Duality principle of the zero-point length of spacetime and Generalized Uncertainty Principle

Abstract

A strong theoretical motivation persists, in quantum gravity, to consider the existence of a fundamental lower bound on the physically measurable length, namely the Planck length . The Planck length can be considered as a zero-point length of spacetime. For a relativistic particle with action , the invariance of its path integral amplitude under the duality transformation encodes the information of such “zero-point length” of spacetime (Padmanabhan T., Phys. Rev. Lett., 78 (1997) 1854). We show that, when we take the non-relativistic limit, this duality principle deforms the canonical commutation relation similarly to a certain variation of the Generalized Uncertainty Principle (GUP). We interpret this as an equivalence between the principle of path integral duality and GUP.