Noncommutative Coordinates Invariant under Rotations and Lorentz Transformations

TL;DR

This paper develops a framework for noncommutative coordinates invariant under rotations and Lorentz transformations, revealing modifications to quantum mechanics and implications for black hole physics.

Contribution

It introduces noncommutative coordinates as boost operators in SO(1,3) and SO(2,3), exploring their effects on quantum localization and field solutions.

Findings

Modification of Heisenberg algebra at small distances

Lower limit on wave packet localization

Discrete eigenvalues in timelike directions

Abstract

Dynamics with noncommutative coordinates invariant under three dimensional rotations or, if time is included, under Lorentz transformations is developed. These coordinates turn out to be the boost operators in SO(1,3) or in SO(2,3) respectively. The noncommutativity is governed by a mass parameter $M$. The principal results are: (i) a modification of the Heisenberg algebra for distances smaller than 1/M, (ii) a lower limit, 1/M, on the localizability of wave packets, (iii) discrete eigenvalues of coordinate operator in timelike directions, and (iv) an upper limit, $M$, on the mass for which free field equations have solutions. Possible restrictions on small black holes is discussed.