New non-commutative and higher derivatives quantum mechanics from GUPs

We explore a new class of Non-linear GUPs (NLGUP) showing the emergence of a new noncommutative and higher derivatives of quantum mechanics. Within it, we introduce the shortest fundamental scale as a UV fixed point in the NLGUP commutators [X, P] = if (P), having in mind a fundamental highest energy threshold related to the Planck scale. We show that this leads to lose commutativity of space coordinates, which starts to be dependent on the angular momenta of the system. On the other hand, non-linear GUP must lead to a redefinition of the Schrödinger equation to a new non-local integral-differential equation. We also discuss the modification of the Dyson series in time-dependent perturbative approaches. This may suggest that, in NLGUPs, non-commutativity and higher derivatives may be intimately interconnected within a unified and coherent algebra. We also show that Dirac and Klein–Gordon equations are extended with higher space-derivatives according to the NLGUP. We compute momenta-dependent corrections to the dispersion velocity, showing that the Lorentz invariance is deformed. We comment on possible implications in tests of light dispersion relations from Gamma-Ray-Bursts or Blazars, with potential interests for future experiments such as LHAASO, HAWC and CTA.

GUP, non-commutative geometry, quantum mechanics foundation, quantum gravity