Calculus of the Free Fock Space: Orthonormal Monomials, Adjoints, Commutation Relations and the Vacuum

In the setting of Noncommutative Laws, Couplings and the Wasserstein Distance: Standing Notation, let $n\in\mathbb{N}$, let $\mathcal{F}_{n}$, $[p]$ and $\Omega_{\mathcal{F}}$ be the free Fock space, its classes and its vacuum vector, and let $l_{j},r_{j},\bar{l}_{j},\bar{r}_{j}$ be the creation and annihilation operators ($j\in[n]$). Write $S_{j}=l_{j}+\bar{l}_{j}$ and $D_{j}=r_{j}+\bar{r}_{j}$ (the left and right semicircular operators), and $S=(S_{1},\dots,S_{n})$, $D=(D_{1},\dots,D_{n})$. For $w\in W_{n}$, $S_{w}$ and $D_{w}$ are the products along $w$, and $w^{\mathrm{rev}}$ is the reversal of $w$. The words $w_{<s}$, $w_{>s}$ and the symbol $\delta_{jk}$ are as in The Free Difference Quotient and the Conjugate Variables of a Noncommutative Law. Let $P:\mathcal{F}_{n}\to\mathcal{F}_{n}$ be the map $P\zeta=\langle\Omega_{\mathcal{F}},\zeta\rangle\Omega_{\mathcal{F}}$.

1. (Orthonormal monomials)¶ For all $u,v\in W_{n}$, $\langle[x_{u}],[x_{v}]\rangle=1$ if $u=v$ and $0$ otherwise; in particular $\lVert\Omega_{\mathcal{F}}\rVert=1$, and $P\in\mathcal{L}(\mathcal{F}_{n})$.

2. (Adjoints)¶ For all $j,k\in[n]$: $l_{j}^{*}=\bar{l}_{j}$ and $r_{j}^{*}=\bar{r}_{j}$; $l_{j}^{*}l_{k}=r_{j}^{*}r_{k}=\delta_{jk}I$; and $l_{j}^{*}\Omega_{\mathcal{F}}=r_{j}^{*}\Omega_{\mathcal{F}}=0$.

3. (Commuting creation operators)¶ For all $j,k\in[n]$: $l_{j}r_{k}=r_{k}l_{j}$ and $l_{j}^{*}r_{k}^{*}=r_{k}^{*}l_{j}^{*}$.

4. (Vacuum commutators)¶ For all $j,k\in[n]$: $r_{j}^{*}l_{k}-l_{k}r_{j}^{*}=\delta_{jk}P$ and $l_{j}^{*}r_{k}-r_{k}l_{j}^{*}=\delta_{jk}P$.

5. (Semicircular operators)¶ For every $j\in[n]$, $S_{j}$ and $D_{j}$ are self-adjoint, $\lVert S_{j}\rVert_{\mathrm{op}}\le2$ and $\lVert D_{j}\rVert_{\mathrm{op}}\le2$.

6. (Left and right commute)¶ $S_{j}D_{k}=D_{k}S_{j}$ for all $j,k\in[n]$.

7. (Reversal)¶ $S_{w}\Omega_{\mathcal{F}}=D_{w^{\mathrm{rev}}}\Omega_{\mathcal{F}}$ for every $w\in W_{n}$.

8. (Right annihilation of the vacuum orbit)¶ Let $j\in[n]$ and let $w\in W_{n}$ have length $k\in\mathbb{N}$, and let $L_{j}(w)=\{s\in[k]:\ w_{s}=j\}$. Then

$$r_{j}^{*}S_{w}\Omega_{\mathcal{F}}=\sum_{s\in L_{j}(w)}\langle\Omega_{\mathcal{F}},S_{w_{>s}}\Omega_{\mathcal{F}}\rangle\,S_{w_{<s}}\Omega_{\mathcal{F}},$$

the sum being $0$ when $L_{j}(w)$ is empty.