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Soler model
Type of 3+1 dimensional quantum field theory
Type of 3+1 dimensional quantum field theory
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{{ cite journal
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.{{cite journal |name-list-style=amp |title=Quantization of the localized solutions in two-dimensional field theories of massive fermions
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 k0, 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.{{cite journal
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.{{cite journal |name-list-style=amp |title=Existence of localized solutions for a classical nonlinear Dirac field
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.{{cite journal |name-list-style=amp |article-number=214101
References
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