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

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?

• Another way to see problems is by noting that EM field has only transverse polarizations. But there are no transverse directions in 1+1. Aug 10, 2011 at 10:24
• @Marek: Yes, this is a good point.
– MKO
Aug 10, 2011 at 11:02

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.)