Schwinger model

Example

The Schwinger model is quantum electrodynamics in two spacetime dimensions with one massless Dirac fermion. Work on $\R^{1,1}$ with signature $(+,-)$. Let $\gamma^0, \gamma^1$ be $2 \times 2$ gamma matrices with $\gamma^\mu\gamma^\nu + \gamma^\nu\gamma^\mu = 2\eta^{\mu\nu} I_2$. Let $\psi$ be a field with values in $\C^2$, and let $A_\mu$ be a real gauge field. Its Lagrangian density is $$\mathcal{L} = \bar\psi \, i\gamma^\mu (\partial_\mu + i e A_\mu) \psi - \frac{1}{4} F_{\mu\nu} F^{\mu\nu}.$$ In two dimensions the coupling $e$ has the dimension of a mass.

Check of the main fact: the model can be solved exactly. Its spectrum consists of a single free boson of mass $$m = \frac{e}{\sqrt{\pi}}$$ in units with $\hbar = c = 1$. The classical theory has a massless photon and a massless fermion, and the axial current $j^\mu_5 = \bar\psi\gamma^\mu\gamma^5\psi$ is conserved classically. In the quantum theory the axial current has an anomaly, $\partial_\mu j^\mu_5 = \frac{e}{\pi} \epsilon^{\mu\nu} \partial_\mu A_\nu$ up to sign convention, and this anomaly is what gives the photon its mass. In the language of bosonization, the fermion is equivalent to a free scalar, and the gauge field gives that scalar the mass $e/\sqrt\pi$.