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Schwinger model

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]

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