Equal-current spatial polarizations

Summary: Complete ten-dimensional fields can share density and circulation yet respond differently to the same six-dimensional partner, because reciprocal rotation dynamics retains polarization information beyond the current tensor.

Put the additional polarization into an actual spatial field

The complete polarization tensors identify information absent from a ten-dimensional circulation tensor. That information can be carried by full spatial solutions, rather than only by vectors with no specified spatial profile.

Use the declared flat scalar Dirac moment model on ordinary macroscopic bound coordinates $x\in\mathbb R^d$. All sectors belong to the one nested physical manifold. With spatial Clifford matrices $\gamma_i$, define

$$\alpha_i=\sigma_1\otimes\gamma_i,\qquad \beta=\sigma_3\otimes I,\qquad H_b=-i\alpha_i\partial_i+\beta A r^2. \tag{1}$$

For $w=(z,0)^T$ and a real normalized radial solution $F,G$,

$$\boxed{\Phi_z(x)= \frac{F(r)w+iG(r)\alpha(\widehat x)w}{\sqrt{\Omega_{d-1}}}, \qquad H_b\Phi_z=M\Phi_z.} \tag{2}$$

The radial source obeys $A=g_0T_\sigma/d$, with $T_\sigma=\int r^2\Phi_z^\dagger\beta\Phi_z\,dx$. The scalar moment branch supplies the equations, normalization and separate stability qualifications. Its pointwise bilinears are

$$\boxed{\rho=\frac{F^2+G^2}{\Omega_{d-1}},\qquad \sigma=\frac{F^2-G^2}{\Omega_{d-1}},\qquad j_i=-\frac{2FG}{\Omega_{d-1}}J_{ij}\widehat x_j.} \tag{3}$$

These identities follow by multiplying the complete upper and lower components, using $\beta w=w$ and $J_{ij}=w^\dagger(-i\alpha_i\alpha_j)w$. The scalar density and every self-binding moment are independent of normalized $z$.

Two fields have the same complete Dirac current

At $d=10$, take the two same-chirality polarizations

$$\boxed{z_+=\frac{|0\rangle+|1234\rangle}{\sqrt2},\qquad z_-=\frac{|0\rangle-|1234\rangle}{\sqrt2}.} \tag{4}$$

The component basis and its phase convention are defined in Complete spatial polarization data. Both have only $J_{9,10}=1$ nonzero. Equations (2)–(3) therefore give the same number density, scalar binding density and every bound component of the Dirac number current, at every spatial point.

Nevertheless,

$$\langle\Phi_+,\Phi_-\rangle=0,\qquad \Phi_+^\dagger(x)\Phi_-(x)=0. \tag{5}$$

The local identity follows from $W^\dagger W=\rho I$ for the frame in (2). A four-form distinguishes them. On the ordinary spatial axis $x=r e_9$,

$$\boxed{\Phi_\pm^\dagger \alpha_1\alpha_3\alpha_5\alpha_7\Phi_\pm=\pm\rho(r).} \tag{6}$$

The operator in (6) commutes with $\alpha_9$, so both radial components carry the same sign there. The field still occupies all ten bound coordinates through (2). One nonzero current plane does not reduce its physical spatial dimensionality.

Complete spatial rotations distinguish the fields

The total generator of a rotation in the $ij$ coordinate plane is

$$\mathscr J_{ij}=-i(x_i\partial_j-x_j\partial_i) -\frac i2\alpha_i\alpha_j. \tag{7}$$

It contains the orbital derivative and the field's spin action, as derived for complete spatial rotations. For a common physical rotation $R$ and its spin lift $S(R)$,

$$S(R)\Phi_z(R^Tx)=\Phi_{S_{\rm rest}(R)z}(x),\qquad \mathscr J_{ij}\Phi_z=\Phi_{B_{ij}z/2}. \tag{8}$$

This is a rotation of the spatial field. Its integrated kinetic norm is the same as the norm of its rest coefficient because (2) is an isometry.

Consider the particular spatial generator

$$G=\frac12(B_{13}+B_{57}). \tag{9}$$

Direct Clifford multiplication gives

$$\boxed{Gz_+=0,\qquad G^2z_-=z_-,\qquad z_-^\dagger Gz_-=0.} \tag{10}$$

Thus the two fields respond differently to the same rotation, despite having identical mean rotation tensors. Their overlaps with their initial spatial modes after rotation by $\vartheta$ are

$$\boxed{|\langle\Phi_+,\Phi_{e^{-i\vartheta G}z_+}\rangle|^2=1, \qquad |\langle\Phi_-,\Phi_{e^{-i\vartheta G}z_-}\rangle|^2 =\cos^2\vartheta.} \tag{11}$$

The difference is the rotation variance $z^\dagger G^2z-(z^\dagger Gz)^2$, which contains four-form information. A vanishing change in the mean current tensor is insufficient to conclude that the full spatial polarization is unchanged.

