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## Noncommutative probability

The traditional mathematical axiomatization of probability, due to Kolmogorov, begins with a probability space $P$ and constructs random variables as certain functions $P \to \mathbb{R}$. But start doing any probability and it becomes clear that the space $P$ is de-emphasized as much as possible; the real focus of probability theory is on the algebra of random variables. It would be nice to have an approach to probability theory that reflects this.

Moreover, in the traditional approach, random variables necessarily commute. However, in quantum mechanics, the random variables are self-adjoint operators on a Hilbert space $H$, and these do not commute in general. For the purposes of doing quantum probability, it is therefore also natural to look for an approach to probability theory that begins with an algebra, not necessarily commutative, which encompasses both the classical and quantum cases.

Happily, noncommutative probability provides such an approach. Terence Tao’s notes on free probability develop a version of noncommutative probability approach geared towards applications to random matrices, but today I would like to take a more leisurely and somewhat scattered route geared towards getting a general feel for what this formalism is capable of talking about.

In the previous post we described the Heisenberg picture of quantum mechanics, which can be phrased quite generally as follows: given a noncommutative algebra $A$ (the algebra of observables of some quantum system) and a Hamiltonian $H \in A$, we obtain a derivation $[-, H]$, which is (up to some scalar multiple) the infinitesimal generator of time evolution. This is a natural and general way to start with an algebra and an energy function and get a notion of time evolution which automatically satisfies conservation of energy.
However, if $A$ is commutative, all commutators are trivial, and yet classical mechanics somehow takes a Hamiltonian $H \in A$ and produces a notion of time evolution. How does that work? It turns out that for algebras of observables $A$ of a classical system, we can think of $A$ as the classical limit $\hbar \to 0$ of a family $A_{\hbar}$ of noncommutative algebras. While $A$ is commutative, the noncommutativity of the family $A_{\hbar}$ equips $A$ with the extra structure of a Poisson bracket, and it is this Poisson bracket which allows us to describe time evolution.