Generalized uncertainty principles and quantum field theory

Abstract

Quantum mechanics with a generalized uncertainty principle arises through a representation of the commutator $[\hat{x},\hat{p}]=if(\hat{p})$. 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.

5 pages, 5 figures

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