I am facing difficulties to grasp why indefinite metric is introduced from nowhere in QED. After searching internet, I found that this is a problem in QED, because one needs it to preserve theory's covariance, and also to solve ultraviolet divergence problem.
But in the same time it raise other different problems such as negative probabilities. Even from pure mathematical point of view, it is can't be introduce a proper topology for a Hilbert space with such metric. (those points I got from N. Nakanishi: Indefinite-Metric Quantum Field Theory. Progress of Theoretical Physics Supplement No.51 (1972) pp. 1-95, doi:10.1143/PTPS.51.1)
However, because those claims may be old, I would like to know the main stream position of this problem, and if it considered as solved, especially that there is some papers that refuses this kind of metric and some usual books on QED doesn't mention it all what makes me even more confused.