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The notions of minimum geometrical length and minimum length scale arediscussed with reference to correlation functions obtained from in-in andin-out amplitudes in quantum field theory. Whereas the in-in propagator formetric perturbations does not admit the former, the in-out Feynman propagatorshows the emergence of the latter. A connection between the Feynman propagatorof quantum field theories of gravity and the deformation parameter $\delta_0$of the generalised uncertainty principle (GUP) is then exhibited, which allowsto determine an exact expression for $\delta_0$ in terms of the residues of thecausal propagator. A correspondence between the non-renormalisability of (some)theories (of gravity) and the existence of a minimum length scale is thenconjectured to support the idea that non-renormalisable theories areself-complete and finite. The role played by the sign of the deformationparameter is further discussed by considering an implementation of the GUP onthe lattice.

Minimum length (scale) in Quantum Field Theory, Generalized Uncertainty Principle and the non-renormalisability of gravity