Schwinger model explained

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 defines the usual QED Lagrangian

l{L} = - (1)/(4 g ^{2}) F _{\mu} F ^{\mu} + \bar{\psi} (i \gamma ^{\mu} D _{\mu} - m) \psi

over a spacetime with one spatial dimension and one temporal dimension. Where

F _{\mu} = \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.^{[2] [3]}

Notes and References

1. Schwinger . Julian . Gauge Invariance and Mass. II . Physical Review . Physical Review, Volume 128 . 1962 . 128 . 5 . 2425–2429 . 10.1103/PhysRev.128.2425 . 1962PhRv..128.2425S.
2. Schwinger . Julian . The Theory of Quantized Fields I . Physical Review . Physical Review, Volume 82 . 1951 . 82 . 6 . 914–927 . 10.1103/PhysRev.82.914 . 1951PhRv...82..914S. 121971249 .
3. Schwinger . Julian . The Theory of Quantized Fields II . Physical Review . Physical Review, Volume 91 . 1953 . 91 . 3 . 713–728 . 10.1103/PhysRev.91.713 . 1953PhRv...91..713S.