Internal degrees of freedom

We give the matrix-generation step explicitly. A Hermitian matrix multiplier $M(q)$ is a matrix annihilator if $D\Pcal_\Psi[-iM\Psi]=0$ for every spinor. Scalar phases and scalar coordinate transport have exactly the previous current transformations. Working first with zero Zeeman control, section 3, section 4 therefore supplies all compact scalar annihilators and full compact coordinate covariance, without using scalar uniqueness.

Subtracting a Zeeman control from zero gives its matrix annihilator. Commuting its phase identity with coordinate covariance yields

for compact smooth $u$; this follows by the same symmetric-second-derivative cancellation as in lemma 3.2, since the commutator with a first-order transport differentiates the multiplier. On the control cell (6.2), the choices $u=h(q)e_z$ and $u=h(q)e_x$ give $h(q)J_i^x$ and $h(q)J_i^z$, respectively, for arbitrary compact full-configuration $h$ supported over that cell. Coordinate covariance relocates small supports; a partition of unity gives these multipliers for every compact $h$.

The matrix identities have both Lie and Jordan closure. Commuting two phase identities gives $i[M,N]$. To prove the needed Jordan operation, suppose all compact coefficients of a constant Hermitian matrix $A$ are annihilators. The assignment is invariant under $e^{itfA}$. For the scalar Hamiltonian,

The norm density is unchanged and the current changes only linearly in $t$, by $t(\Psi^\dagger A\Psi)\nabla f$. Comparison of consistency identities gives $|\nabla f|^2A^2$ as an annihilator. Choose $f$ so that a derivative of $|\nabla f|^2$ is nonzero on a prescribed small ball. Equation (6.3), with a vector field dividing by that derivative, then gives $hA^2$ for every $h$ supported in the ball. Partitioning gives arbitrary compact coefficients. Polarization of $(A+B)^2$ yields $AB+BA$; compact cutoffs equal to one on the required supports justify products and commutators throughout.

These operations generate all Hermitian matrices on each irreducible spin factor. Indeed, powers of $J^z$ give its spectral projections by polynomial interpolation. The nonzero adjacent entries of $J^x$ link consecutive eigenvalues. If $P_m,P_n$ are distinct spectral projections, their Jordan products with $J^x$ isolate

and a commutator with $P_m$ gives the corresponding imaginary Hermitian matrix. Adjacent links generate all off-diagonal matrix units by further products and commutators. Diagonal projections supply the rest. Matrices on different factors commute, so their Jordan product is twice their tensor product. We have therefore obtained every compact Hermitian multiplier on the full internal space. This is the point at which irreducibility and spatial variation of the physical controls are used.

Fix a nowhere-zero spinor and a compact field $u$ with $\diver(ru)=0$, where $r=\Psi^\dagger\Psi$. Put $z=\Psi$, $\eta=T_u\Psi$. Then $\operatorname{Re}(z^\dagger\eta)=0$. The compactly supported matrix

is smooth and Hermitian and satisfies $-iMz=\eta$. Indeed, $c=z^\dagger\eta$ is purely imaginary, so $\eta^\dagger z=-c$. The first numerator in (6.4) is Hermitian, its second coefficient $-ic$ is real, and direct multiplication gives $Mz=i\eta$. Division by $r$ is harmless on the compact support of $\eta$. Matrix annihilation and transport covariance imply $\diver(p^\Psi u)=0$. The weighted circulations in proposition 3.3 now give $p^\Psi/r$ constant. Normalization, and then the approximation in section A, finish the proof.