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

Each result cited is universally quantified over the data in its own statement.

Throughout, $\langle\cdot,\cdot\rangle$ is the inner product of $\mathcal{F}_{n}$, $\Omega=\Omega_{\mathcal{F}}$, and $D=(D_{1},\dots,D_{n})$ with $D_{j}=r_{j}+\bar{r}_{j}$ is the tuple of right semicircular operators of Calculus of the Free Fock Space: Orthonormal Monomials, Adjoints, Commutation Relations and the Vacuum. For $u\in W_{n}$, $x_{u}(S)=S_{u}$ by Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §evaluation, so $\lambda_{S}(x_{u})=\langle\Omega,S_{u}\Omega\rangle$. Every $A\in\mathcal{L}(\mathcal{F}_{n})$ has an adjoint $A^{*}$ by Bounded Linear Maps between Complex Inner Product Spaces: the Least Bound, Operations, the Underlying Real Structure, Adjoints, Completeness and the Quadratic-Form Bound §adjoint, and by Adjoint of a Linear Map between Complex Inner Product Spaces §adjoint and conjugate symmetry, $\langle A^{*}\eta,\xi\rangle=\langle\eta,A\xi\rangle$ and $\langle A\xi,\eta\rangle=\langle\xi,A^{*}\eta\rangle$ for all $\xi,\eta\in\mathcal{F}_{n}$. A linear map $\varphi:\mathcal{P}_{n}\to\mathbb{C}$ with $\varphi(x_{w})=0$ for every $w\in W_{n}$ is $0$, by the uniqueness in Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §linear-extension (a) (the zero map has the same values on monomials); we refer to this as (F1).

Clause 1 (The vacuum law). By Calculus of the Free Fock Space: Orthonormal Monomials, Adjoints, Commutation Relations and the Vacuum §self-adjoint each $S_{j}$ is self-adjoint with $\lVert S_{j}\rVert_{\mathrm{op}}\le2$, and $\lVert\Omega\rVert=1$ by Calculus of the Free Fock Space: Orthonormal Monomials, Adjoints, Commutation Relations and the Vacuum §orthonormal. So The Law of a Tuple of Bounded Self-Adjoint Operators in a Tracial Vector State applies with $H=\mathcal{F}_{n}$, the vector $\Omega$, $R=2$ and $T=S$, and its map $\lambda_{T}$ is $\lambda_{S}$. By The Law of a Tuple of Bounded Self-Adjoint Operators in a Tracial Vector State §law it suffices to show $\langle\Omega,S_{u}S_{v}\Omega\rangle=\langle\Omega,S_{v}S_{u}\Omega\rangle$ for all $u,v\in W_{n}$. We use three facts.

(a) $S_{u}^{*}=S_{u^{\mathrm{rev}}}$ and $D_{u}^{*}=D_{u^{\mathrm{rev}}}$ for every $u\in W_{n}$. Indeed each $S_{j}$ and each $D_{j}$ is self-adjoint by Calculus of the Free Fock Space: Orthonormal Monomials, Adjoints, Commutation Relations and the Vacuum §self-adjoint, so by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §adjoint with $p=x_{u}$, and since $x_{u}^{*}=x_{u^{\mathrm{rev}}}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint, $S_{u}^{*}=x_{u}(S)^{*}=x_{u^{\mathrm{rev}}}(S)=S_{u^{\mathrm{rev}}}$, and likewise for $D$.

(b) $S_{u}D_{v}=D_{v}S_{u}$ for all $u,v\in W_{n}$. First let $T=(T_{1},\dots,T_{n})$ be an $n$-tuple in $\mathcal{L}(\mathcal{F}_{n})$ and $B\in\mathcal{L}(\mathcal{F}_{n})$ with $T_{i}B=BT_{i}$ for every $i\in[n]$; then $T_{u}B=BT_{u}$ for every $u\in W_{n}$. For $u=\varnothing$ this holds as $T_{\varnothing}=I$ (Evaluation of Noncommutative Polynomials at a Tuple of Bounded Operators §word-products). For words of positive length we induct on the length $m\in\mathbb{N}$, the statement for $m$ being that $T_{u}B=BT_{u}$ for all $u$ of length $m$: for $m=1$, $u=(i)$ is a letter by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §last-letter and $T_{u}=T_{i}$; and a word $u$ of length $m+1$ is $v(i)$ with $v$ of length $m$ by the same clause, so $T_{u}B=T_{v}T_{i}B=T_{v}BT_{i}=BT_{v}T_{i}=BT_{u}$ by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values. Applying this with $T=S$ and $B=D_{k}$, which is allowed by Calculus of the Free Fock Space: Orthonormal Monomials, Adjoints, Commutation Relations and the Vacuum §commute, gives $S_{u}D_{k}=D_{k}S_{u}$ for all $u$ and $k$; applying it again with $T=D$ and $B=S_{u}$ gives $D_{v}S_{u}=S_{u}D_{v}$.

