyle \psi \in \mathbb {C} ^{2}}${\displaystyle \psi \in \mathbb {C} ^{2}}.^[7]

• Dirac equation
• Gross–Neveu model
• Nonlinear Dirac equation
• Thirring model

• Quantum field theory

• CS1 maint: multiple names: authors list
• Articles with short description
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