Schwinger model
In physics, the Schwinger model, named after Julian Schwinger, is the model describing 2D Euclidean quantum electrodynamics with a Dirac fermion. This model exhibits a spontaneous symmetry breaking of the U 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.
The model has a Lagrangian
Where is the photon field strength, is the gauge covariant derivative, is the fermion spinor, is the fermion mass and 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, instead of in 4 dimensions, 3 spatial, 1 time.
