# $S$-matrix in QED in 2d space-time

Let us consider the QED in 2-dimensional space-time. The $S$-matrix can be interpreted as a unitary operator between spaces of In- and Out-states.

In the classical 4d QED In- and Out-states are collections of electrons, positrons and polarized (in exactly two ways) photons.

QUESTION. What happens with photons in 2d space-time? In what form they should be present in In- and Out-states? More precisely, how many polarizations they should have?

My guess is that in the 2d case photons have no physical polarizations at all. But does that mean that there is no photons in In- and Out-states, which consist only of collections of fermions? Or the $S$-matrix is not well defined in 2d?

## 1 Answer

I think, what you first want to make sure is, that you have elektromagnetism in 2 dimensions. This essay[1] shows a way to obtain it, and also shows, that there are only two degrees of freedom, one for the electric and one for the magnetic field. (So in there is no room for a polarization).

Then your "in" and "out" states are presumably fock states, so you would need to quantize the theory first. I don't know whether anyone has bothered to work that out.

edit: according to [1] the photon is then described by a single Klein-Gordon field $A$, i.e. the analog of the vector potential. The modes of this field will then of course occur in the fock states, so one degree of freedom as in the quantisation of the Klein-Gordon field. But since $A$ is a scalar there are no polarization vectors.

[1] Nicholas Wheeler, “Electrodynamics” in 2-dimensional Spacetime, (Notes from a Reed College Physics Seminar, 1997)

• Minor comment to the post (v1): Please consider to mention explicitly author, title, etc. of link, so it is possible to reconstruct link in case of link rot. – Qmechanic Aug 3 '16 at 22:23
• This is exactly my point: in 2d there is no room for a polarization. Hence the question is whether the Fock states should include photons in any form. – MKO Aug 4 '16 at 13:51