# Section 4: Local controls and fixed interactions

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<a id="section-4"></a>

## 4 Local controls and fixed interactions

<a id="sec:local"></a>

Local scalar controls initially give ${C_c^\infty}({\mathbb R}^3)$ annihilators in each particle block. By [lemma 3.2](/quantum-measurement/research/control-consistency/from-phases-to-transport-and-uniqueness#lem:transport), the assignment is covariant under independent compactly supported coordinate flows on the particles. We now use that covariance to expose the interaction.



**Lemma 4.1 (Mixed conjugation).**

<a id="lem:mixed"></a> For an edge $\{i,j\}$, and compactly supported one-body diffeomorphisms $F,G$ generated by flows on particles $i,j$, respectively, the function <a id="eq:mixed"></a>


$$
\begin{aligned}M_{F,G}(q_i,q_j)={}&W_{ij}(F(q_i),G(q_j))
 -W_{ij}(F(q_i),q_j)\\
 &-W_{ij}(q_i,G(q_j))+W_{ij}(q_i,q_j)
 
\end{aligned}
$$

Equation (4.1).

 belongs to ${\mathcal A}$. In particular, for $u,v\in{C_c^\infty}({\mathbb R}^3;{\mathbb R}^3)$, <a id="eq:edge-source"></a>


$$

 \sum_{a,b}u^a(q_i)v^b(q_j)
 \partial_{i,a}\partial_{j,b}W_{ij}(q_i,q_j)\in{\mathcal A}.

$$

Equation (4.2).

 





**Proof.**

Set $H^F=U_F^{-1}HU_F$, using the block half-density action, and similarly for $G,FG$. Covariance [(3.7)](/quantum-measurement/research/control-consistency/from-phases-to-transport-and-uniqueness#eq:cov-fin) transfers the statistical evolution identity for $H$ to $H^F$. To see precisely which current is transferred, a real half-density coordinate conjugation of $-\Delta/2$ has the form 

$$

 -\tfrac12\partial_\alpha(a^{\alpha\beta}\partial_\beta)+V_{\mathrm{real}},
 \qquad a=(DF)^{-1}(DF)^{-T}.

$$

 Its current is $a^{\alpha\beta}\operatorname{Im}(\overline\Psi\partial_\beta\Psi)$. Explicitly, if $v_F^\Psi$ denotes the conjugated velocity, then 

$$

 v_F^\Psi(q)=DF(q)^{-1}v^{U_F\Psi}(F(q))
 =\frac{a(q)\operatorname{Im}(\overline\Psi(q)\nabla\Psi(q))}{|\Psi(q)|^2}.

$$

 The Jacobian half-density factor is real and hence contributes no imaginary phase gradient. Changing variables in the continuity equation transfers the assigned-density identity with this same velocity. Only the metric in the transformed particle block changes.

The operator difference 

$$

 H^{FG}-H^F-H^G+H

$$

 therefore has zero kinetic part. All one-body terms cancel. All pair terms except $W_{ij}$ cancel as well, leaving exactly multiplication by [(4.1)](/quantum-measurement/research/control-consistency/local-controls-and-fixed-interactions#eq:mixed). The same mixed difference of the four currents is zero, block by block. Subtracting the four transferred statistical identities gives $D{\mathcal P}_\Psi[-iM_{F,G}\Psi]=0$.

Take $F=F_s^u$, $G=F_t^v$, divide by $st$, and let $s,t\to0$. The limit is [(4.2)](/quantum-measurement/research/control-consistency/local-controls-and-fixed-interactions#eq:edge-source). The mixed differences and their derivatives have common compact support in the two active blocks; the other coordinates are spectators only in this multiplier estimate. Taylor's formula gives convergence on every Schwartz tangent vector. Thus [(3.2)](/quantum-measurement/research/control-consistency/from-phases-to-transport-and-uniqueness#eq:closure) applies. 

□





**Lemma 4.2 (Edge saturation).**

<a id="lem:edge"></a> If [(2.2)](/quantum-measurement/research/control-consistency/physical-model-and-statistical-assumptions#eq:interaction) holds, every real function in ${C_c^\infty}({\mathbb R}^3\times{\mathbb R}^3)$, viewed as a multiplier in the edge variables, belongs to ${\mathcal A}$. 





**Proof.**

Some component $c=\partial_{i,a}\partial_{j,b}W_{ij}$ is nonzero on a product of sufficiently small balls $B_i\times B_j$. Taking $u=f e_a,v=g e_b$ in [(4.2)](/quantum-measurement/research/control-consistency/local-controls-and-fixed-interactions#eq:edge-source) gives $f(q_i)g(q_j)c(q_i,q_j)\in{\mathcal A}$. For $h$ compactly supported inside that product, $h/c$ is smooth there. Finite sums of separated compact functions approximate it in $C^\infty$, with common compact support. For example, extend it to a box, approximate its smooth periodic extension by Fourier partial sums, and multiply by fixed cutoffs in the two variables. Multiplication by $c$ and [(3.2)](/quantum-measurement/research/control-consistency/from-phases-to-transport-and-uniqueness#eq:closure) give $h\in{\mathcal A}$.

