Minimum length (scale) in quantum field theory, generalized uncertainty principle and the non-renormalisability of gravity

The notions of minimum geometrical length and minimum length scale are discussed with reference to correlation functions obtained from in-in and in-out amplitudes in quantum field theory. Whereas the in- in propagator for metric perturbations does not admit the former, the in-out Feynman propagator shows the emergence of the latter. A connection between the Feynman propagator of quantum field theories of gravity and the deformation parameter δ0 of the generalised uncertainty principle (GUP) is then exhibited, which allows to determine an exact expression for δ0 in terms of the residues of the causal propagator. A correspondence between the non-renormalisability of (some) theories (of gravity) and the existence of a minimum length scale is then conjectured to support the idea that non-renormalisable theories are self-complete and finite. The role played by the sign of the deformation parameter is further discussed by considering an implementation of the GUP on the lattice.