Observable Quotients and Exact Projected Dynamics: Closure, Memory, Minimal Dynamical Completion, and Effective Field Operators

Descent of observable functionals

Jeremy RodgersSection 4 of 14Original PDF ↗Publication DOI ↗

Descent of observable functionals

Definition 3.1 (Exact observability of a functional).

Let $V$ be a set and $A:\mathcal{H}_{\mathrm{src}}\to V$ . We say $A$ is exactly observable through $K$ if there exists $\bar A:\operatorname{Ran}K\to V$ with $A=\bar A\circ K .\tag{3.1}$

Proposition 3.2 (Functional descent).

$A$ is exactly observable through $K$ if and only if $A(\Psi+n)=A(\Psi) \qquad\text{for all }\Psi\in\mathcal{H}_{\mathrm{src}},\ n\in\ker K .\tag{3.2}$ When (3.2) holds, $\bar A$ is unique.

Proof. If (3.1) holds and $n\in\ker K$ , then $A(\Psi+n)=\bar A(K\Psi+Kn)=\bar A(K\Psi)=A(\Psi)$ , which is (3.2).

Conversely, assume (3.2). Given $\omega\in\operatorname{Ran}K$ , choose any $\Psi$ with $K\Psi=\omega$ and set $\bar A(\omega):=A(\Psi)$ . If $K\Psi=K\Psi'=\omega$ then $\Psi'=\Psi+n$ with $n\in\ker K$ , so (3.2) gives $A(\Psi')=A(\Psi)$ ; the definition is therefore independent of the chosen preimage, and (3.1) holds by construction. Uniqueness is forced because $K$ is onto $\operatorname{Ran}K$ : any $\bar A'$ satisfying (3.1) agrees with $\bar A$ at every $\omega=K\Psi$ . ◻

Proposition 3.2 is the fiberwise descent statement of [22] specialized to the affine fibers $\Psi+\ker K$ ; we have proved it here so that no external result is load-bearing.

Corollary 3.3 (Linear descent and the adjoint).

Let $A\in\mathcal{H}_{\mathrm{src}}^{*}$ be a continuous linear functional. Then $A$ descends through $K$ , in the sense of Definition 3.1, if and only if $A|_{\ker K}=0 .\tag{3.3}$ Identifying $\mathcal{H}_{\mathrm{src}}^{*}\cong\mathcal{H}_{\mathrm{src}}$ by the Riesz representation $A=\langle a , \cdot \rangle$ , condition (3.3) says $a\in(\ker K)^{\perp}$ , and $(\ker K)^{\perp}=\overline{\operatorname{Ran}K^{*}} .\tag{3.4}$

Proof. Since $A$ is linear, (3.2) reads $A(n)=0$ for all $n\in\ker K$ , which is (3.3). For (3.4): $\Psi\in\ker K$ iff $K\Psi=0$ iff $\langle K\Psi , \omega \rangle=0$ for all $\omega\in\mathcal{H}_{\mathrm{obs}}$ iff $\langle \Psi , K^{*}\omega \rangle=0$ for all $\omega$ , i.e. iff $\Psi\perp\operatorname{Ran}K^{*}$ . Hence $\ker K=(\operatorname{Ran}K^{*})^{\perp}$ , and taking orthogonal complements and using $M^{\perp\perp}=\overline{M}$ for a subspace $M$ gives $(\ker K)^{\perp}=\overline{\operatorname{Ran}K^{*}}$ . ◻

It is essential to notice what Definition 3.1 does and does not require. It requires a factorization, $A=\bar A\circ K$ , and nothing more: $\bar A$ is a map on the set $\operatorname{Ran}K$ , and no continuity of $\bar A$ with respect to any particular norm on $\operatorname{Ran}K$ has been demanded. Corollary 3.3 therefore characterizes algebraic descent. For a continuous linear $A=\langle a , \cdot \rangle$ with $a\in(\ker K)^{\perp}$ , the induced $\bar A$ is automatically continuous in the transported quotient norm $\lVert\cdot\rVert_{\sharp}$ of Remark 2.4 (because $\bar A\circ\widetilde{K}$ is the continuous functional $[\Psi]\mapsto\langle a , \Psi \rangle$ on the Hilbert space $\mathcal{H}_{K}$ ). But continuity in the inherited observable norm, the norm an experimenter actually has access to on $\mathcal{H}_{\mathrm{obs}}$ , is a strictly stronger demand, and it is governed by a different, sharper condition.

Theorem 3.4 (Stable observability of a linear functional).

