Noncommutative coordinates invariant under rotations and Lorentz transformations

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 the 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 are discussed.