  Quantum mechanics with a generalized uncertainty principle arises through a
representation of the commutator $[\hat{x}, \hat{p}] = i f(\hat{p})$. We apply
this deformed quantization to free scalar field theory for $f_\pm =1\pm \beta
p^2$. The resulting quantum field theories have a rich fine scale structure.
For small wavelength modes, the Green's function for $f_+$ exhibits a
remarkable transition from Lorentz to Galilean invariance, whereas for $f_-$
such modes effectively do not propagate. For both cases Lorentz invariance is
recovered at long wavelengths.
