[Slaviks]

You do not need to go into the speculative domains of modern phsyics to find non-commuattivity.

It was the brilliant realization of Heisenberg (in 1925!) that all phsyical quantities on microscopic level must be dercibed by quantities (operators or matrices) that are not commutative under multiplication. It is corner stone of quantum theory, and of everything that incorporates it (including solid state theory, string theories, Standrad Model, Big Bag calculations, you name...)

The celebrated Heisenberg uncertainty relation is just a way to say that coordinate and momentum of a particle do not commute.