Section 3 4 October 2026

From phases to transport and uniqueness

Jeremy Rodgers · Independent Researcher
doi:10.5281/zenodo.23131058

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3 From phases to transport and uniqueness

The unitary change of variables $x_i=\sqrt{m_i}q_i$ , including its constant half-density factor, makes the kinetic energy $-\Delta_x/2$ . It preserves particle blocks, compact support, the Schwartz topology, and condition (2.2). We use these mass-scaled coordinates in the proofs.

3.1 Statistical annihilators

Call a real smooth Schwartz multiplier $f$ a scalar annihilator if

D\Pcal_\Psi[-if\Psi]=0\quad\hbox{for every }\Psi.

(3.1)

A Schwartz multiplier is a smooth function whose multiplication operator preserves $\Sch$ continuously; all bounded smooth functions with bounded derivatives, and the polynomial functions used below, qualify. The identities are always local off nodes. Denote their real vector space by $\Acal$ .

Subtracting (2.3) for two controls shows that every admitted control difference belongs to $\Acal$ , because a scalar potential does not change the current at a fixed state. This is the initial source of annihilators. There is also a useful closure rule:

f_n\in\Acal,\quad f_n\Psi\longrightarrow f\Psi\ \hbox{in }\Sch \ \hbox{for each fixed }\Psi \quad\Longrightarrow\quad f\in\Acal.

(3.2)

It follows from continuity of the first derivative in A2. All limits below have exactly this meaning; no uniform operator-norm closure is assumed.

For a smooth vector field $u$ , put

T_u\Psi=-u\cdot\nabla\Psi-\tfrac12(\diver u)\Psi, \qquad A_u p=-\diver(up).

(3.3)

The first is the infinitesimal action on half-densities, and the second is the action on densities.

Lemma 3.1 (Phase and gradient identities)

If $f\in\Acal$ and its phase multipliers preserve $\Sch$ , then

D\Pcal_\Psi[T_{\nabla f}\Psi]=A_{\nabla f}p^\Psi, \qquad |\nabla f|^2\in\Acal.

(3.4)

Consequently $f,g\in\Acal$ imply

\Gamma(f,g):=\nabla f\cdot\nabla g\in\Acal

(3.5)

whenever the displayed multipliers are admissible on $\Sch$ .

Proof

Integrating (3.1) along $e^{itf}\Psi$ gives $p^{e^{itf}\Psi}=p^\Psi$ , since the nonzero set does not change. Differentiating this identity in the state also intertwines the corresponding first derivatives. For the scalar Hamiltonian $H$ ,

-i(e^{-itf}He^{itf}-H)\Psi =tT_{\nabla f}\Psi-\frac{it^2}{2}|\nabla f|^2\Psi.

The velocity of $e^{itf}\Psi$ is $v^\Psi+t\nabla f$ . Comparing (2.3) at these two states therefore gives a polynomial identity in $t$ . Its linear and quadratic coefficients give (3.4). Polarizing the second identity proves (3.5).

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Lemma 3.2 (Transport generation)

If every compactly supported smooth scalar function on $\R^d$ is in $\Acal$ , then

D\Pcal_\Psi[T_u\Psi]=A_up^\Psi

(3.6)

for every $u\in\Cc(\R^d;\R^d)$ . It follows that

\Pcal(U_F\Psi)=F_*\Pcal(\Psi),\qquad (U_F\Psi)(q)=|\det DF^{-1}(q)|^{1/2}\Psi(F^{-1}(q)),

(3.7)

on corresponding nonzero patches, for every finite product $F$ of flows of such vector fields. The same assertion holds within one particle block if only that block's compact multipliers are known.

Proof

If (3.6) holds for $u,v$ , differentiating the two identities in the directions $T_v\Psi,T_u\Psi$ and subtracting cancels the symmetric second derivative. To make the regularity requirement explicit, choose a real smooth test function $\zeta$ supported in a compact nonzero patch. Differentiation of the $u$ -identity and use of the $v$ -identity gives

\left\langle D^2\Pcal_\Psi[T_v\Psi,T_u\Psi]+D\Pcal_\Psi[T_uT_v\Psi],\zeta\right\rangle =\left\langle p^\Psi,v\cdot\nabla(u\cdot\nabla\zeta)\right\rangle.

