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 $\gamma ^{\mu }$ , $0\leq \mu \leq 3$ , 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.^[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]

References

↑ 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.
↑ 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.
↑ 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.

Four-fermion interactions

BCS theory Fermi's interaction Gross–Neveu model Hubbard model Luttinger liquid Nambu–Jona-Lasinio model Thirring model Top quark condensate

Book:Four-fermion interactions

Quantum field theories

Standard	Chern–Simons model Chiral model Kondo model Conformal field theory Local quantum field theory Noncommutative quantum field theory Non-linear sigma model Quartic interaction Quantum chromodynamics Quantum electrodynamics Quantum hadrodynamics Quantum Yang–Mills theory Schwinger model sine-Gordon Standard Model String theory Toda field theory Topological quantum field theory Wess–Zumino model Wess–Zumino–Witten model Yang–Mills theory Yang–Mills–Higgs model Yukawa model Thirring–Wess model

Four-fermion interactions	Thirring model Gross–Neveu model Nambu–Jona-Lasinio model

Related	History of quantum field theory Axiomatic quantum field theory Quantum field theory in curved spacetime

This article is issued from Wikipedia - version of the 6/5/2016. The text is available under the Creative Commons Attribution/Share Alike but additional terms may apply for the media files.