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

and re-introduced and investigated in 1970 by Mario Soler as a toy model of self-interacting electron.

This model is described by the Lagrangian density

:$\mathcal=\overline \left(i\partial\!\!\!/-m \right) \psi + \frac\left(\overline \psi\right)^2$

where $g$ is the coupling constant,
$\partial\!\!\!/=\sum_^3\gamma^\mu\frac$ in the Feynman slash notations, $\overline=\psi^*\gamma^0$.
Here $\gamma^\mu$, $0\le\mu\le 3$, are Dirac gamma matrices.

The corresponding equation can be written as

:$i\frac\psi=-i\sum_^\alpha^j\frac\psi+m\beta\psi-g(\overline \psi)\beta\psi$,

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.

Generalizations

A commonly considered generalization is

:$\mathcal=\overline \left(i\partial\!\!\!/-m \right) \psi + g\frac$

with $k>0$, or even

:$\mathcal=\overline \left(i\partial\!\!\!/-m \right) \psi + F\left(\overline \psi\right)$,

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.

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
$\phi(x)e^,$
where $\phi$ is localized (becomes small when $x$ is large)
and $\omega$ is a real number.

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 $\psi\in\Complex^2$.

See also

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

References

{{Quantum field theories

Quantum field theory