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In Special Relativity, is it mathematically possible for a local, gauge-invariant field theory to have only one vector field $A_\mu$ and to have $U(1)$ symmetry, assuming the vector field $A_\mu$ and/or its derivatives must appear in the field equations (i.e. this vector field $A_\mu$ cannot be eliminated from the field equations by using the antisymmetric field strength tensor, or by setting a source term to zero)?

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    $\begingroup$ I don't understand what "must appear in the field equations" is supposed to mean. Free electromagnetism is a U(1) gauge theory with only $A_\mu$ as a dynamical field. $\endgroup$ Commented Aug 20 at 21:44
  • $\begingroup$ That’s explained in the parenthetical. In standard classical electrodynamics, A_mu can be eliminated from the field equations by instead using the field strength tensor F_mu_nu. $\endgroup$ Commented Aug 20 at 21:46
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    $\begingroup$ How are the field equations supposed to be gauge-invariant if $A$ appears explicitly and cannot be replaced by gauge-invariant quantities like $F$? The question seems to ask for an oxymoron: A gauge theory with the condition that its not gauge-invariant. $\endgroup$ Commented Aug 20 at 21:49
  • $\begingroup$ Several vectors, including the electromagnetic potential “A” from classical electrodynamics, appear explicitly and without differentiation in the tree-level field equations of electroweak theory, but electroweak theory is nevertheless considered to be gauge invariant. $\endgroup$ Commented Aug 20 at 22:04
  • $\begingroup$ I guess a good thing to do would be to start with clearing up what "gauge-invariant field theory" means. Which things in the theory are required to be gauge-invariant? Obviously, not all things, since $A^\mu$ is not, and obviously, we expect more things to be invariant than just $F^{\mu\nu}$ . Defining this is not easy. Demonstrating that predictions of quantity values are gauge-invariant in a theory which works just with $A^\mu$ does not seem easy, and may be based on general arguments or just a belief. $\endgroup$ Commented Aug 21 at 0:43

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