The state of Indefinite metric in Quantum Electrodynamics

Question

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.

Answer 1

Warning: Students, stay away from antiquities. The aim to learn is to survive.

There's usually reasons that old materials are not cited. In this case, Nakanishi is obsolete.

Nakanishi is not wrong, but unsatisfactory in two points. First, what happened to the gauge symmetry? And it's limited to QED, and not applicable to non-Abelian symmetries.

The answer is known since late 70s, given by BRST. The gauge symmetry is conserved even after gauge fixing, where the seemingly lost symmetry lies in zero-norm or ghosts, and they're unphysical i.e. harmless. This also means the Lorentz covariance is preserved too in non-covariant gauges. You know, you definitely need ghosts (negative-norm states) for practical calculation for non-Abelian gauges, unlike QED where ghosts are detached.

I recommend Weinberg's QFT book, vol 2, sec 15.7 for BRST introduction. Even if you don't need non-Abelian gauge, the cited section of Weinberg is not difficult at all. (There you encounter the structure constant $C^{\alpha\beta\gamma}$. You can safely think of it as the Levi-Civita symbol $\epsilon^{ijk}$, and $t_{\alpha}$ as Pauli matrices.) Or rather, you'll be stricken to find how easy it is in the functional quantization, compared to Nakanishi's machinery, (Fourier expansion of the field and heavy use of $\delta(x)$ and its derivative.) - BRST wins also in pragmatism.

As a free material, see for example Sredinicki's QFT book, sec 74, but prerequisites sections may be a bit more cumbersome than Weinberg. Peskin & Schroeder doesn't help.