It is at the heart of gauge theory that one cannot find potentials for
the massless Wigner +/-h field strength tensors which are (i) local,
(ii) covariant, (iii) have 2 degrees of freedom and (iv) live on the
Wigner-Fock Hilbert space of their field strength. Typically, one keeps
locality, introduces unphysical degrees of freedom and works on an
indefinite Krein space in intermediate steps. However, one can also
introduce non-local potentials which satisfy (ii)-(iv), for example
"string-localized" potentials.
I present recent results on the self- and matter-interaction of
string-localized graviton potentials. The structure of these
interactions is determined by the requirements that the physical
quantities, in this case the S-matrix, are local and string independent.
Although only derived at tree level, the results are "purely quantum" in
the sense that they do not refer to the gauge invariance of an
underlying classical theory.