Giovanni Cambria said:
someone could explain better the crucial part
The basic idea is that, if you take any field theory (classical or quantum) that is written down assuming a flat spacetime, and try to use it in a curved spacetime, there will always be possible additional terms in the Lagrangian that couple to spacetime curvature. For example, MTW discusses this in relation to Maxwell's Equations; in flat spacetime, in 4-d tensor formalism, they look like this:
$$
\partial_c F^{ab} + \partial_b F^{ca} + \partial_a F^{bc} = 0
$$
$$
\partial_a F^{ab} = 4 \pi J^b
$$
If we now try to use these equations in curved spacetime, we run into a problem. The standard recipe for doing this is to just change all partial derivatives to covariant derivatives; that would give this:
$$
\nabla_c F^{ab} + \nabla_b F^{ca} + \nabla_a F^{bc} = 0
$$
$$
\nabla_a F^{ab} = 4 \pi J^b
$$
The problem, though, is that ##F^{ab}## itself involves partial derivatives, since the field tensor is the exterior derivative of the 4-potential; i.e., in flat spacetime we have
$$
F^{ab} = \partial^a A^b - \partial^b A^a
$$
But if we apply the "partial to covariant derivative" rule to this and then substitute into, say, the second Maxwell Equation, we get:
$$
\nabla_a \left( \nabla^a A^b - \nabla^b A^a \right) = 4 \pi J^b
$$
Expanding this out gives
$$
\nabla_a \nabla^a A^b - \nabla_a \nabla^b A^a = 4 \pi J^b
$$
And now we can see the problem: covariant derivatives in curved spacetime do not commute, so there is an ambiguity in how we do the translation to curved spacetime above; we could have perfectly consistently included a curvature coupling term ##R^b{}_a A^a## (the commutator of the covariant derivatives ##\nabla^a A^b## and ##\nabla^b A^a##) to obtain
$$
R^b{}_a A^a + \nabla_a \nabla^a A^b - \nabla_a \nabla^b A^a = 4 \pi J^b
$$
There is no general rule for deciding which one is right; but the argument in the CERN article you linked to is basically that, other things being equal, we should expect that curvature coupling term to be there, and we should expect it to have a significant magnitude--significant enough to affect experimental results--when the spacetime curvature length scale is of the same order as the photon length scale, i.e., the Compton wavelength of the electron. The above was written in terms of the classical Maxwell Equation, but the same equation also turns out to be the field equation for the photon in QED (or, to put it another way, the same Lagrangian that gives rise to Maxwell's Equations in classical electrodynamics also gives rise to the photon propagator in QED). So we should expect the same curvature coupling term to be present if we do QED in curved spacetime.
