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

:$\mathcal = - \fracF_F^ + \bar (i \gamma^\mu D_\mu -m) \psi$

over a spacetime with one spatial dimension and one temporal dimension. Where $F_ = \partial_\mu A_\nu - \partial_\nu A_\mu$ 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.

References

Quantum field theory
Quantum electrodynamics
Exactly solvable models
Quantum chromodynamics

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