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

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

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

The corresponding equation can be written as

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 $\alpha ^{j}$ , $1\leq j\leq 3$ , 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

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

with $k>0$ , or even

{\mathcal {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 3 the 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 solutions of the form $\phi (x)e^{-i\omega t},$ 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 scalar and $J^{\mu }=(\psi ^{*}\psi ,\psi ^{*}\sigma _{1}\psi ,\psi ^{*}\sigma _{2}\psi )$ the charge-current density. The relation follows from the identity $(\psi ^{*}\sigma _{1}\psi )^{2}+(\psi ^{*}\sigma _{2}\psi )^{2}+(\psi ^{*}\sigma _{3}\psi )^{2}=(\psi ^{*}\psi )^{2}$ , for any $\psi \in \mathbb {C} ^{2}$ .[7]

References

Dmitri Ivanenko (1938). "Notes to the theory of interaction via particles" (PDF). Zh. Eksp. Teor. Fiz. 8: 260–266.
Mario Soler (1970). "Classical, Stable, Nonlinear Spinor Field with Positive Rest Energy". Phys. Rev. D. 1 (10): 2766–2769. Bibcode:1970PhRvD...1.2766S. doi:10.1103/PhysRevD.1.2766.
Gross, David J. and Neveu, André (1974). "Dynamical symmetry breaking in asymptotically free field theories". Phys. Rev. D. 10 (10): 3235–3253. Bibcode:1974PhRvD..10.3235G. doi:10.1103/PhysRevD.10.3235.CS1 maint: multiple names: authors list (link)
S.Y. Lee & A. Gavrielides (1975). "Quantization of the localized solutions in two-dimensional field theories of massive fermions". Phys. Rev. D. 12 (12): 3880–3886. Bibcode:1975PhRvD..12.3880L. doi:10.1103/PhysRevD.12.3880.
Galindo, A. (1977). "A remarkable invariance of classical Dirac Lagrangians". Lettere al Nuovo Cimento. 20 (6): 210–212. doi:10.1007/BF02785129.
Thierry Cazenave & Luis Vàzquez (1986). "Existence of localized solutions for a classical nonlinear Dirac field". Comm. Math. Phys. 105 (1): 35–47. Bibcode:1986CMaPh.105...35C. doi:10.1007/BF01212340.
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:1512.03973. Bibcode:2016PhRvL.116u4101C. doi:10.1103/PhysRevLett.116.214101.

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[1] Dmitri Ivanenko (1938). "Notes to the theory of interaction via particles" (PDF). Zh. Eksp. Teor. Fiz. 8: 260–266.

[2] Mario Soler (1970). "Classical, Stable, Nonlinear Spinor Field with Positive Rest Energy". Phys. Rev. D. 1 (10): 2766–2769. Bibcode:1970PhRvD...1.2766S. doi:10.1103/PhysRevD.1.2766.

[3] Gross, David J. and Neveu, André (1974). "Dynamical symmetry breaking in asymptotically free field theories". Phys. Rev. D. 10 (10): 3235–3253. Bibcode:1974PhRvD..10.3235G. doi:10.1103/PhysRevD.10.3235.CS1 maint: multiple names: authors list (link)

[4] S.Y. Lee & A. Gavrielides (1975). "Quantization of the localized solutions in two-dimensional field theories of massive fermions". Phys. Rev. D. 12 (12): 3880–3886. Bibcode:1975PhRvD..12.3880L. doi:10.1103/PhysRevD.12.3880.

[5] Galindo, A. (1977). "A remarkable invariance of classical Dirac Lagrangians". Lettere al Nuovo Cimento. 20 (6): 210–212. doi:10.1007/BF02785129.

[6] Thierry Cazenave & Luis Vàzquez (1986). "Existence of localized solutions for a classical nonlinear Dirac field". Comm. Math. Phys. 105 (1): 35–47. Bibcode:1986CMaPh.105...35C. doi:10.1007/BF01212340.

[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:1512.03973. Bibcode:2016PhRvL.116u4101C. doi:10.1103/PhysRevLett.116.214101.