Generalized uncertainty principles and quantum field theory

Understand

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.