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Quantum mechanics with a generalized uncertainty principle arises through a representation of the commutator [^x,^p]=if(^p). We apply this deformed quantization to free scalar field theory for f_±=1±βp². 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.
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Husain et al. (2013) studied this question.
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