The reversed identity has $u,v$ interchanged. Thus every spatial differentiation can be placed on the test function. With $[u,v]=u\cdot\nabla v-v\cdot\nabla u$ , one has $[T_u,T_v]=-T_{[u,v]}$ and $[A_u,A_v]=-A_{[u,v]}$ . Hence the identity also holds for $[u,v]$ . The density commutators are interpreted distributionally: this calculation needs functional $C^2$ , but not spatial $C^2$ , regularity of $p^\Psi$ .

For smooth real $\phi,\gamma$ , the Euclidean gradient-bracket identity is

\begin{align}\phi|\nabla\gamma|^2\nabla\gamma ={}&-\tfrac14[\nabla(\phi\gamma^2),\nabla\gamma] -\tfrac1{12}[\nabla\phi,\nabla(\gamma^3)] \notag\\ &-\tfrac14[\nabla(\gamma^2),\nabla(\phi\gamma)]. \tag{3.8}\end{align}

This is the specialization of [1, Proposition 1, equation (10)]; expansion by the product rule also verifies it directly. For each component $u_j$ of a compact vector field, choose $\gamma_j\in\Cc$ equal to $q_j$ on a neighborhood of its support and take $\phi=u_j$ . The left side is $u_j e_j$ . Every function on the right is compactly supported. lemma 3.1, bracket closure, and summation prove (3.6).

For completeness, writing $a=\nabla\phi$ , $b=\nabla\gamma$ , $P=D^2\phi$ , and $G=D^2\gamma$ , the three brackets on the right of (3.8), in their displayed order, are

\begin{aligned}&\gamma^2(Ga-Pb)-2\gamma(a\cdot b)b-2\gamma|b|^2a-2\phi|b|^2b,\\ &3\gamma^2(Ga-Pb)+6\gamma(a\cdot b)b,\\ &-2\gamma^2(Ga-Pb)+2\gamma|b|^2a-2\phi|b|^2b. \end{aligned}

Multiplication by $-1/4,-1/12,-1/4$ cancels every term except $\phi|b|^2b$ . This verifies the precise local identity used here independently of a transport-group theorem.

Along the half-density flow, (3.6) is the linear transport equation for the assigned density. Its unique local distributional solution is the pushforward: testing against the inverse-transported test function makes its derivative zero. Compact support gives a complete smooth coordinate flow. This proves (3.7); the one-block proof is identical with other coordinates as parameters.

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These transports are auxiliary actions on states and probability assignments. We have not claimed that $U_F$ is a Schrödinger propagator produced by the physical controls.

Proposition 3.3 (Full scalar-control criterion)

For scalar states on $\R^d$ , $d\ge2$ , assumptions A1–A3 and $\Cc(\R^d;\R)\subset\Acal$ imply (1.1) on every nowhere-zero state.

Proof

Fix such a state, put $r=|\Psi|^2$ , and let $u$ be compactly supported with $\diver(ru)=0$ . Then $2\operatorname{Re}(\overline\Psi T_u\Psi)=-\diver(ru)=0$ . Since the state is scalar and nonzero, $T_u\Psi=-if\Psi$ for a real compactly supported smooth $f$ . The annihilator identity and lemma 3.2 give

0=\diver(p^\Psi u)=r\,u\cdot\nabla(p^\Psi/r).

Such weighted divergence-free fields span every tangent space. Indeed, in a ball around $q_0$ , choose $j\ne k$ , a compact $\chi$ equal to $r(q_0)(q_k-q_{0k})$ near $q_0$ , and set

u=r^{-1}\big((\partial_k\chi)e_j-(\partial_j\chi)e_k\big).

This field has $\diver(ru)=0$ and $u(q_0)=e_j$ . Therefore $\nabla(p^\Psi/r)=0$ . Connectedness of $\R^d$ and normalization prove the assertion.

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Remark 3.4 (The line exception)

For one scalar coordinate let $r=|\Psi|^2/\norm{\Psi}_2^2$ and $F^\Psi(x)=\int_{-\infty}^x r(s)\,ds$ . For any positive smooth $g$ on $[0,1]$ with integral one,

p^\Psi(x)=r(x)g(F^\Psi(x))

is normalized, projective, and has the stated functional regularity. The continuity equation with vanishing flux at infinity gives $(\partial_t+v\partial_x)F^\Psi=0$ , proving equivariance for every scalar potential and positive mass. The assignment also has the approximation continuity above. A nonconstant $g$ , for example $1+\alpha\cos(2\pi s)$ with $0<|\alpha|<1$ , gives a non-Born law. This is the exception discussed in [6, Section 8]. Local circulation in dimension at least two is essential to proposition 3.3.