The Vacuum Law of the Semicircular Operators is the Unique Noncommutative Law Satisfying the Schwinger-Dyson Equation

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $n\in\mathbb{N}$, and let $\mathcal{F}_{n}$ and $\Omega_{\mathcal{F}}$ be the free Fock space and its vacuum vector, and let $S=(S_{1},\dots,S_{n})$ with $S_{j}=l_{j}+\bar{l}_{j}$, where $l_{j}$ and $\bar{l}_{j}$ are the left creation and annihilation operators; each $S_{j}$ is self-adjoint by Calculus of the Free Fock Space: Orthonormal Monomials, Adjoints, Commutation Relations and the Vacuum §self-adjoint. For $p\in\mathcal{P}_{n}$, $p(S)$ is the value of $p$ at $S$.

1. (The vacuum law)¶ The map $\lambda_{S}:\mathcal{P}_{n}\to\mathbb{C}$, $\lambda_{S}(p)=\langle\Omega_{\mathcal{F}},p(S)\Omega_{\mathcal{F}}\rangle$, belongs to $\Sigma_{n,2}$, and hence to $\Sigma_{n}$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law.

2. (Schwinger-Dyson)¶ $\lambda_{S}$ satisfies the Schwinger-Dyson equation.

3. (Uniqueness)¶ Every $\lambda\in\Sigma_{n}$ that satisfies the Schwinger-Dyson equation equals $\lambda_{S}$.

4. (Conjugate variables)¶ A law $\lambda\in\Sigma_{n}$ satisfies the Schwinger-Dyson equation if and only if it has conjugate variables and these are $(\widehat{x_{1}},\dots,\widehat{x_{n}})$, the classes of the variables in $\mathcal{H}_{\lambda}$.