# Soler model

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Type of 3+1 dimensional quantum field theory

The **soler model** is a [quantum field theory](/source/Quantum_field_theory) model of [Dirac fermions](/source/Dirac_fermion) interacting via [four fermion interactions](/source/Four_fermion_interaction) in 3 spatial and 1 time dimension. It was introduced in 1938 by [Dmitri Ivanenko](/source/Dmitri_Ivanenko) [1] and re-introduced and investigated in 1970 by [Mario Soler](https://en.wikipedia.org/w/index.php?title=Mario_Soler&action=edit&redlink=1)[2] as a [toy model](/source/Toy_model) of self-interacting [electron](/source/Electron).

This model is described by the [Lagrangian density](/source/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](/source/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](/source/Feynman_slash_notation), 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](/source/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](/source/Dirac_matrices). In one dimension, this model is known as the massive [Gross–Neveu model](/source/Gross%E2%80%93Neveu_model).[3][4]

## 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](/source/Internal_symmetry).[5]

### Renormalizability

The Soler model is [renormalizable](/source/Renormalizability) 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](/source/Soliton_wave) of the form ϕ ( x ) e − i ω t , {\displaystyle \phi (x)e^{-i\omega t},} where ϕ {\displaystyle \phi } is localized (becomes small when x {\displaystyle x} is large) and ω {\displaystyle \omega } is a [real number](/source/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 ( ψ ¯ ψ ) 2 = J μ J μ {\displaystyle ({\bar {\psi }}\psi )^{2}=J_{\mu }J^{\mu }} , with ψ ¯ ψ = ψ ∗ σ 3 ψ {\displaystyle {\bar {\psi }}\psi =\psi ^{*}\sigma _{3}\psi } the relativistic scalar and J μ = ( ψ ∗ ψ , ψ ∗ σ 1 ψ , ψ ∗ σ 2 ψ ) {\displaystyle J^{\mu }=(\psi ^{*}\psi ,\psi ^{*}\sigma _{1}\psi ,\psi ^{*}\sigma _{2}\psi )} the charge-current density. The relation follows from the identity ( ψ ∗ σ 1 ψ ) 2 + ( ψ ∗ σ 2 ψ ) 2 + ( ψ ∗ σ 3 ψ ) 2 = ( ψ ∗ ψ ) 2 {\displaystyle (\psi ^{*}\sigma _{1}\psi )^{2}+(\psi ^{*}\sigma _{2}\psi )^{2}+(\psi ^{*}\sigma _{3}\psi )^{2}=(\psi ^{*}\psi )^{2}} , for any ψ ∈ C 2 {\displaystyle \psi \in \mathbb {C} ^{2}} .[7]

## See also

- [Dirac equation](/source/Dirac_equation)

- [Gross–Neveu model](/source/Gross%E2%80%93Neveu_model)

- [Nonlinear Dirac equation](/source/Nonlinear_Dirac_equation)

- [Thirring model](/source/Thirring_model)

## References

1. **[^](#cite_ref-1)** Dmitri Ivanenko (1938). ["Notes to the theory of interaction via particles"](http://istina.msu.ru/media/publications/articles/079/c1a/1049479/Ivanenko-nonlinear.pdf) (PDF). *Zh. Eksp. Teor. Fiz*. **8**: 260–266.

1. **[^](#cite_ref-2)** Mario Soler (1970). ["Classical, Stable, Nonlinear Spinor Field with Positive Rest Energy"](http://prd.aps.org/abstract/PRD/v1/i10/p2766_1). *Phys. Rev. D*. **1** (10): 2766–2769. [Bibcode](/source/Bibcode_(identifier)):[1970PhRvD...1.2766S](https://ui.adsabs.harvard.edu/abs/1970PhRvD...1.2766S). [doi](/source/Doi_(identifier)):[10.1103/PhysRevD.1.2766](https://doi.org/10.1103%2FPhysRevD.1.2766).

1. **[^](#cite_ref-3)** [Gross, David J.](/source/David_Gross) and [Neveu, André](/source/Andr%C3%A9_Neveu) (1974). "Dynamical symmetry breaking in asymptotically free field theories". *Phys. Rev. D*. **10** (10): 3235–3253. [Bibcode](/source/Bibcode_(identifier)):[1974PhRvD..10.3235G](https://ui.adsabs.harvard.edu/abs/1974PhRvD..10.3235G). [doi](/source/Doi_(identifier)):[10.1103/PhysRevD.10.3235](https://doi.org/10.1103%2FPhysRevD.10.3235).{{[cite journal](https://en.wikipedia.org/wiki/Template:Cite_journal)}}: CS1 maint: multiple names: authors list ([link](https://en.wikipedia.org/wiki/Category:CS1_maint:_multiple_names:_authors_list))

1. **[^](#cite_ref-4)** S.Y. Lee & A. Gavrielides (1975). ["Quantization of the localized solutions in two-dimensional field theories of massive fermions"](http://prd.aps.org/abstract/PRD/v12/i12/p3880_1). *Phys. Rev. D*. **12** (12): 3880–3886. [Bibcode](/source/Bibcode_(identifier)):[1975PhRvD..12.3880L](https://ui.adsabs.harvard.edu/abs/1975PhRvD..12.3880L). [doi](/source/Doi_(identifier)):[10.1103/PhysRevD.12.3880](https://doi.org/10.1103%2FPhysRevD.12.3880).

1. **[^](#cite_ref-5)** Galindo, A. (1977). "A remarkable invariance of classical Dirac Lagrangians". *Lettere al Nuovo Cimento*. **20** (6): 210–212. [doi](/source/Doi_(identifier)):[10.1007/BF02785129](https://doi.org/10.1007%2FBF02785129). [S2CID](/source/S2CID_(identifier)) [121750127](https://api.semanticscholar.org/CorpusID:121750127).

1. **[^](#cite_ref-6)** Thierry Cazenave & Luis Vàzquez (1986). ["Existence of localized solutions for a classical nonlinear Dirac field"](http://projecteuclid.org/getRecord?id=euclid.cmp/1104115255). *Comm. Math. Phys*. **105** (1): 35–47. [Bibcode](/source/Bibcode_(identifier)):[1986CMaPh.105...35C](https://ui.adsabs.harvard.edu/abs/1986CMaPh.105...35C). [doi](/source/Doi_(identifier)):[10.1007/BF01212340](https://doi.org/10.1007%2FBF01212340). [S2CID](/source/S2CID_(identifier)) [121018463](https://api.semanticscholar.org/CorpusID:121018463).

1. **[^](#cite_ref-7)** J. Cuevas-Maraver; P.G. Kevrekidis; A. Saxena; A. Comech & R. Lan (2016). "Stability of solitary waves and vortices in a 2D nonlinear Dirac model". *Phys. Rev. Lett*. **116** (21) 214101. [arXiv](/source/ArXiv_(identifier)):[1512.03973](https://arxiv.org/abs/1512.03973). [Bibcode](/source/Bibcode_(identifier)):[2016PhRvL.116u4101C](https://ui.adsabs.harvard.edu/abs/2016PhRvL.116u4101C). [doi](/source/Doi_(identifier)):[10.1103/PhysRevLett.116.214101](https://doi.org/10.1103%2FPhysRevLett.116.214101). [PMID](/source/PMID_(identifier)) [27284659](https://pubmed.ncbi.nlm.nih.gov/27284659). [S2CID](/source/S2CID_(identifier)) [15719805](https://api.semanticscholar.org/CorpusID:15719805).

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Adapted from the Wikipedia article [Soler model](https://en.wikipedia.org/wiki/Soler_model) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Soler_model?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.