Schwinger model

 ( Quantum Field Theory ) 

In quantum field theory, the Schwinger model is a model describing 1+1D (time + 1 spatial dimension) quantum electrodynamics (QED) which includes Dirac equation, coupled to photons. It is named after named after Julian Schwinger who developed it in 1962.

The model defines the usual QED Lagrangian density

$\backslash mathcal\{L\}\; =\; -\; \backslash frac\{1\}\{4g^2\}F\_\{\backslash mu\; \backslash nu\}F^\{\backslash mu\; \backslash nu\}\; +\; \backslash bar\{\backslash psi\}\; (i\; \backslash gamma^\backslash mu\; D\_\backslash mu\; -m)\; \backslash psi$

over a spacetime with one spatial dimension and one temporal dimension. Where $F\_\{\backslash mu\; \backslash nu\}\; =\; \backslash partial\_\backslash mu\; A\_\backslash nu\; -\; \backslash partial\_\backslash nu\; A\_\backslash mu$ is the photon field strength with symmetry group $\backslash mathrm\{U\}(1)$ (unitary group), $D\_\backslash mu\; =\; \backslash partial\_\backslash mu\; -\; iA\_\backslash mu$ is the gauge covariant derivative, $\backslash psi$ is the Dirac spinor, $m$ is the fermion mass and $\backslash gamma^0,\; \backslash gamma^1$ form the two-dimensional representation of the Clifford algebra.

This model exhibits confinement of the fermions and as such, is a toy model for quantum chromodynamics. 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 . 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.