On the fractional minimal length Heisenberg-Weyl uncertainty relation from fractional Riccati generalized momentum operator

It was showed that the minimal length Heisenberg-Weyl uncertainty relation may be obtained if the ordinary momentum differentiation operator is extended to its fractional counterpart, namely the generalized fractional Riccati momentum operator of order 0 < β ⩽ 1. Some interesting consequences are exposed in concordance with the UV/IR correspondence obtained within the framework of non-commutative C-space geometry, string theory, Rovelli loop quantum gravity, Amelino-Camelia doubly special relativity, Nottale scale relativity and El-Naschie Cantorian fractal spacetime. The fractional theory integrates an absolute minimal length and surprisingly a non-commutative position space.