Obtain the rotation from a polarized lower partner

A reciprocal example uses the existing bilateral rotation-moment energy. Specify normalized field channels on $Q_6\subset Q_{10}$, with a fixed local spatial inclusion $I:\mathbb R^6\to\mathbb R^{10}$, and energy

$$\boxed{U_{\rm rot}=\frac{\kappa}{2} \langle I J_a I^T,J_b\rangle,\qquad \langle A,B\rangle=\sum_{i<j}A_{ij}B_{ij}.} \tag{12}$$

The coefficient $\kappa$ is one specified interaction parameter for this benchmark. It is not an electromagnetic coupling derived from particle data. The channel projection and rotational moment term are assumptions of the declared reduced action for modes of the same field.

Give both full spatial channels their ordinary first-order kinetic terms. To define (12) away from the radial family, take $J_{ij}=2\langle\Phi,\mathscr J_{ij}\Phi\rangle$ as its integrated rotation-moment observable. Equation (8) makes this equal to the rest two-form on that family. Variation of this specified full-field energy gives the source operators

$$\boxed{ \delta H_a=\kappa\sum_{\mu<\nu}(I^TJ_bI)_{\mu\nu}\mathscr J^a_{\mu\nu}, \quad \delta H_b=\kappa\sum_{i<j}(I J_a I^T)_{ij}\mathscr J^b_{ij}.} \tag{13}$$

In the invariant radial family these become

$$\boxed{ i\dot z_a=\frac{\kappa}{2} \sum_{\mu<\nu}(I^TJ_bI)_{\mu\nu}B^a_{\mu\nu}z_a, \quad i\dot z_b=\frac{\kappa}{2} \sum_{i<j}(I J_a I^T)_{ij}B^b_{ij}z_b.} \tag{14}$$

Both sources follow from the same energy, and both polarizations remain dynamical. Each channel may have its own radial carrier; it is factored out in (14).

Choose the six shared directions as $(e_1,e_3,e_5,e_7,e_9,e_{10})$ and a lower polarization whose embedded tensor has unit planes $13$, $57$ and $9\,10$. This is a particular relative orientation in the nested space. It makes no direction intrinsically different to either particle.

With either $z_b=z_+$ or $z_b=z_-$, the higher tensor has only its $9\,10$ plane. The lower source is therefore its local $56$ rotation. Choose the lower unit state $z_{a0}$ with all three local plane weights $+1$. The exact pair solution is

$$\boxed{ z_a(t)=e^{-i\kappa t/2}z_{a0},\qquad z_b(t)=e^{-i\kappa t/2}e^{-i\kappa tG}z_b(0).} \tag{15}$$

The lower tensor remains fixed. The higher $J_b$ also remains fixed: the rotation generated by $G$ acts in directions where that tensor is zero and commutes with its $9\,10$ plane. Substitution into (14) therefore reproduces the same generators at every time, proving reciprocal closure. Both norms and $U_{\rm rot}=\kappa/2$ are conserved.

The second field has nontrivial spatial polarization motion that a model following only $J_a,J_b$ misses. Even an energy expressed entirely in their two-forms requires the larger polarization dynamics to recover that motion.

Scope of the spatial solution

Multiplying (15) by the complete profiles and radial carrier phases gives solutions of the declared nonlinear spatial channel equations including (13). Rotations preserve the radial scalar density, so their original self-binding sources remain valid. The derivative part of (13) also adds the corresponding rotational advection to the local conserved number current. That drift is the same in the two compared preparations; (3) names the Dirac current rather than silently including this extra term.

The example uses two individual spatial field channels. It does not introduce an atom, a mediator particle, or a new physical dimension. It also does not identify the lower field as an actual electron or the higher field as an actual tau merely from their depths. Their occupied mode, physical interaction operator and stability remain separate tasks. The scalar branch's known radial instabilities are not removed by (15).

The normal polarization waves provide a different exact motion with a preserved scalar source. Allowing polarization to vary across the bound coordinates instead requires the full kinetic remainder derived in Spatial polarization textures.

Verification

spatial_polarization_structure_checks.py solves normalized radial fields at one common scalar coefficient $g_0=6.5$ and two radial resolutions. Spatial quadrature reconstructs their binding sources; independent derivatives check the original Dirac equations.

It compares all pointwise number currents, the four-form in (6), and actual orbital-plus-spin rotations. Independent complex energy variations and nonlinear integration verify both equations in (14), their conserved quantities and (15). Independent spatial and temporal derivatives then verify the full driven field equations, including the orbital source terms. Common rotation of the higher field and spatial inclusion preserves the pair energy. Reports are timestamped and checks remain active under Python optimization.