Is it possible that in a theory of gravitons, i.e., a quantum field theory of gravitation, general relativity's principle of the equivalence of gravitational mass and inertial mass, no longer holds?
On the contrary Deser and others [1, and refs therein] have argued that trying to construct a theory of a graviton, that is, a massless spin-2 field in a flat background, consistent with special relativity, then "[c]onsistency [leads] us to universal coupling, which implies the equivalence principle" . The argument is summarized by MTW [2, Box 17.2.5, see also Box 18.2, and refs therein]. More concretely, the conclusion is that the field equations for the graviton field must be the Einstein field equations with the source being the stress-energy tensor.
- Deser, S. (1970). Self-interaction and gauge invariance. General Relativity and gravitation, 1(1), 9-18.
- Misner, C. W., Thorne, K. S., & Wheeler, J. A. (1973). Gravitation. Macmillan.
Yes and no. It depends which of the equivalence principles you have in mind. There are two formulations.
So called Einstein equivalence principle reads:
The outcome of any local non-gravitational experiment in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime.
This is nothing, but a compact reformulation (by Haugan and Lammerzahl) of the original one presented by Einstein in:
A. Einstein, Annalen Phys. 35, 898 (1911) [Annalen Phys. 14, 425 (2005)]. doi:10.1002/andp.200590033
For instance, considering a quantum theory of the gravitational field, you don't necessary have to worry about the equivalence principle if you are interested only in the semiclassical limit of your theory. This may happen when you are going to find a curved background on which you will later rigorously quantize a matter field. Personally I disagree with this interpretation, but a lot of physicists follow it. A good example here is the model called Loop Quantum Cosmology.
So called strong equivalence principle reads (again, I'm citing Haugan and Lammerzahl):
The outcome of any local experiment (gravitational or not) in a freely falling laboratory is independent of the velocity of the laboratory and its location in spacetime.
In this case you have to consider the equivalence principle in the case of any classical or quantum fields.
Two more comments. Even if you don't assume the equivalence principle for gravitational field, you have to consider general covariance for any field, also for the gravitational one. Finally, in the case of the Einstein-Hilbert action and related equations of motion, classically the equivalence principle is satisfied.
Writing the answer, I've cited this article: Mark P. Haugan and C. Lammerzahl, Lect. Notes Phys., 562:195–212, 2001. [arXiv:gr-qc/0103067]