  A large body of work over several decades indicates that, in the presence of
gravitational interactions, there is loss of localization resolution within a
fundamental ( $\sim$ Planck) length scale $\ell$. We develop a general
formalism based on wavelet decomposition of fields that takes this UV
`opaqueness' into account in a natural and mathematically well-defined manner.
This is done by requiring fields in a local Lagrangian to be expandable in only
the scaling parts of a (complete or, in a more general version, partial)
wavelet Multi-Resolution Analysis. This delocalizes the interactions, now
mediated through the opaque regions, inside which they are rapidly decaying.
The opaque regions themselves are capable of discrete excitations of $\sim
1/\ell$ spacing. The resulting effective Feynman rules, which give UV regulated
and (perturbatively) unitary physical amplitudes, resemble those of string
field theory.