Let $A=\langle a , \cdot \rangle\in\mathcal{H}_{\mathrm{src}}^{*}$ with $a\in(\ker K)^{\perp}$ , so that $A$ descends algebraically, with unique $\bar A:\operatorname{Ran}K\to\mathbb{C}$ satisfying $A=\bar A\circ K$ . Then $\bar A\ \text{is bounded on }(\operatorname{Ran}K,\lVert\cdot\rVert_{\mathcal{H}_{\mathrm{obs}}}) \qquad\Longleftrightarrow\qquad a\in\operatorname{Ran}K^{*} ,\tag{3.5}$ i.e. if and only if there exists an exact observable probe $\omega\in\mathcal{H}_{\mathrm{obs}}$ with $a=K^{*}\omega$ . In that case $\bar A$ extends to the bounded functional $\langle \omega , \cdot \rangle$ on all of $\mathcal{H}_{\mathrm{obs}}$ , and $A(\Psi)=\langle \omega , K\Psi \rangle_{\mathcal{H}_{\mathrm{obs}}}$ for every $\Psi$ : the source quantity is read off the observable state by a bounded measurement.

Proof. ( $\Leftarrow$ ) Suppose $a=K^{*}\omega$ with $\omega\in\mathcal{H}_{\mathrm{obs}}$ . Then for every $\Psi$ , $A(\Psi)=\langle a , \Psi \rangle=\langle K^{*}\omega , \Psi \rangle=\langle \omega , K\Psi \rangle_{\mathcal{H}_{\mathrm{obs}}},$ so $\bar A(y)=\langle \omega , y \rangle_{\mathcal{H}_{\mathrm{obs}}}$ for $y\in\operatorname{Ran}K$ , which satisfies $\lvert\bar A(y)\rvert\le\lVert\omega\rVert\,\lVert y\rVert_{\mathcal{H}_{\mathrm{obs}}}$ and is therefore bounded in the inherited norm.

( $\Rightarrow$ ) Suppose $\bar A$ is bounded on $(\operatorname{Ran}K,\lVert\cdot\rVert_{\mathcal{H}_{\mathrm{obs}}})$ , say $\lvert\bar A(y)\rvert\le M\lVert y\rVert_{\mathcal{H}_{\mathrm{obs}}}$ for all $y\in\operatorname{Ran}K$ . Since $\operatorname{Ran}K$ is a subspace of $\mathcal{H}_{\mathrm{obs}}$ , the Hahn–Banach theorem [1, Cor. 1.2] extends $\bar A$ to a bounded linear functional $\Lambda$ on all of $\mathcal{H}_{\mathrm{obs}}$ with the same norm. By the Riesz representation theorem there is $\omega\in\mathcal{H}_{\mathrm{obs}}$ with $\Lambda=\langle \omega , \cdot \rangle$ . Then for every $\Psi\in\mathcal{H}_{\mathrm{src}}$ , $\langle a , \Psi \rangle=A(\Psi)=\bar A(K\Psi)=\Lambda(K\Psi)=\langle \omega , K\Psi \rangle =\langle K^{*}\omega , \Psi \rangle ,$ so $\langle a-K^{*}\omega , \Psi \rangle=0$ for all $\Psi$ , i.e. $a=K^{*}\omega\in\operatorname{Ran}K^{*}$ . ◻

Remark 10 (Algebraic descent versus stable readout: exact preimage and closure).

Equation (3.4) contains a closure and one must not delete it, and Theorem 3.4 says exactly what the closure costs. Combining the two:

$a\in\overline{\operatorname{Ran}K^{*}}=(\ker K)^{\perp}$ $\iff$ $A$ descends algebraically through $K$ (Corollary 3.3), with $\bar A$ continuous in the transported quotient norm;
$a\in\operatorname{Ran}K^{*}$ $\iff$ $A$ descends to a functional that is bounded in the inherited observable norm (Theorem 3.4).

The two conditions coincide precisely when $\operatorname{Ran}K^{*}$ is closed, which by the closed range theorem is equivalent to $\operatorname{Ran}K$ being closed [1, Thm. 2.19], the same dichotomy as in Theorem 2.3. So for a compact $K$ with infinite-dimensional range, for instance Example 4.5, $\operatorname{Ran}K^{*}$ is dense in $(\ker K)^{\perp}$ but proper, and there exist source functionals which descend perfectly well as maps of the observable state, yet possess no exact representing probe $\omega$ with $a=K^{*}\omega$ . Such a functional is a limit of representable ones, but the approximating probes $\omega_{k}$ have $\lVert\omega_{k}\rVert\to\infty$ : reading the quantity off the observable state amplifies observational error without bound. This is (D2) reappearing on the dual side. Accordingly we say that the elements of $(\ker K)^{\perp}$ are the observable linear source directions, and reserve stably observable for those in $\operatorname{Ran}K^{*}$ ; the phrase should never be read as asserting inherited-readout stability where only algebraic descent has been established.