How to quantize the free electro-magnetic field in 2d?

Question

I am wondering how one can quantize the free electro-magnetic field in the two dimensional space-time. The standard method of fixing the Coulomb gauge in 4d does not seem to generalize immediately to 2d.

If one tries to generalize it directly then, as in 4d, after a gauge transformation one may assume that the scalar potential vanishes. Thus $A_\mu=(0,A_1)$. Then the Maxwell equations imply easily that $$\frac{\partial A_1}{\partial x^0}\equiv C=const$$ where $x^0$ is the time coordinate. Hence $A_1(x)=C x^0+h(x^1)$. Now one can apply a gauge transformation such that $h$ will vanish: $A_\mu(x)=(0,Cx^0)$. Thus there is no degrees of freedom to quantize!!

Another problem is that the expression $\frac{1}{q^2+i0}$ which appears in the free photon propagator in 4d, is not well defined in 2d for purely mathematical reasons if I understand correctly (there is no such generalized function: the problem in at $q=0$).

Is there a treatment of this in literature?

Actually the above question was motivated by my attempt to read J. Schwinger's paper "Gauge invariance and mass, II", Phys. Review, 128, number 5 (1962). There he studies QED in 2d (so called Schwinger's model). Is there a more modern and/or detailed exposition of this paper?

Answer 1

2d Electromagnetism is a special case of 2d Yang-Mills theory, which is the trivial theory on flat $\mathbb{R}^2$. However, this theory becomes entertaining when the spacetime has non-trivial topology -- e.g., is a cylinder or a Riemann surface -- because then you get a countable number of degrees of freedom. The gauge field can have non-trivial holonomies around the non-contractible loops, like in the Aharonov-Bohm effect. (The Hamiltonian is non-trivial on a cylinder of radius R, but there's a mass gap proportional to R; as you decompactify the cylinder, these degrees of freedom become arbitrarily massive.)

One nice place to read about this is Witten's "On Quantum Gauge Theories in Two Dimensions". Also, Chapter 11 of Volume 2 of the IAS QFT lectures.