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

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

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

The corresponding equation can be written as

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

where ${\displaystyle \alpha ^{j}}$, ${\displaystyle 1\leq j\leq 3}$, and ${\displaystyle \beta }$ 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

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

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

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

where ${\displaystyle F}$ 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. ^[5]

Renormalizability

The Soler model is renormalizable by the power counting for ${\displaystyle k=1}$ and in one dimension only, and non-renormalizable for higher values of ${\displaystyle k}$ and in higher dimensions.

Solitary wave solutions

The Soler model admits solitary wave solutions of the form ${\displaystyle \phi (x)e^{-i\omega t},}$ where ${\displaystyle \phi }$ is localized (becomes small when ${\displaystyle x}$ is large) and ${\displaystyle \omega }$ 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 ${\displaystyle ({\bar {\psi }}\psi )^{2}=J_{\mu }J^{\mu }}$, with ${\displaystyle {\bar {\psi }}\psi =\psi ^{*}\sigma _{3}\psi }$ the relativistic scalar and ${\displaystyle J^{\mu }=(\psi ^{*}\psi ,\psi ^{*}\sigma _{1}\psi ,\psi ^{*}\sigma _{2}\psi )}$ the charge-current density. The relation follows from the identity ${\displaystyle (\psi ^{*}\sigma _{1}\psi )^{2}+(\psi ^{*}\sigma _{2}\psi )^{2}+(\psi ^{*}\sigma _{3}\psi )^{2}=(\psi ^{*}\psi )^{2}}$, for any ${\displaystyle \psi \in \mathbb {C} ^{2}}$. ^[7]

See also

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

References

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.
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. S2CID   121750127.
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. S2CID   121018463.
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. PMID   27284659. S2CID   15719805.