F(p) quantum mechanics

We explore the possibility to extend the Heisenberg’s uncertainty principle to a nonlinear extension of the quantum algebra related to a functional operator of the momenta as F(P). We show that such an extension of quantum mechanics is intimately connected to the non-commutative space-time algebra and the Lorentz symmetry deformations. We show that a large class of F(P) models can introduce superluminal modes in the quantized theories. We also show that the Hoˇrava–Lifshitz theory is related to a large class of F(P) Quantum Mechanics.

GUP; non-commutative geometry; quantum mechanics foundation; quantum gravity.