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We propose a picture for the UV properties of Galileon field theories. Weconjecture that Galileons, and all theories incorporating the Vainshteinmechanism, fall into Jaffe's class of `non-localizable' field theoriescharacterized by an exponential growth in their Kallen-Lehmann spectraldensities. Similar properties have been argued to arise for Little StringTheories and M-theory. For such theories, the notion of micro-causality and thetime ordering used to define the S-matrix and correlation functions must bemodified, and we give a Lorentz invariant prescription for how this can beachieved. In common with General Relativity (GR), the scattering amplitudes forGalileons are no longer expected to satisfy polynomial boundedness away fromthe forward scattering or fixed physical momentum transfer limits. This is areflection of the fact that these theories are fundamentally gravitational andnot local field theories. We attribute this to the existence of a localitybound 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 theenergy of all states are positive definite. We perform an explicit computationof the Wightman functions for this theory, and demonstrate the exponentialgrowth associated with the locality bound. In analogy with GR, the bound iscorrelated with the absence of Galileon Duality (i.e. Diffeomorphism) invariantlocal observables. We argue that these theories can be quantized in a mannerwhich preserves Lorentz invariance and macro-causality and that the latterensures that the superluminalities found in the low energy effective theory areabsent in the full theory.

UV properties of Galileons: Spectral Densities