I faced difficulties to grasp why indefinite metric is introduced from no where in QED, after searching internet I found that this is a problem in QED, because one needs it to preserve theory's covariance, and to solve ultraviolet divergence problem, but in the same time it raise other different problems such as negative probabilities, and even from mathematical point of view that it's can't be introduced a proper topology for a Hilbert space with such metric. (those point I get 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)
Thus those information may be old, I would like to know the official (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.