Summary: A spherical scalar Dirac family retains independent complex radial polarization vectors, tangential spatial currents and reciprocal source variations in both odd and even dimensions, including six and ten.

A spatial family available in even dimensions

The localized axial family uses odd-dimensional kinetic chiral blocks. A separate spherical family works for every D≥2D\ge2, including the proposed depths six and ten. It extends the fixed-polarization radial construction by allowing every radial polarization component to have its own complex profile. The spatial domain here is a specified flat chart, not a selected global particle geometry.

Let the Hermitian matrices α1,…,αD,β\alpha_1,\ldots,\alpha_D,\beta obey the complete Dirac Clifford relations. Their minimal complex size is n=2⌊(D+1)/2⌋n=2^{\lfloor(D+1)/2\rfloor}. Choose an orthonormal matrix WW whose P=n/2P=n/2 columns span the entire β=+1\beta=+1 eigenspace. For r=∣x∣>0r=|x|>0, put αr=α⋅x^\alpha_r=\alpha\cdot\widehat x and set

Ψ(x,t)=WF(r,t)+αrWG(r,t),F,G∈CP.(1) \boxed{\Psi(x,t)=W F(r,t)+\alpha_r W G(r,t), \qquad F,G\in\mathbb C^P.} \tag{1}

The vectors FF and GG are independent functions of radius and time. They need not be scalar functions multiplying one fixed polarization. Their components describe a field on DD ordinary spatial coordinates; they introduce no additional physical dimensions.

At the origin, sufficient smoothness conditions are that FF is a smooth function of r2r^2 and G=rG~(r2)G=r\widetilde G(r^2) with smooth G~\widetilde G. Then αrG=α⋅x G~\alpha_rG=\alpha\cdot x\,\widetilde G is regular in the original Cartesian coordinates. Decay or appropriate boundary conditions must also support the stated integrations by parts.

Spatial dimension DD Complete Dirac size nn Components of each radial vector PP
3 4 2
5 8 4
6 8 4
10 32 16

These counts belong to (1); they are not extra interaction multipliers.

The scalar and positive density are radial

Because βαr=−αrβ\beta\alpha_r=-\alpha_r\beta, the columns of αrW\alpha_rW span the opposite β\beta eigenspace. Thus W†αrW=0W^\dagger\alpha_rW=0 and αr2=I\alpha_r^2=I give pointwise

ρ=Ψ†Ψ=F†F+G†G,σ=Ψ†βΨ=F†F−G†G.(2) \boxed{\rho=\Psi^\dagger\Psi=F^\dagger F+G^\dagger G, \qquad \sigma=\Psi^\dagger\beta\Psi=F^\dagger F-G^\dagger G.} \tag{2}

No angular averaging is used to obtain either identity. If two radial patterns are superposed, their complete vector sums enter (2). In particular their scalar includes 2Re⁡(F1†F2−G1†G2)2\operatorname{Re}(F_1^\dagger F_2-G_1^\dagger G_2). Their phases and polarization contractions remain present.

Exact closure under the local scalar equation

Consider the same specified local interaction,

i∂tΨ=−iα⋅∇Ψ+[m+gσ]βΨ,m≥0,(3) i\partial_t\Psi=-i\alpha\cdot\nabla\Psi +[m+g\sigma]\beta\Psi,\qquad m\ge0, \tag{3}

and define M(r,t)=m+gσ(r,t)M(r,t)=m+g\sigma(r,t). The two Cartesian identities

α⋅∇(WF)=αrWF′,α⋅∇(αrWG)=W(G′+D−1rG)(4) \alpha\cdot\nabla(WF)=\alpha_rWF',\qquad \alpha\cdot\nabla(\alpha_rWG) =W\left(G'+\frac{D-1}{r}G\right) \tag{4}

follow from ∂ix^j=(δij−x^ix^j)/r\partial_i\widehat x_j=(\delta_{ij}-\widehat x_i\widehat x_j)/r and the Clifford anticommutators. Substitution gives the exact vector equations

i∂tF=−i(∂r+D−1r)G+MF,i∂tG=−i∂rF−MG.(5) \begin{aligned} i\partial_tF&=-i\left(\partial_r+\frac{D-1}{r}\right)G+MF,\\ i\partial_tG&=-i\partial_rF-MG. \end{aligned} \tag{5}

Every regular solution of (5) reconstructs a solution of the full DD-dimensional equation through (1). The nonlinearity preserves this angular family because its scalar (2) is radial. It still couples all occupied vector components through the common MM. This is an invariant spatial family, not a claim that arbitrary angular patterns solve (5).

