I read this article by Terry Tao about non-commutative probability.
I honestly don't really get it. Apparently, we can generalize Kolmogorov's system of probability theory, in such a way that it is no longer "commutative" (I wasn't even aware that probability theory is "commutative", and I'm not sure why it is. Does it mean that for two random variables $X$ and $Y$, $E(XY)=E(YX)$?).
Could you explain what the point is of non-commutative probability?
What makes Kolmogorov's system commutative, and what makes non-commutative probability non-commutative?
Why would we want to drop commutativity?
Are there simple examples of things we can understand with non-commutative probability theory, that we cannot understand with Kolmogorov's system?
Edit: here is another article on the topic, one I also don't really get as it goes into quantum mechanics. But it makes the confusing but enticing statement that: "Although noncommutative spaces don’t have a good notion of point, quantum probability spaces suggest that they have a good notion of measure (which we can think of as a “smeared-out” point", and talks about "random algebra's".