Schwinger model

In quantum field theory, the Schwinger model is a model describing 1+1D (time + 1 spatial dimension) quantum electrodynamics (QED) which includes electrons, coupled to photons. It is named after named after Julian Schwinger who developed it in 1962.

The model defines the usual QED Lagrangian density

${ℒ}=-\frac{1}{4g^2}{F}_{\mu \nu }{F}^{\mu \nu }+\overline{\psi }(i{\gamma }^{\mu }{D}_{\mu }-m)\psi$

over a spacetime with one spatial dimension and one temporal dimension. Where $F_{\mu \nu }={\partial }_{\mu }{A}_{\nu }-{\partial }_{\nu }{A}_{\mu }$ is the photon field strength with symmetry group $\mathrm{U}(1)$ (unitary group), $D_{\mu }={\partial }_{\mu }-i{A}_{\mu }$ is the gauge covariant derivative, $\psi$ is the fermion spinor, $m$ is the fermion mass and ${\gamma }^0,{\gamma }^1$ form the two-dimensional representation of the Clifford algebra.

This model exhibits confinement of the fermions and as such, is a toy model for quantum chromodynamics. A handwaving argument why this is so is because in two dimensions, classically, the potential between two charged particles goes linearly as $r$, instead of $1/r$ in 4 dimensions, 3 spatial, 1 time. This model also exhibits a spontaneous symmetry breaking of the U(1) symmetry due to a chiral condensate due to a pool of instantons. The photon in this model becomes a massive particle at low temperatures. This model can be solved exactly and is used as a toy model for other more complex theories.

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