How comes negative norm is such a big thing for fermions that Dirac equation was chosen over KG, while the same kind of problem for photons is just ok? In fact, many books in QED do not even mention that the norm has to be indefinite to preserve covariance.


Those are two very different situations.

Dirac equation was chosen over KG only in the early days of QFT, when people haven't discovered second quantization yet. The problem was the apparent negative probability of observing a KG particle. Nowadays it is generally accepted that both KG and Dirac give consistent QFTs of particles with spins $0$ and $1/2$ respectively. All states in the 2nd-quantized theory are positive norm.

The negative norm in gauge theories is entirely different. It is a norm of a 2nd-quantized state, and cannot be given a physical interpretation. It is also not real, meaning that we don't actually get negative-norm states in the gauge theory where we have correctly accounted for the constraint equations (you don't need to justify Gupta-Bleuler by getting rid of the negative-norm states, it is just a quantum-mechanical way of imposing constraint equations. Those constraints are real, physical, exist in both classical and quantum theories and are crucial to the theory's dynamics. Negative-norm states are purely an artifact which shows up before the constraints are imposed. They don't show up in the actual theory.)

  • $\begingroup$ Interesting. Do you in general agree with the redefinition of the inner product in order to preserve covariance (e.g. Akhiezer, and Berestetskii. Quantum electrodynamics. No. AEC-TR-2876. 1986., Ch: "quantization of EM field")? The second quantized state, from what I understand, does not come out with indefinite norm, if one follows the original prescription. The quantization procedure had to be modified (solely for photon fields) to fit the Maxwell's equations. I am sorry if I am wrong, I am just testing my understanding. $\endgroup$ – MsTais May 4 '18 at 15:06
  • $\begingroup$ I feel like they hit the topological effect in quantization procedure and didn't find a better way out rather than to hide it into the norm... I am sorry for my ignorance-related skepticism... $\endgroup$ – MsTais May 4 '18 at 15:11
  • $\begingroup$ @MsTais Could you provide either a link to the pdf or a screenshot of the relevant page please? $\endgroup$ – Prof. Legolasov May 5 '18 at 6:41
  • $\begingroup$ I have added the link, but my version is russian. If it would be helpful, let me know and I will make a screenshot. $\endgroup$ – MsTais May 7 '18 at 15:59

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