revisited {#subsec:finite_Fisher}
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Let $(M,\tau)$ be a tracial $W^\ast$-probability space and let $X_1,\dots,X_n \in M$ be self-adjoint with $\Phi^\ast(X_1,\dots,X_n) < \infty$.

As we have seen in Section \[algebraic\_relations\], those conditions guarantee that, for $j=1,\dots,n$, there exists a unique derivation $$\partial_j:\ {\mathbb{C}}\langle X_1,\dots,X_n\rangle \rightarrow {\mathbb{C}}\langle X_1,\dots,X_n\rangle \otimes {\mathbb{C}}\langle X_1,\dots,X_n\rangle,$$ which is determined by the condition $\partial_j X_i = \delta_{i,j} 1 \otimes 1$ for $i=1,\dots,n$. For each $j=1,\dots,n$, we may consider $\partial_j$ as a densely defined unbounded linear operator $$\partial_j:\ L^2(X_1,\dots,X_n,\tau) \supseteq D(\partial_j) \rightarrow L^2(X_1,\dots,X_n, \tau \otimes \tau)$$ with domain $D(\partial_j) := {\mathbb{C}}\langle X_1,\dots,X_n\rangle$. Note that as $L^2(X_1,\dots,X_n,\tau)$ stands for the closure of ${\mathbb{C}}\langle X_1,\dots,X_n\rangle$ in $L^2(M,\tau)$, the closure of ${\mathbb{C}}\langle X_1,\dots,X_n\rangle^{\otimes 2}$ in the $L^2$-space $L^2(M \overline{\otimes} M,\tau \otimes \tau)$, constructed over the von Neumann algebra tensor product $M \overline{\otimes} M$ of $M$ with itself, is denoted here by $L^2(X_1,\dots,X_n,\tau \otimes \tau)$.

Since due to $\Phi^\ast(X_1,\dots,X_n) < \infty$ a conjugate system $(\xi_1,\dots,\xi_n)$ for $(X_1,\dots,X_n)$ exists, we see by that $1 \otimes 1$ belongs to the domain of definition of the adjoints $\partial_1^\ast,\dots,\partial_n^\ast$ and that $\xi_j = \partial_j^\ast(1 \otimes 1)$ holds for $j=1,\dots,n$.

The subsequent considerations will be based on several well-known facts about those operators $\partial_j$, which we collect here for 