Annihilator identities are invariant under pullback by the one-body coordinate transports: differentiate [(3.7)](/quantum-measurement/research/control-consistency/from-phases-to-transport-and-uniqueness#eq:cov-fin) along a phase variation. Independent compact diffeomorphisms can move any sufficiently small pair of balls to this nonvanishing product. Pullback therefore gives all compact multipliers in every such product cell. A finite partition of unity on the support of a target function proves the result globally. 

□



The remaining step combines overlapping edge variables. We include the elementary local factorization that makes this step exact.



**Lemma 4.3 (Gradient-product factorization).**

<a id="lem:factor"></a> On a Euclidean open set, every real compact smooth function is a finite sum of terms $\nabla a\cdot\nabla b$, with $a,b$ compactly supported in that open set. 





**Proof.**

A finite partition reduces the question to a small box with room to spare in a larger box. Write its coordinates as $(x_1,x')$, let $H(x')=\int f(t,x')\,dt$, and choose a unit-integral bump $\alpha(x_1)$ in a spare interval disjoint from the $x_1$-projection of ${\operatorname{supp}} f$. Then 

$$

 a(x_1,x')=\int_{-\infty}^{x_1}
       \big(f(t,x')-\alpha(t)H(x')\big)\,dt

$$

 is compactly supported. Choose a compact function $\beta(x_1)$ whose derivative is one on the projection of ${\operatorname{supp}} f$, zero on ${\operatorname{supp}}\alpha$, and has its compensating integral in another spare interval. Let $\chi(x')$ be one near the transverse support of $a$ and put $b=\beta\chi$. Transverse derivative products vanish, whereas $\partial_1a\,\partial_1b=(f-\alpha H)\beta'=f$. All supports fit in the larger box. Summation proves the lemma. 

□





**Proposition 4.4 (Connected-graph saturation).**

<a id="prop:graph"></a> Under the hypotheses of [theorem 2.1](/quantum-measurement/research/control-consistency/physical-model-and-statistical-assumptions#thm:main), 

$$

 {C_c^\infty}({\mathbb R}^{3N};{\mathbb R})\subset{\mathcal A}.

$$

 





**Proof.**

Induct along a spanning tree. One-body and edge multipliers are available. Suppose all compact multipliers on a connected collection $S$ of particle blocks are available, and attach a new particle $j$ by an edge at $i\in S$. For compact one-body factors $f_k$, apply [lemma 4.3](/quantum-measurement/research/control-consistency/local-controls-and-fixed-interactions#lem:factor) to the desired factor on block $i$, writing $f_i=\sum_\ell\nabla a_\ell\cdot\nabla b_\ell$. Then 

$$

 \Gamma\left(a_\ell(q_i)\!\!\prod_{k\in S\setminus\{i\}}\!\!f_k(q_k),
                 b_\ell(q_i)f_j(q_j)\right)
 =(\nabla a_\ell\cdot\nabla b_\ell)(q_i)
                  \prod_{k\in(S\cup\{j\})\setminus\{i\}}f_k(q_k).

$$

 Only the shared block contributes. The two inputs are available by induction and edge saturation, so [lemma 3.1](/quantum-measurement/research/control-consistency/from-phases-to-transport-and-uniqueness#lem:phase) gives the desired separated product after summation. Separated compact functions are $C^\infty$-dense, with common blockwise compact supports, in compact functions on these blocks, by the Fourier-cutoff argument in [lemma 4.2](/quantum-measurement/research/control-consistency/local-controls-and-fixed-interactions#lem:edge). This is convergence after multiplication by each fixed Schwartz state, including spectator variables. Thus [(3.2)](/quantum-measurement/research/control-consistency/from-phases-to-transport-and-uniqueness#eq:closure) completes the induction. 

□





**Proof of [theorem 2.1](/quantum-measurement/research/control-consistency/physical-model-and-statistical-assumptions#thm:main).**

 Use [proposition 4.4](/quantum-measurement/research/control-consistency/local-controls-and-fixed-interactions#prop:graph), [proposition 3.3](/quantum-measurement/research/control-consistency/from-phases-to-transport-and-uniqueness#prop:saturation) in dimension $3N\ge2$. For $N=1$, one-body controls already give full scalar saturation. [lemma A.1](/quantum-measurement/research/control-consistency/appendix-a-nonzero-approximation-and-nodes#lem:nodal-density) below supplies normalized nowhere-zero approximations; [A4](/quantum-measurement/research/control-consistency/physical-model-and-statistical-assumptions#ass:nodes) and [(2.4)](/quantum-measurement/research/control-consistency/physical-model-and-statistical-assumptions#eq:L2TV) then extend the equality of measures to every state. Undoing mass scaling gives [(1.1)](/quantum-measurement/research/control-consistency/introduction#eq:born). 

□



The proof has generated identities for a statistical functional, including identities associated with joint multipliers. None of the mixed conjugations, brackets, or density limits is a claim that an additional joint potential has been physically implemented. Only the controls quantified in [(2.3)](/quantum-measurement/research/control-consistency/physical-model-and-statistical-assumptions#eq:cc) are physical inputs.