My understanding is that massive and massless QED share some key physical features including (see this PSE post and 8-4 of Ref. 1):
- renormalizability
- charge conservation
The key differences of massive QED being:
- A massive photon mediates a short range force.
- A massive photon has a 3rd, longitudinal polarization.
- Massive QED is not gauge invariant.
It seems to me that while renormalizability and charge conservation are features we might naturally like to impose on a QED, the consequences of a massive photon seem to be experimental questions, rather than theoretical constraints.
We can contrast this with non-Abelian gauge theory, where gauge invariance is required for renormalizability; so a massless gauge boson expected in the absence of a Higgs mechanism (see this PSE post and section 12-5-2 of Ref. 1).
One "benefit" of local gauge invariance is that it implies a global symmetry and thus a conserved Noether current. But I don't see the benefit to imposing a local gauge invariance instead of simply imposing the global symmetry directly, since either either way you get a conserved current.
Also, since gauge symmetry is a custodial symmetry for the photon mass, the photon mass is technically natural (see Ref. 2 section 22.6). So it seems that there is no fundamental problem with QED having a very small photon mass.
So my two related questions are:
- Is there a theoretical reason to "prefer" that QED is massless?
- Is there a theoretical reason to "prefer" a gauge invariant interaction for a matter field with an abelian global symmetry?
Please correct any misunderstandings I have in my discussion above!
References:
- Itzykson & Zuber's Quantum Field Theory
- Schwartz's Quantum Field Theory & the Standard Model
