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 < b 6 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.