Schwinger model

In quantum field theory, the Schwinger model is a model describing 1+1D (time + 1 spatial dimension) quantum electrodynamics (QED) which includes electrons, coupled to photons. It is named after named after Julian Schwinger who developed it in 1962.<ref>Template:Cite journal</ref>

The model defines the usual QED Lagrangian density

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

over a spacetime with one spatial dimension and one temporal dimension. Where <math> F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu</math> is the photon field strength with symmetry group <math> \mathrm{U}(1) </math> (unitary group), <math> D_\mu = \partial_\mu - iA_\mu </math> is the gauge covariant derivative, <math> \psi </math> is the fermion spinor, <math> m </math> is the fermion mass and <math> \gamma^0, \gamma^1 </math> form the two-dimensional representation of the Clifford algebra.

This model exhibits confinement of the fermions and as such, is a toy model for quantum chromodynamics. A handwaving argument why this is so is because in two dimensions, classically, the potential between two charged particles goes linearly as <math>r</math>, instead of <math>1/r</math> 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.<ref>Template:Cite journal</ref><ref>Template:Cite journal </ref>

References

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