non-commutative probability for dummies?

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?

1. What makes Kolmogorov's system commutative, and what makes non-commutative probability non-commutative?

2. Why would we want to drop commutativity?

3. 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".

In a very tiny, very densely packed nutshell, here's a partial answer to your question 1 (as far as I understand it, which is not very far at all; and I really cannot say much of anything about your questions 2 and 3).

Regarding what it means for probability to be "commutative", it's not just that $E(XY)=E(YX)$, it's that $XY=YX$. In other words, multiplication of random variables is a commutative operation on the set of random variables. This is supposed to be "obviously true", because random variables are real valued, and multiplication of real numbers is commutative.

Regarding the meaning of "noncommutative probability", the first idea is to think of "commutative probability" as being a theory of the commutative algebra of random variables. In other words, one focusses not on the state space, nor on the events in the state space, but instead one focusses solely on the abstract set of random variables, and on the operations of addition and multiplication and scalar multiplication on this set. A set with two binary operations $+$ and $\times$ and with a scalar multiplication operation, and satisfying various axioms that govern the behavior of those operations (in particular the commutativity of multiplication), is called an "algebra" or more precisely a "commutative algebra".

One then makes a vast leap from commutative algebra into noncommutative algebra, to see what properties of "commutative algebras of random variables" can be generalized in the setting of noncommutative algebras.

What's going on here is that one conveniently "forgets" where random variables came from, in particular one forgets that they are real valued functions on measure spaces. One then proceeds to formally generalize the properties of commutative algebras of random variables to the setting of noncommutative algebras. Then one attempts to "remember" where random variables came from in this more abstract setting. That confusing but enticing statement you point out is an example of the "forgetting-remembering" process. This turns out magically to be a very fruitful process and results in some very interesting and important mathematics.

In short,

1. Multiplication of random variables is commutative in standard probability theory but not in noncommutative probability.

2. Terence Tao uses noncommutative probability to understand random matrices; I wrote the blog post you linked to in order to understand quantum mechanics, and in particular quantum probability, which is noncommutative.

3. The uncertainty principle. This is explained in my blog post that you linked to.

• followup to your point 1: multiplication of real random variables is indeed commutative. But this is not a fundamental property of the kolmogorov system, is it? We can still use the Kolmogorov system for random variables to arbitrary (i.e. non-$\mathbb R^1$) measure spaces, such as the space of $n\times m$ matrices. It seems to me that dropping commutativity is therefore not necessarily dropping Kolmogorov's system. – user56834 Jan 7 '18 at 17:47
• @Programmer2134: yes, you can still consider commutative probability on matrices, and yes, that's not what noncommutative probability means. In noncommutative probability, instead of talking about probability measures we talk about linear functionals on random variables which in the commutative case correspond to expectations wrt to probability measures. (These linear functionals take values in $\mathbb{C}$ even when we talk about matrices, so they are not expectations wrt to a classical probability measure on matrices.) In the noncommutative case they are genuinely something else. – Qiaochu Yuan Jan 7 '18 at 19:34
• Anyway, all of this is carefully explained in painstaking detail in the blog posts you linked to. – Qiaochu Yuan Jan 7 '18 at 19:34