  We propose a picture for the UV properties of Galileon field theories. We
conjecture that Galileons, and all theories incorporating the Vainshtein
mechanism, fall into Jaffe's class of `non-localizable' field theories
characterized by an exponential growth in their Kallen-Lehmann spectral
densities. Similar properties have been argued to arise for Little String
Theories and M-theory. For such theories, the notion of micro-causality and the
time ordering used to define the S-matrix and correlation functions must be
modified, and we give a Lorentz invariant prescription for how this can be
achieved. In common with General Relativity (GR), the scattering amplitudes for
Galileons are no longer expected to satisfy polynomial boundedness away from
the forward scattering or fixed physical momentum transfer limits. This is a
reflection of the fact that these theories are fundamentally gravitational and
not local field theories. We attribute this to the existence of a locality
bound for Galileons, analogous to the Giddings-Lippert locality bound for GR.
We utilize the recently developed Galileon duality to define a UV finite,
Lorentz invariant, quantization of a specific Galileon theory for which the
energy of all states are positive definite. We perform an explicit computation
of the Wightman functions for this theory, and demonstrate the exponential
growth associated with the locality bound. In analogy with GR, the bound is
correlated with the absence of Galileon Duality (i.e. Diffeomorphism) invariant
local observables. We argue that these theories can be quantized in a manner
which preserves Lorentz invariance and macro-causality and that the latter
ensures that the superluminalities found in the low energy effective theory are
absent in the full theory.
