In gauge theories such as QED or QCD, are the rest energies, and therefore the invariant masses, of various bound states, such as those of positronium or charmonium, gauge-invariant observable quantities?
It seems obvious to me that they must be, but I need confirmation from reliable experts in quantum field theory to refute arguments in other threads that energy is in general gauge-dependent and that only energy differences are observable.
If these energies are not gauge-invariant observables, then I would like to understand why not.
For context, the arguments have been in the following threads:
The reasons I have for believing that rest energies of bound states are gauge-independent observables are:
It is common to talk about the invariant mass of such states, implying well-defined-ness, gauge-invariance, and observability of the rest-energy.
There are well-known gauge-invariant energy-momentum-stress tensors for gauge field theories.
Energy curves spacetime and therefore is obviously well-defined, gauge-invariant, and observable.
Once we settle how this works in full quantum field theories, I will post a follow-on question about the gauge-invariance of energies of bound states in non-relativistic quantum mechanics, where I think there are more subtleties since the electromagnetic field isn't quantized.