Schwinger model

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

The model defines the usual QED Lagrangian


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

over a spacetime with one spatial dimension and one temporal dimension. Where $$ F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu$$ is the $$ U(1) $$ photon field strength, $$ D_\mu = \partial_\mu - iA_\mu $$ 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 QCD. 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.