A Note on Polarization Vectors in Quantum Electrodynamics

A photon of momentum k can have only two polarization states, not three. Equivalently, one can say that the magnetic vector potential A must be divergence-free in the Coulomb gauge. These facts are normally taken into account in QED by introducing two polarization vectors (k) with {1,2}, which are orthogonal to the wave-vector k. These vectors must be very discontinuous functions of k and, consequently, their Fourier transforms have bad decay properties. Since these vectors have no physical significance there must be a way to eliminate them and their bad decay properties from the theory. We propose such a way here.Dedicated to Freeman Dyson on the occasion of his eightieth birthdayWork partially supported by U.S. National Science Foundation grant PHY 01-39984.Work partially supported by U.S. National Science Foundation grant DMS 03-00349. 2003 by the authors. This paper may be reproduced, in its entirety, for non-commercial purposes.Acknowledgement We thank Herbert Spohn and Jakob Yngvason for many useful discussions about this work. After completing this work and submitting it to CMP it was brought to our attention that the last section, 10.3, of the paper 2] by Fröhlich, Griesemer and Schlein contains the same idea in the context of Rayleigh scattering in the dipole approximation. The three-component concept enables them to extend the results in the rest of their paper from scalar fields to vector fields, but, as we see here, the concept works in much greater generality.