In QED the field strength $F_{\mu\nu}$ is gauge-invariant. This is reasonable since its components are physical fields $\vec{E}$, $\vec{B}$, so it doesn't matter in which gauge you express it.

However, in non-abelian gauge theories, $F_{\mu\nu}$ is not gauge invariant $F_{\mu\nu}'=VF_{\mu\nu}V^{-1}$. A gauge-invariant object would be $\text{Tr}(F_{\mu\nu})$, but the number of independent components of this object has nothing to do with the gauge group. Is there any notion of "physical fields" (gauge-invariant,then) in non-abelian gauge theories?

  • $\begingroup$ @CosmasZachos It's better now. You're right, there's a lot I have to read about. $\endgroup$
    – AFG
    Jun 15, 2022 at 15:31
  • 2
    $\begingroup$ The quantitities $F^a_{\mu\nu}$ are not gauge invariants, but when we consider the "gauge charges" $Q_a$ then the quantity $Q_a F^a_{\mu\nu}$ IS gauge invariant. In electromagnetism also $qF_{\mu\nu}$ is gauge invariant, because both electrical charge $q$ and the field $F_{\mu\nu}$ are gauge invariant by themselves. $\endgroup$
    – Davius
    Jul 23, 2022 at 17:34

1 Answer 1


In confining theories you only observe gauge singlets, as in QCD (hadrons), and you might think of the gauge fields (gluons) as "merely" brilliant indirect calculational devices, leading up to phenomena involving gauge singlet observables--but just try to describe gluon jets without nonsinglet gauge fields!

But what about SSB theories such as the EW interactions?

That is a very long story you have to read about... the "principle of complementarity" of Susskind. The confining phase and the Higgs phase are not as distinct as they appear... So you may define singlet composite operators, hence physical and gauge invariant, corresponding 1-to-1 to the nonsinglet spectrum particles of the SM.

See Sec 3 of T Banks and E Rabinovici (1979), "Finite-temperature behavior of the lattice abelian Higgs model", Nucl Phys B160 (2), pp 349-379 for a quick-and-dirty illustration. (I am sure there are nicer reviews out there, but I haven't paid due diligence to find them. The recondite technical underpinning is in E Fradkin and S Shenker (1979) "Phase diagrams of lattice gauge theories with Higgs fields", Phys Rev D19 (12), p3682.)

Basically, there is a "theoretical" cottage industry out there, fussing the conceptual quandaries you might fear you have stumbled on, Elitzur's Theorem, etc, but, in practical terms, the gauge-fixing monkey-see-monkey-do of standard QFT texts is still the best way to handle the SM and keep track of symmetries and degrees of freedom!

  • $\begingroup$ This is a popular overview. $\endgroup$ Mar 11, 2023 at 16:43

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