Axial spatial Dirac packets

Summary: An exact angular family retains complete complex polarization vectors and transverse localization in odd-dimensional scalar Dirac dynamics, with a variational radial and axial regulator for reciprocal packet encounters.

An ordinary spatial family with explicit angular dependence

Consider the same local scalar equation used by the positive-frequency vertices, now in odd ordinary spatial dimension $D\ge3$:

$$i\partial_t\Psi=-i\alpha\cdot\nabla\Psi+g\sigma\beta\Psi, \qquad \sigma=\Psi^\dagger\beta\Psi. \tag{1}$$

This is a specified flat-space scalar term. The following packets are unbound fields, not identified photons, neutrinos or leptons. Choosing a spatial symmetry of their preparation does not introduce compactified or internal dimensions.

Write $x=(x_\perp,z)\in\mathbb R^{D-1}\times\mathbb R$, $r=|x_\perp|$, and $s=D-1$. Let $\Gamma_j$ be a minimal complex Clifford representation for the $D$ spatial directions. Choose

$$\alpha_j=\begin{pmatrix}\Gamma_j&0\\0&-\Gamma_j\end{pmatrix}, \qquad \beta=\begin{pmatrix}0&I\\I&0\end{pmatrix}, \qquad \Gamma_r=\sum_{j=1}^{s}\Gamma_j\frac{x_j}{r}. \tag{2}$$

The two blocks are the opposite complete kinetic chiralities. Let $W$ contain an entire orthonormal basis of the $+1$ eigenspace of $\Gamma_z$. Its dimension is $P=2^{(D-3)/2}$. Since $\{\Gamma_z,\Gamma_r\}=0$,

$$W^\dagger W=I_P, \qquad W^\dagger\Gamma_r W=0, \qquad (\Gamma_r W)^\dagger\Gamma_r W=I_P. \tag{3}$$

At each nonzero transverse position, $(W,\Gamma_r W)$ spans the complete chiral spinor space. Take the full spatial field

$$\boxed{ \Psi_u=W F_u(t,r,z)+\Gamma_r W G_u(t,r,z),\qquad \Psi_v=W F_v(t,r,z)+\Gamma_r W G_v(t,r,z), } \tag{4}$$

where every $F_b,G_b$ is a complex $P$-component vector. Their polarization can vary with radius and axial position. The retained angular dependence is the actual matrix $\Gamma_r$. This is a restricted angular family, not a representation of arbitrary angular textures. Its pointwise completeness does not remove that restriction on how different positions are related.

For $D=3$, choose $W=(1,0)^T$. Equation (4) reads

$$\Psi=(F_u,e^{i\varphi}G_u,F_v,e^{i\varphi}G_v)^T. \tag{5}$$

The axial angular structure is familiar from the three-dimensional Dirac separation described by Fillion-Gourdeau, Lorin and Bandrauk, section 2, after a constant change of spinor basis. The nonlinear closure and higher-dimensional formulas below follow directly from (1)–(4). For smooth Cartesian fields at the axis, $F_b$ is smooth in $r^2$, and $G_b=r$ times a function smooth in $r^2$.

The full local equation closes exactly

The scalar is independent of transverse angle, but retains the complete complex interference:

$$\boxed{\sigma=2\operatorname{Re}(F_u^\dagger F_v+G_u^\dagger G_v).} \tag{6}$$

This expression follows from the complete field before any spatial integration. It is not a density average replacing its polarization. The Clifford derivative gives

$$\Gamma_\perp\cdot\nabla_\perp(WF)=\Gamma_r W\partial_rF, \qquad \Gamma_\perp\cdot\nabla_\perp(\Gamma_r WG) =W\left(\partial_r+\frac{s-1}{r}\right)G. \tag{7}$$

Indeed, $\sum_j\Gamma_j\partial_j\Gamma_r=(s-1)I/r$. Together with $\Gamma_zW=W$ and $\Gamma_z\Gamma_rW=-\Gamma_rW$, this yields

$$\boxed{ \begin{aligned} i\partial_tF_u&=-i\partial_zF_u -i\left(\partial_r+\frac{D-2}{r}\right)G_u+g\sigma F_v,\\ i\partial_tG_u&=+i\partial_zG_u-i\partial_rF_u+g\sigma G_v,\\ i\partial_tF_v&=+i\partial_zF_v +i\left(\partial_r+\frac{D-2}{r}\right)G_v+g\sigma F_u,\\ i\partial_tG_v&=-i\partial_zG_v+i\partial_rF_v+g\sigma G_u. \end{aligned}} \tag{8}$$

Every source and derivative stays in (4). Thus any sufficiently regular solution of (8) reconstructs a solution of (1), with its full angular spinor factor. Both incoming patterns evolve. A common rotation of the physical coordinates, axis and spinor frame transports the same solution.

