Quantisation of $\theta$-expanded non-commutative QED

We analyse two new versions of -expanded non-commutative quantum electrodynamics up to first order in and first loop order. In the first version we expand the bosonic sector using the Seiberg-Witten map, leaving the fermions unexpanded. In the second version we leave both bosons and fermions unexpanded. The analysis shows that the Seiberg-Witten map is a field redefinition at first order in . However, at higher order in the Seiberg-Witten map cannot be regarded as a field redefinition. We find that the initial action of any -expanded massless non-commutative QED must include one extra term proportional to which we identify by loop calculations. Received: 3 July 2002 / Published online: 7 October 2002 RID="a" ID="a" e-mail: jesper@hep.itp.tuwien.ac.at Work supported by The Danish Research Agency. RID="b" ID="b" e-mail: raimar.wulkenhaar@mis.mpg.de Schloe?mann fellow