The full scalar-source variation is also retained. For independent variations δF,δG\delta F,\delta G,

δσ=2Re⁡(F†δF−G†δG),δ(σF−σG)=(δσF+σδF−δσG−σδG).(6) \delta\sigma=2\operatorname{Re} (F^\dagger\delta F-G^\dagger\delta G),\quad \delta\binom{\sigma F}{-\sigma G} =\binom{\delta\sigma F+\sigma\delta F} {-\delta\sigma G-\sigma\delta G}. \tag{6}

Both ordinary and conjugate-dependent responses remain in this real-linear derivative. No fixed partner or additional pair coefficient is inserted.

A radial density can carry a tangential current

The full spatial number current is not determined by ρ\rho alone. For a unit direction x^\widehat x, define

Ki(x^)=W†αiαrW−x^iIP.(7) K_i(\widehat x)=W^\dagger\alpha_i\alpha_rW-\widehat x_iI_P. \tag{7}

These matrices are anti-Hermitian and obey ∑ix^iKi=0\sum_i\widehat x_iK_i=0. Direct reconstruction of ji=Ψ†αiΨj_i=\Psi^\dagger\alpha_i\Psi yields

ji=2x^iRe⁡(F†G)+2Re⁡(F†KiG).(8) \boxed{j_i=2\widehat x_i\operatorname{Re}(F^\dagger G) +2\operatorname{Re}(F^\dagger K_iG).} \tag{8}

The second term is tangential. Its coefficients retain the complete spatial rotation matrices and complex polarization texture. For fixed radius it is a linear combination of tangential rotation fields on the sphere; its divergence is zero. The radial component jr=2Re⁡(F†G)j_r=2\operatorname{Re}(F^\dagger G) therefore gives

∂tρ+1rD−1∂r(rD−1jr)=0.(9) \partial_t\rho+\frac1{r^{D-1}}\partial_r(r^{D-1}j_r)=0. \tag{9}

Thus a radial scalar and density do not imply absence of angular current. Nor is this number current already an identified electromagnetic current.

Canonical norm and the full polarization matrix

Let ΩD−1=2πD/2/Γ(D/2)\Omega_{D-1}=2\pi^{D/2}/\Gamma(D/2). The physical norm and its matrix refinement are

Q=ΩD−1∫0∞(FF†+GG†)rD−1dr,Q=tr⁡Q.(10) \mathcal Q=\Omega_{D-1}\int_0^\infty (FF^\dagger+GG^\dagger)r^{D-1}dr, \qquad Q=\operatorname{tr}\mathcal Q. \tag{10}

The convention in (1) includes no factor 1/ΩD−11/\sqrt{\Omega_{D-1}}; that physical area consequently remains in (10). Using (5), the matrix integrand obeys

∂t(FF†+GG†)=−(∂r+D−1r)(GF†+FG†).(11) \partial_t(FF^\dagger+GG^\dagger) =-\left(\partial_r+\frac{D-1}{r}\right) (GF^\dagger+FG^\dagger). \tag{11}

The real scalar terms cancel, including the nonlinear part of MM. Vanishing endpoint flux proves conservation of every entry of Q\mathcal Q, not just its trace. This is a property of the stated radial family; it does not assert a new unrestricted internal symmetry of every spatial Dirac field. Local polarization profiles and tangential currents can still change while their integrated matrix is conserved.

For a common physical rotation OO with spin lift SS, SS commutes with β\beta and induces the unitary matrix U=W†SWU=W^\dagger SW. Transforming F,GF,G to UF,UGUF,UG reconstructs SΨ(O−1x)S\Psi(O^{-1}x), so (2)–(11) are covariant under rotation of all physical data.

Verification and scope

radial_polarization_fields.py implements reconstruction, the vector time derivatives and the complete spatial current. radial_polarization_checks.py runs 234 checks of complete Clifford modules in every dimension from two through ten. Independent Cartesian derivatives verify (5) on complex profiles whose polarizations vary with radius. The checks also reconstruct every current component, rotate the full field, retain the complete scalar Hessian and integrate the derivative of the full conserved matrix (10).

This supplies a controlled spatial family at the even particle depths as well as at odd depths. Its patterns share one radial center. General separated-center encounters, arbitrary angular textures, stable occupied particle states and physically normalized cross-depth coupling require additional calculations. The family does not select the global geometry of our three-dimensional space or of a higher sector.