Spatial polarization textures

Summary: Allowing polarization to vary across a bound particle can preserve its scalar density while creating a full spatial kinetic residual, whose omitted components must be solved before the texture is a physical mode.

Allow the polarization to vary over the actual bound coordinates

A constant vector of complex amplitudes describes one part of a particle's polarization; Complete spatial polarization data identifies its full tensor description. Its phase and orientation can also vary across its spatial profile. That possibility must be tested in the field equation rather than accepted because it preserves a selected density.

Use the complete stationary radial frame of Equal-current spatial polarizations:

$$H_bW=MW,\qquad W^\dagger W=\rho(x)I,\qquad W^\dagger\beta W=\sigma(x)I. \tag{1}$$

The bound coordinates $x$ are ordinary macroscopic directions in the particle's submanifold. There is only the one infinite nested physical manifold. The frame's column labels describe field components over it.

Consider

$$\boxed{\Psi(x,t)=e^{-iMt}W(x)z(x,t),\qquad z^\dagger z=1.} \tag{2}$$

At every point, the number and scalar densities remain $\rho$ and $\sigma$. For a scalar self-binding source depending only on the latter density's centered moments, its coefficient is unchanged. This statement holds regardless of how rapidly $z$ varies. It establishes the source value, not the full spatial equation.

The spatial derivative supplies an additional response

With $H_b=-i\alpha_i\partial_i+\beta m_{\rm eff}(x)$, direct application of the product rule gives

$$\boxed{ H_b(Wz)=MWz-i\sum_i\alpha_iW\,\partial_i z.} \tag{3}$$

Consequently the remaining field equation after removing the carrier is

$$\boxed{iW\partial_tz=-i\sum_i\alpha_iW\,\partial_i z.} \tag{4}$$

The right side need not lie in the image of $W(x)$. Where $\rho(x)>0$, define the actual local projector and compressed current matrices

$$\boxed{ P_W=\frac{WW^\dagger}{\rho},\qquad A_i=\frac{W^\dagger\alpha_iW}{\rho}.} \tag{5}$$

Equation (4) is equivalent to the projected equation together with its complementary condition:

$$\boxed{ i\partial_tz=-i\sum_i A_i\partial_i z,\qquad 0=-i\sum_i(I-P_W)\alpha_iW\,\partial_i z.} \tag{6}$$

The second equality is essential. Solving only the first can manufacture an apparent polarization mode while omitting a nonzero spatial response. For a real evolving field, additional profile components can be excited, and its source must then be recomputed.

The radial frame gives the full matrix identity

$$\boxed{W^\dagger\alpha_iW =-\frac{2FG}{\Omega_{d-1}} \sum_j B_{ij}\widehat x_j,} \tag{7}$$

where $B_{ji}=-B_{ij}$ and $B_{ii}=0$. The diagonal bilinear of (7) recovers the ordinary Dirac current. Its entire matrix is needed for the varying amplitudes in (6).

The first equation also carries the number continuity condition. Because $\partial_i(\rho A_i)=0$ for this fixed radial frame,

$$\partial_t(\rho\,z^\dagger z) +\partial_i(\rho\,z^\dagger A_i z)=0 \tag{8}$$

follows from it. Maintaining the pointwise constraint $z^\dagger z=1$ requires the resulting current to have zero divergence. An arbitrary initial texture need not preserve that constraint under its dynamics.

A concrete texture lies outside the retained equation

Take a normalized constant $z_0$ and a Hermitian spatial rotation matrix $B$ with $B^2=I$, and set

$$z(x)=e^{ikx_1B}z_0. \tag{9}$$

Only $\partial_1z=ikBz$ is nonzero, so its stationary kinetic remainder is

$$\boxed{\mathcal R=k\alpha_1WBz,\qquad \|\mathcal R(x)\|^2=k^2\rho(x).} \tag{10}$$

On the physical axis $x=r e_1$, equation (7) gives $A_1=0$. Thus

$$\boxed{P_W\mathcal R=0,\qquad \|(I-P_W)\mathcal R\|^2=k^2\rho.} \tag{11}$$

At that point the entire nonzero response is missed by the projected polarization equation. The example identifies precisely which part of the full spatial equation must be retained; it does not claim that the imposed texture is a stationary particle.

Even a common spatial phase can conceal the missing motion

Let $\Phi$ be a unit-normalized stationary field of the same model. For a smooth real function $\theta(x)$, put

$$\widetilde\Phi(x)=e^{i\theta(x)}\Phi(x). \tag{12}$$

Every local nondifferential bilinear is unchanged:

$$\widetilde\Phi^\dagger O\widetilde\Phi=\Phi^\dagger O\Phi \tag{13}$$

for a local component operator $O$. This includes the number density, the scalar source and every local Dirac current component. The spatial equation nevertheless gives

$$\boxed{ (H_b-M)\widetilde\Phi =e^{i\theta}\alpha(\nabla\theta)\Phi,\qquad \|(H_b-M)\widetilde\Phi\|_{L^2}^2 =\int\rho\,|\nabla\theta|^2\,dx.} \tag{14}$$

The last equality uses the full Clifford identity $\alpha(v)^\dagger\alpha(v)=|v|^2I$ for real $v$. For $\theta=kx_1$, the residual norm is exactly $|k|$ for a normalized field. This is a physical spatial phase gradient, despite all the unchanged bilinears in (13).

The original scalar Dirac equation has a global phase symmetry. A local phase with its connection held fixed changes the field dynamics. If a gauge connection is part of a different specified action, transform that connection too before declaring two configurations gauge-equivalent.

What a complete polarization calculation must add

The constant polarization waves supply exact solutions with their own normal partner and unchanged scalar source. The scalar envelope derives a separate remainder when the bound profile changes with its amplitude. Equations (3)–(14) add the bound-coordinate texture response.

A proposed richer particle pattern must therefore supply its spatial amplitudes and phases, satisfy the complete kinetic equation, and retain the profile components and source changes generated by that equation. The neutral-doublet transport provides another example where a projected derivative omits a necessary Dirac complement. Neither a density plot nor the number of polarization parameters is sufficient to establish a physical mode.

Verification

spatial_polarization_structure_checks.py uses complete self-consistent spatial fields at depths three, five, six and ten. It tests their local metrics and projectors, then differentiates the textured fields in every actual bound coordinate.

Independent full Dirac evaluations verify (3), the projected and complementary terms in (6), the wholly omitted response in (11), and the spatial-phase norm in (14). The scalar source and every Dirac current are recomputed rather than inferred from a plotted profile. Reports are timestamped and all checks remain active under Python optimization.