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 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. 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]}

## References

**^**Schwinger, Julian (1962). "Gauge Invariance and Mass. II".*Physical Review*. Physical Review, Volume 128.**128**(5): 2425–2429. Bibcode:1962PhRv..128.2425S. doi:10.1103/PhysRev.128.2425.**^**Schwinger, Julian (1951). "The Theory of Quantized Fields I".*Physical Review*. Physical Review, Volume 82.**82**(6): 914–927. Bibcode:1951PhRv...82..914S. doi:10.1103/PhysRev.82.914.**^**Schwinger, Julian (1953). "The Theory of Quantized Fields II".*Physical Review*. Physical Review, Volume 91.**91**(3): 713–728. Bibcode:1953PhRv...91..713S. doi:10.1103/PhysRev.91.713.