(c) $(w^{\mathrm{rev}})^{\mathrm{rev}}=w$ and $(vu)^{\mathrm{rev}}=u^{\mathrm{rev}}v^{\mathrm{rev}}$ for all words, by Basic Properties of Words: Associativity, Reversal, Finitely Many Factorisations, and Countability §reversal.

Now let $u,v\in W_{n}$. Using the adjoint identities recalled above, (a), the reversal clause (applied to $v$ and to $v^{\mathrm{rev}}$), (b), (c) and Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values:

$$\begin{aligned} \langle\Omega,S_{u}S_{v}\Omega\rangle&=\langle S_{u^{\mathrm{rev}}}\Omega,S_{v}\Omega\rangle=\langle S_{u^{\mathrm{rev}}}\Omega,D_{v^{\mathrm{rev}}}\Omega\rangle=\langle D_{v}S_{u^{\mathrm{rev}}}\Omega,\Omega\rangle=\langle S_{u^{\mathrm{rev}}}D_{v}\Omega,\Omega\rangle\\ &=\langle S_{u^{\mathrm{rev}}}S_{v^{\mathrm{rev}}}\Omega,\Omega\rangle=\langle S_{(vu)^{\mathrm{rev}}}\Omega,\Omega\rangle=\langle S_{vu}^{*}\Omega,\Omega\rangle=\langle\Omega,S_{vu}\Omega\rangle=\langle\Omega,S_{v}S_{u}\Omega\rangle . \end{aligned}$$

Hence $\lambda_{S}\in\Sigma_{n,2}$, and $\lambda_{S}\in\Sigma_{n}$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law.

Clause 2 (Schwinger-Dyson). Let $j\in[n]$. By The Law of a Tuple of Bounded Self-Adjoint Operators in a Tracial Vector State §state $\lambda_{S}$ is linear, $p\mapsto x_{j}p$ is linear by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §algebra, and $\partial^{\lambda_{S}}_{j}$ is linear by The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §difference-quotient; so $p\mapsto\lambda_{S}(x_{j}p)-\partial^{\lambda_{S}}_{j}(p)$ is linear, and by (F1) it suffices to show that it vanishes at $p=x_{w}$ for every $w\in W_{n}$.

For $w=\varnothing$: $x_{j}x_{\varnothing}=x_{j}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials, and $S_{j}\Omega=D_{j}\Omega=r_{j}\Omega+r_{j}^{*}\Omega=r_{j}[1]=[x_{j}]$ by Calculus of the Free Fock Space: Orthonormal Monomials, Adjoints, Commutation Relations and the Vacuum §reversal, Calculus of the Free Fock Space: Orthonormal Monomials, Adjoints, Commutation Relations and the Vacuum §adjoints and The Free Fock Space: Creation, Annihilation and Semicircular Operators §operators; hence $\lambda_{S}(x_{j})=\langle[x_{\varnothing}],[x_{(j)}]\rangle=0$ by Calculus of the Free Fock Space: Orthonormal Monomials, Adjoints, Commutation Relations and the Vacuum §orthonormal, which equals $\partial^{\lambda_{S}}_{j}(1)=0$.

Let $w$ have length $k\in\mathbb{N}$, and let $L_{j}(w)=\{s\in[k]:\ w_{s}=j\}$. Then $x_{j}x_{w}=x_{(j)w}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials and $S_{(j)w}=S_{j}S_{w}$ by Evaluation of Noncommutative Polynomials at Bounded Operators: a Unital Homomorphism Compatible with Adjoints, Substitution, Representations and the GNS Multiplication Operators §values. As $S_{j}$ is self-adjoint and $S_{j}\Omega=r_{j}\Omega$ (shown above), the adjoint identities and Calculus of the Free Fock Space: Orthonormal Monomials, Adjoints, Commutation Relations and the Vacuum §annihilation give

$$\lambda_{S}(x_{j}x_{w})=\langle S_{j}\Omega,S_{w}\Omega\rangle=\langle r_{j}\Omega,S_{w}\Omega\rangle=\langle\Omega,r_{j}^{*}S_{w}\Omega\rangle=\sum_{s\in L_{j}(w)}\langle\Omega,S_{w_{>s}}\Omega\rangle\langle\Omega,S_{w_{<s}}\Omega\rangle=\sum_{s\in L_{j}(w)}\lambda_{S}(x_{w_{<s}})\,\lambda_{S}(x_{w_{>s}}),$$

using linearity of the inner product in its second argument. In the sum defining $\partial^{\lambda_{S}}_{j}(x_{w})$ in The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §difference-quotient, the terms with $w_{l}\neq j$ vanish and those with $w_{l}=j$ are the terms above, so $\lambda_{S}(x_{j}x_{w})=\partial^{\lambda_{S}}_{j}(x_{w})$. Thus $\lambda_{S}$ satisfies the Schwinger-Dyson equation.

