New Non-commutative and Higher Derivatives Quantum Mechanics from GUPs

TL;DR

This paper introduces a new class of non-linear generalized uncertainty principles leading to non-commutative quantum mechanics with higher derivatives, affecting fundamental equations and potentially observable in high-energy astrophysical phenomena.

Contribution

It proposes a novel framework connecting non-commutativity, higher derivatives, and GUPs, extending fundamental quantum equations and analyzing their physical implications.

Findings

Non-commutative space coordinates depend on angular momentum.

Modified Schrödinger, Dirac, and Klein-Gordon equations include higher derivatives.

Deformed dispersion relations suggest Lorentz invariance violation.

Abstract

We explore a new class of Non-linear GUPs (NLGUP) showing the emergence of a new non-commutative and higher derivatives quantum mechanics. Within it, we introduce the shortest fundamental scale as a UV fixed point in the NLGUP commutators [X, P] = i\hbar 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, that start to be dependent by the angular momenta of the system. On the other hand, non-linear GUP must lead to a redefinition of the Schrodinger 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.