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 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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