TITLE: Is there a noncommutative Gaussian?
QUESTION [20 upvotes]: In classical probability theory, the (multivariate) Gaussian is in some sense the "nicest quadratic" random variable, i.e. with second moment a specified positive-definite matrix. I do not know how to make this precise, but non-precisely what I mean is that 1. Gaussian shows up everywhere, and 2. it is universal/canonical/... in some sense, e.g. as in the central limit theorem.
My question is whether for many noncommutative probability spaces (an algebra $A$ over $\mathbf{C}$ and a map $E:A\to\mathbf{C}$, with conditons), there also exists a "nicest quadratic" random variable $X\in A$, satisfying analogous properties to the Gaussian.

REPLY [5 votes]: If one does not insist on having a notion of "independence" in the background, but takes, as asked in the question, the Wick/Isserlis formula (which expresses general moments in terms of second moments via a nice combinatorial formula) as guiding principle, then there are also other nice non-commutative versions of multivariate Gaussian distributions. A prominent one is given by the q-Gaussian distribution, which interpolates between the classical Gaussian and the free Gaussian (i.e., semicircular) distribution.