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 1970 by Mario Soler^[1] 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 $\gamma^\mu$, $0\le\mu\le 3$, 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 $\alpha^j$, $1\le j\le 3$, and $\beta$ are the Dirac matrices. In one dimension, this model is known as the massive Gross-Neveu model.^{[2] [3]}

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

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.^[4]

See also

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

References

1. 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.
2. 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.
3. S.Y. Lee and 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.
4. Thierry Cazenave and 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.