Schwinger model

In physics, the Schwinger model, named after Julian Schwinger, is the model[1] describing 1+1D (1 spatial dimension + time) Lorentzian quantum electrodynamics which includes electrons, coupled to photons.

The model defines the usual QED Lagrangian

${\displaystyle {\mathcal {L}}=-{\frac {1}{4g^{2}}}F_{\mu \nu }F^{\mu \nu }+{\bar {\psi }}(i\gamma ^{\mu }D_{\mu }-m)\psi }$

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

This model exhibits confinement of the fermions and as such, is a toy model for QCD. A handwaving argument why this is so is because in two dimensions, classically, the potential between two charged particles goes linearly as ${\displaystyle r}$, instead of ${\displaystyle 1/r}$ 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.[2][3]