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.



Is there a noncommutative Gaussian?
The theory of classical independence and classical convolution can be generalised to noncommutative settings in several ways. The most famous one is that of [free independence and free convolution](https://en.wikipedia.org/wiki/Free_probability) (introduced by Voiculescu), but there is also boolean independence and boolean convolution (introduced by Speicher and Woroudi in [Boolean convolution](https://mathscinet.ams.org/mathscinet-getitem?mr=1426845)); monotone independence and monotone convolution (introduced by Muraki in [Monotonic independence, monotonic central limit theorem and monotonic law of small numbers](https://mathscinet.ams.org/mathscinet-getitem?mr=1824472)); and anti-monotone independence and anti-monotone convolution (the order-reversal of the previous notion). There are classification results of Speicher ([On universal products](https://mathscinet.ams.org/mathscinet-getitem?mr=1426844)) and Muraki ([The five independences as natural products](https://mathscinet.ams.org/mathscinet-getitem?mr=2016316)) that show that these are the only notions of independence (or convolution) that obey some natural set of axioms. (Speicher's classification assumed that convolution is commutative, so omitted the monotone and anti-monotone cases that were later discovered by Muraki.)


For each such concept of independence, there is a central limit theorem. Classically, the limiting distribution is the gaussian; in free probability it is the semicircular law; in the boolean case it is the Bernoulli distribution; and in the monotone and anti-monotone cases it is the arcsine law. See Section 9.2.1 of the recent thesis [Evolution equations in non-commutative probability](https://escholarship.org/content/qt8n39f7mt/qt8n39f7mt.pdf?t=qbrc2a) of David Jekel (and Chapter 5 of that thesis contains a more detailed history of the development of these notions of independence). For the classical and free independence concepts, at least, there is also an associated notion of entropy, and these distributions extremise the entropy amongst all distributions of a fixed mean and variance; again, Jekel's thesis has further information. (For the free case, of course, pretty much any introduction to free probability will contain these facts.)


EDIT: There is also finite free convolution (see [Marcus, Spielman, and Srivastava - Finite free convolutions of polynomials](https://arxiv.org/abs/1504.00350)), in which the analogue of the gaussian is the distribution of zeroes of Hermite polynomials.


