Generalized uncertainty principles and quantum field theory

Quantum mechanics with a generalized uncertainty principle arises through a representation of the commutator $[\stackrel{^}{x},\stackrel{^}{p}]=if(\stackrel{^}{p})$. We apply this deformed quantization to free scalar field theory for ${f}_{\ifmmode\pm\else\textpm\fi{}}=1\ifmmode\pm\else\textpm\fi{}\ensuremath{\beta}{p}^{2}$. 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 ${f}_{\ensuremath{-}}$ such modes effectively do not propagate. For both cases Lorentz invariance is recovered at long wavelengths.

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