Soler model explained

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

l{L}=\overline{\psi} \left(i\partial/ - m \right) \psi + (g)/(2) \left(\overline{\psi} \psi\right) ^{2}

where

g is the coupling constant,\partial/ = \sum ^{3 \gamma} _{\mu =0} ^{\mu (\partial)/(\partial x ^{\mu})} in the Feynman slash notations, \overline{\psi} = \psi ^{*\gamma} ^{0} .Here \gamma ^{\mu} , 0 \le\mu\le 3 , are Dirac gamma matrices.

The corresponding equation can be written as

i (\partial)/(\partial t) \psi =- i\sum ^{3} _{j =1} \alpha ^{j (\partial)/(\partial x ^{j})} \psi + m\beta\psi - g(\overline{\psi} \psi)\beta\psi ,

where

\alpha ^{j} , 1 \le j\le 3 ,and \beta are the Dirac matrices.In one dimension,this model is known as the massive Gross–Neveu model.^{[3] [4]}

Generalizations

A commonly considered generalization is

l{L}=\overline{\psi} \left(i\partial/ - m \right) \psi + g (\left(\overline{\psi)/(\psi\right)) ^{k+1} }

with

k> 0 , or evenl{L}=\overline{\psi} \left(i\partial/ - m \right) \psi + F\left(\overline{\psi} \psi\right) ,

where

F is a smooth function.

Features

Internal symmetry

Besides the unitary symmetry U(1),in dimensions 1, 2, and 3the equation has SU(1,1) global internal symmetry.^[5]

Renormalizability

The Soler model is renormalizable by the power counting for

k =1 and in one dimension only,and non-renormalizable for higher values of k and in higher dimensions.

Solitary wave solutions

The Soler model admits solitary wave solutionsof the form

\phi(x)e ^{-i\omega} , where \phi is localized (becomes small when x is large)and \omega is a real number.^[6]

Reduction to the massive Thirring model

In spatial dimension 2, the Soler model coincides with the massive Thirring model,due to the relation

(\bar\psi\psi) ^{2= J} _{\mu} J ^{\mu} ,with\bar\psi\psi = \psi ^{*\sigma} _{3 \psi} the relativistic scalarandJ ^{\mu=(\psi} ^{*\psi,\psi} ^{*\sigma} ^{2 \psi)} the charge-current density.The relation follows from the identity(\psi ^{2+ (\psi} _{1 \psi)} ^{2+ (\psi} ^{2 \psi)} ^{2 = (\psi} ^{3 \psi)} ^{*\psi)} ^{2} ,for any \psi\in\Complex ^{2} .^[7]

See also

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

References

1. Dmitri Ivanenko. Notes to the theory of interaction via particles. Zh. Eksp. Teor. Fiz.. 8. 260–266. 1938.
2. Mario Soler. Classical, Stable, Nonlinear Spinor Field with Positive Rest Energy. Phys. Rev. D. 1. 10. 2766–2769. 1970. 10.1103/PhysRevD.1.2766. 1970PhRvD...1.2766S .
3. Gross, David J. and Neveu, André. Dynamical symmetry breaking in asymptotically free field theories. Phys. Rev. D. 10. 10. 3235–3253. 1974. 10.1103/PhysRevD.10.3235. 1974PhRvD..10.3235G .
4. S.Y. Lee . A. Gavrielides . amp . Quantization of the localized solutions in two-dimensional field theories of massive fermions. Phys. Rev. D. 12. 12. 3880–3886. 1975. 10.1103/PhysRevD.12.3880. 1975PhRvD..12.3880L .
5. Galindo, A.. A remarkable invariance of classical Dirac Lagrangians. Lettere al Nuovo Cimento. 20. 6. 210–212. 1977. 10.1007/BF02785129. 121750127.
6. Thierry Cazenave . Luis Vàzquez. amp . Existence of localized solutions for a classical nonlinear Dirac field. Comm. Math. Phys.. 105. 1. 35–47. 1986. 10.1007/BF01212340. 1986CMaPh.105...35C . 121018463.
7. J. Cuevas-Maraver. P.G. Kevrekidis. A. Saxena. A. Comech. R. Lan. amp. Stability of solitary waves and vortices in a 2D nonlinear Dirac model. Phys. Rev. Lett.. 116. 21. 214101. 2016. 10.1103/PhysRevLett.116.214101 . 27284659. 1512.03973. 2016PhRvL.116u4101C. 15719805.