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In SM, both for the gauge field $A_{\mu}$ associated with photon and for the gauge fields $W^{j}_{\mu}$, to restore the invariance of the lagrangian after the changing of the global transformation to local gauge transformation you introduce under the assumption of minimal coupling between fermion fields and gauge boson fields the covariant derivative $$D_{\mu}=\partial_{\mu}+iqA_{\mu}$$ $$D_{\mu}=\partial_{\mu}-ig\frac{\sigma_j}{2}W^j_{\mu}$$

I read on Wikipedia that minimal coupling between fields assume that only the charge distribution and not higher multipole moments is involved in the interaction.

  1. Maybe I have gaps in classic electromagnetism but I don't know intuitively what this means?

  2. What is the physical assumption behind the minimal coupling hypothesis and how is it verifiable experimentally?

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Stefano Barone
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    Possibly this answer is of relevance; non minimal-couplings typically involve magnetic moment g s of the Pauli type, i.e. with values different than the Dirac value (minimal coupling). – Cosmas Zachos Feb 01 '20 at 19:59
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    "minimal coupling" means absence of non-renormalizable terms such as mass dimension 5 terms delineated in the link provided by Cosmas Zachos above and mass dimension 6 terms here: https://physics.stackexchange.com/questions/513424/mass-dimension-6-qed-lagrangian/513553#513553 – MadMax Feb 03 '20 at 15:10

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For a point-like charge (electron) in an external field $A_{\mu}^{\text{ext}}$ this works to certain limitations (Classic Electrodynamics = CED), but it fails even in CED when one applies this way of interaction to the "self-action" $A_{\mu}$.

In SM people do the same thing by analogy and they encounter similar "catastrophes". "Forgetting" these difficulties is a regrettable way of teaching today.

Vladimir Kalitvianski
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  • Thank you very much for your interest, however, your answer does not answer either question 1 or question 2. Also, I don't think that all QFT books start from a completely wrong hypothesis, you could go deeper. – Stefano Barone Feb 01 '20 at 19:11
  • Yes, I have answerred correctly to both your questions. You could learn the problems (catastrophes) encountered by physicists before asking such questions. – Vladimir Kalitvianski Feb 02 '20 at 07:45