Clause 3 (Uniqueness). Let $\lambda\in\Sigma_{n}$ satisfy the Schwinger-Dyson equation; $\lambda$ is linear by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state, and so is $\lambda_{S}$, which also satisfies it by Clause 2. We first show $\lambda(x_{u})=\lambda_{S}(x_{u})$ for every $u\in W_{n}$. For $u=\varnothing$ both are $1$ by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §tracial-state (a), since $\lambda_{S}\in\Sigma_{n}$ by Clause 1. For the other words we induct on $m\in\mathbb{N}$, the statement for $m$ being that $\lambda(x_{u})=\lambda_{S}(x_{u})$ for every word $u$ of length at most $m$. For $m=1$, $u=(j)$ is a letter and $\lambda(x_{j})=\lambda(x_{j}1)=\partial^{\lambda}_{j}(1)=0$, and likewise $\lambda_{S}(x_{j})=0$. Suppose the statement holds for $m$ and let $u$ have length $m+1$. Let $j=u_{1}$ and let $v$ be the word of length $m$ with $v_{i}=u_{i+1}$ ($i\in[m]$), so that $u=(j)v$ by Words over a Finite Alphabet: the Empty Word, Concatenation and Reversal §concatenation and $x_{u}=x_{j}x_{v}$ by Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §monomials. By the Schwinger-Dyson equation and The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §difference-quotient,

$$\lambda(x_{u})=\partial^{\lambda}_{j}(x_{v})=\sum_{l=1}^{m}\delta_{v_{l}j}\,\lambda(x_{v_{<l}})\,\lambda(x_{v_{>l}}).$$

For $l\in[m]$ the words $v_{<l}$ and $v_{>l}$ are empty or have lengths $l-1\le m$ and $m-l\le m$, so $\lambda$ and $\lambda_{S}$ agree on $x_{v_{<l}}$ and $x_{v_{>l}}$ by the case $u=\varnothing$ and the induction hypothesis. Hence $\lambda(x_{u})=\partial^{\lambda_{S}}_{j}(x_{v})=\lambda_{S}(x_{j}x_{v})=\lambda_{S}(x_{u})$, by Clause 2. This completes the induction. Now $\lambda-\lambda_{S}$ is linear and vanishes on every monomial, so $\lambda=\lambda_{S}$ by (F1).

Clause 4 (Conjugate variables). Let $\lambda\in\Sigma_{n}$; by Noncommutative Laws of Finitely Many Self-Adjoint Variables with a Norm Bound §law, $\lambda\in\Sigma_{n,r}$ for some real $r>0$. For $j\in[n]$ and $p\in\mathcal{P}_{n}$, Left and Right Multiplication Operators on the Complex GNS Space of a Noncommutative Law: Boundedness, Algebra Rules, Adjoints, Commutation, the Vacuum and the Conjugation §vacuum and $x_{j}^{*}=x_{j}$ (Noncommutative Polynomials Form a Unital Complex Algebra with Involution: Linear Extension from Monomials, Products, Adjoints and Self-Adjoint Parts §adjoint) give

$$\langle\widehat{x_{j}},\widehat{p}\rangle_{\mathcal{H}_{\lambda}}=\lambda(x_{j}^{*}p)=\lambda(x_{j}p).$$

If $\lambda$ satisfies the Schwinger-Dyson equation, then $\langle\widehat{x_{j}},\widehat{p}\rangle_{\mathcal{H}_{\lambda}}=\partial^{\lambda}_{j}(p)$ for all $j$ and $p$, so $\xi_{j}=\widehat{x_{j}}$ satisfy the defining relation of The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §conjugate; thus $\lambda$ has conjugate variables, and by the uniqueness stated there they are $(\widehat{x_{1}},\dots,\widehat{x_{n}})$. Conversely, if $\lambda$ has conjugate variables and they are $(\widehat{x_{1}},\dots,\widehat{x_{n}})$, then by The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law §conjugate and the display, $\lambda(x_{j}p)=\langle\widehat{x_{j}},\widehat{p}\rangle_{\mathcal{H}_{\lambda}}=\partial^{\lambda}_{j}(p)$ for all $j\in[n]$ and $p\in\mathcal{P}_{n}$, which is the Schwinger-Dyson equation.