Free probability: motivation for Voiculescu's free gaussian functor

Question

Given a Hilbert space, let us denote by $\mathcal{T}(\mathcal{H})$ the full Fock space $$ \mathcal{T}(\mathcal{H}) = \mathbb{C} \Omega \oplus \bigoplus_{n \geq 1} \mathcal{H}^{\otimes n}. $$ For $\xi \in \mathcal{H}$ we the so-called creation operator, $$ l(\xi): \mathcal{T}(\mathcal{H}) \rightarrow \mathcal{T}(\mathcal{H}): \begin{cases} \eta \mapsto \xi \otimes \eta \\ \Omega \mapsto \xi \end{cases}.$$ In any notes I can find about free probability theory, the operator $$ s(\xi) = \frac{l(\xi)+l(\xi)^\ast}{2},$$ which is the real part of $l(\xi)$, plays a somewhat central role. I know that the von Neumann algebra it generates is isomorphic to $L^\infty(\mathbb{R})$ and that free products of these von Neumann algebras thus give a model for the von Neumann algebra of the free groups $L(\mathbb{F}_n)$. I also know that the distribution of $s(\xi)$ w.r.t. to the vacuum state $x \mapsto \langle x \Omega, \Omega \rangle$ is the semi-circular law. Although I undererstand these facts, I don't have the intuition and I don't know what Voiculescu motivated to look at the operators $l(\xi)$ and $s(\xi)$ above. I'd like to have an answer containing the reason why we study these.

Answer 1

I do not know precisely what motivated Voiculescu, but I can give some reasons why one might consider this setup.

One way to think of the operators $l$ and $l^*$, is as a variation on the shift operators $R$ and $L$ on $\ell^2(\mathbb{N})$, where for a sequence $x = (x_1,x_2,\dots) \in \ell^2(\mathbb{N})$, $Rx = (0,x_1,x_2,\dots)$ and $Lx = (x_2,x_3,\dots)$. The operators $R$ and $L$ are canonical (anti-)examples in function analysis, and hence interesting in their own right (for instance $R$ is an injective linear map that is not surjective, an impossibility in a finite dimensional vector space). Given $\mathcal{H}$ and $\xi \in \mathcal{H}$, $\ell(\xi)$ is then the operator that 'shifts by $\xi$' on $\mathcal{T}(\mathcal{H})$, etcetera.

As pointed out in the comments, Fock spaces originate in physics, where they model many-body quantum systems. In this context, $\mathcal{H}$ is intepreted as the Hilbert space of single-particle quantum states and the vacuum state $\Omega$ represents an empty system. The operator $\ell(\xi)$ then has the interpretation of creating a particle in the state $\xi$, and $\ell^*(\xi)$ removes (or annihilates) a particle in this state. However, physicists impose relations on creation/annihilation operators coming from different vectors of $\mathcal{H}$ (modelling 'particle exchange'). Let $e_1,\dots,e_n$ be an ONB for $\mathcal{H}$ (I will assume that $\mathcal{H}$ is finite dimensional for simplicity), and write $\ell(e_i) = \ell_i$ for brevity. Then the two most common such relations are the 'canonical commutation relations' $\ell_i^* \ell_j - \ell_j \ell_i^* = \delta_{ij}$ and the 'canonical anti-commutation relations' $\ell_i^* \ell_j + \ell_j \ell_i^* = \delta_{ij}$ (these model 'Bosons' and 'Fermions' respectively). You can check that in our case, the operators satisfy instead $\ell_i^* \ell_j = \delta_{ij}$, which is the 'average' of the previous two relations.

Now for the operators $s_i = s(e_i)$.

Mathematically, the relation just derived implies that the $\ell_i$ are not normal: $\ell_i^*\ell_i = \mathrm{id}$, but $\ell_i \ell_i^* \neq \mathrm{id}$ since $\ell_i \ell_i^* \Omega = 0$. Hence, we do not have a priori access to the powerful tools of spectral theory. Additionally, the vacuum state is not faithful on the von Neumann algebra generated by the $\ell_i$, again since $\ell_i \ell_i^* \Omega = 0$, but one can check that $\ell_i \ell_i^* \neq 0$. Considering the self-adjoint part of $\ell_i$ will certainly fix the first problem, and one can check that it also fixes the second (the vacuum state even becomes a faithful trace when considered on the von Neumann algebra generated by the $s_i$, and $\Omega$ itself a cyclic and separating vector). By for instance solving the associated moment problem, one can then show that the measure on $\sigma(s_i)$ given by sandwiching the resolution of the identity induced by $s_i$ with the vacuum $\Omega$ is precisely the semi-circle law (as you pointed out). When considering joint moments of the $s_1,\dots,s_n$, one will notice that they can always be expressed in terms of moments of order 2 (Wick's/Isserlis' theorem), which is also true for an $n$-tuple of 'classical' random variables with a joint Gaussian distribution (hence the name free Gaussian functor). This observation allows for a simple proof that $s$ operators associated to orthogonal vectors in $\mathcal{H}$ are free.

Physically, in the cases of the canonical (anti-)commutation relations, it is true that the $s_i$ satisfy the same relations as the $\ell_i$. For the canonical anti-commutation relations, the $s_i$ are the famous Majorana Fermions. While it does not work for our case, it might still be motivation to consider the $s_i$.