That means that (with any given set of axes), it is impossible to accurately measure the position of a particle with respect to more than one axis.
Various lower limits have been claimed for the noncommutative scale, (i.e. how accurately positions can be measured) but there is currently no experimental evidence in favour of such a theory or grounds for ruling them out.
Heisenberg was the first to suggest extending noncommutativity to the coordinates as a possible way of removing the infinite quantities appearing in field theories before the renormalization procedure was developed and had gained acceptance.
The particle physics community became interested in the noncommutative approach because of a paper by Nathan Seiberg and Edward Witten.
Two papers, one by Sergio Doplicher, Klaus Fredenhagen and John Roberts[5] and the other by D. V. Ahluwalia,[6] set out another motivation for the possible noncommutativity of space-time.
The arguments go as follows: According to general relativity, when the energy density grows sufficiently large, a black hole is formed.
A sufficient condition for preventing gravitational collapse can be expressed as an uncertainty relation for the coordinates.
On the other hand, differently from Connes' noncommutative geometry, the proposed model turns out to be coordinate-dependent from scratch.