electromagnetism - Interpretation of QED gauge freedom

In quantum (or classical) electrodynamics we are free to make gauge transformations, which change the form of terms in the Feynman diagrams (or the potentials) without affecting any physical observable. This is sometimes viewed as a flaw in the theory.

A similar freedom exists in general relativity. However, here the gauge freedom admits a compelling physical interpretation: it reflects our freedom to describe events using arbitrary coordinates; i.e. the independence of physics from the marks we choose to put on our rulers.

Is there a similarly physical way of interpreting the QED gauge freedom?

Answer

There is no "physical" interpretation of internal gauge symmetries. Gauge symmetries are reflections of an overcounting of degrees of freedom¹, or, equivalently, of the presence of constraints. While the gauge principle is a powerful theoretical tool, the symmetry itself is not really "physical".

General relativity is a special case - and not a "proper" gauge field theory - because diffeomorphisms of spacetime (which you may view as mere "coordinate changes", but which you may also consider active transformations like actual rotation) naturally induce gauge transformations by the Jacobian: The Jacobian is a "local $\mathrm{GL}(n)$ gauge transformation", if you wish. This gauge symmetry is not internal, it interacts with the parameters (the spacetime coordinates/manifold) of the theory, not merely with aspects of its degrees of freedom (the fields living on the manifold). Hence general relativity gets a "physical" interpretation because the "gauge" transformations are not decoupled from the "physical" choice of coordinates.

"Proper" gauge field theories don't do that. They live "above" spacetime, and whatever you do on spacetime has no gauge-theoretic effect whatsoever. They purely act on the fields, and even there only on those that carry a gauge charge - which is essentially the marker for them containing redundant information, which is nevertheless kept to achieve a more elegant formulation, in particular keep (some other) manifest symmetry or avoid having to deal with locally solving the constraints and running into the Gribov problem as long as possible.²

To say that the $\mathrm{U}(1)$ gauge of quantum electrodynamics "adjusts that phase" is correct - but it is not a physical interpretation! An overall (even local) phase is unphysical to begin with, only relative phases play a role, that is the whole reason we are allowed to make a proper gauge theory out of this. Quantum theories don't care for phases, quantum states are rays in Hilbert spaces - using individual vectors in a Hilbert space is already an overcounting of degrees of freedom, paving the way to have a gauge theory in the "second quantized" description.

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^{1See also Gauge symmetry is not a symmetry?}

^{2This might be the "flaw" your professor alluded to - the quantization of gauge theories is much more difficult than that of unconstrained theories (because their Hamiltonian formulation is much more complicated) and may even fail non-perturbatively in some cases. However, they are the only known way (well, known to me, granted) to express things like the strong and weak force in a way amenable at all to quantization.}