Physical norm, source and conserved polarization

Let $S_{s-1}=2\pi^{s/2}/\Gamma(s/2)$ be the area of the unit transverse sphere. The physical volume element and positive norm are

$$dV=S_{s-1}r^{s-1}dr\,dz, \qquad Q=\int dV\sum_{b=u,v}(F_b^\dagger F_b+G_b^\dagger G_b). \tag{9}$$

The energy is the full Dirac kinetic energy plus $g\int\sigma^2dV/2$. In particular, the radial derivative's adjoint in this norm is

$$(\partial_r)^\dagger=-\partial_r-\frac{s-1}{r}, \tag{10}$$

with the stated regularity and sufficient decay. Dropping the divergence term would change both the dynamics and norm.

The reduced action is invariant under the same constant unitary transformation of all four polarization vectors. Consequently the entire Hermitian matrix

$$\boxed{\mathcal P=\int dV\sum_{b=u,v} (F_bF_b^\dagger+G_bG_b^\dagger)} \tag{11}$$

is conserved. Its trace is $Q$. This is a symmetry of the specified angular family. It does not state that a local spatial overlap or either object's individual polarization matrix is fixed.

Writing $c=F_u^\dagger F_v+G_u^\dagger G_v$ gives the reciprocal law

$$\dot Q_u=2g\operatorname{Im}\int c^2dV, \qquad \dot Q_v=-\dot Q_u. \tag{12}$$

A complete single-chirality field has $\sigma=0$ and evolves freely. Another exact control occurs when all $F_u,G_u$ lie in a fixed polarization subspace orthogonal to the subspace containing $F_v,G_v$. Then $c=0$ pointwise and the free evolution preserves the mismatch. At $P\ge2$ this admits nonzero localized fields on both sides. It neither forces the occupied patterns into those subspaces nor makes five dimensions uniquely noninteracting. Spatially varying textures can span the whole polarization space.

A finite regulator derived from the same local energy

For numerical work, choose a radial scale $w$ and retain $N$ functions in each of the two radial families. Put $x=r^2/w^2$ and $\alpha=s/2-1$. Define

$$\begin{aligned} \phi_n^0(r)&=A_n e^{-x/2}L_n^\alpha(x),& A_n^2&=\frac{2n!}{S_{s-1}w^s\Gamma(n+s/2)},\\ \phi_n^1(r)&=B_n\sqrt{x}e^{-x/2}L_n^{\alpha+1}(x),& B_n^2&=\frac{2n!}{S_{s-1}w^s\Gamma(n+s/2+1)}. \end{aligned} \tag{13}$$

Each family is orthonormal in $S_{s-1}r^{s-1}dr$. This extends the independently normalized radial spatial Dirac basis. The exact derivative identity is

$$\partial_r\phi_n^0=-\frac1w \left[\sqrt{n+s/2}\,\phi_n^1+\sqrt n\,\phi_{n-1}^1\right]. \tag{14}$$

Let $D_{mn}$ be that derivative matrix, with $G$ rows and $F$ columns. For axial Fourier momentum $k$, the upper chiral Galerkin operator is

$$H_u(k)=\begin{pmatrix}kI&iD^\dagger\\-iD&-kI\end{pmatrix}, \qquad H_v(k)=-H_u(k). \tag{15}$$

It is self-adjoint in the canonical coefficient norm. The reverse derivative has a complementary radial component at the finite cutoff; convergence in $N$ is required. The finite regulator is not an assertion that the continuum field occupies only these modes.

If $D=U\Lambda V^\dagger$, transform $F$ by $V^\dagger$ and $G$ by $U^\dagger$. Each free block has frequencies $\pm\sqrt{k^2+\lambda_j^2}$, supplying its exact propagator and positive-frequency projector. Applying that projector prepares an incoming field. The nonlinear evolution retains both free-frequency signs thereafter; neither is assigned an antiparticle interpretation.

The local quartic has exact finite-basis radial quadrature. Write the polynomial parts of $F,G$ as $f,g$, including $\sqrt x$ in the $G$ basis, so $\sigma=e^{-x}\sigma_{\rm pol}$. Changing variable to $u=2x$ gives

$$\boxed{ E_{\rm int}=\frac g2\frac{S_{s-1}w^s}{2^{s/2+1}} \int dz\int_0^\infty du\,u^{s/2-1}e^{-u} \sigma_{\rm pol}(u/2,z)^2. } \tag{16}$$

Every quartic product contains an even number of $G$ factors. The remaining integrand is polynomial in $u$, of degree at most $4N-2$. A generalized Gauss–Laguerre rule of order $2N$ or higher therefore integrates (16) exactly in this retained radial space. The nonlinear coefficient source is its full complex derivative.

The axis uses a finite Fourier interval as numerical storage for localized fields. For an even number $N_z$ of retained complex modes, zero-padding to $2N_z$ before evaluating (16), followed by orthogonal projection of the source, integrates the quartic without Fourier aliasing. This preserves the finite action's axial translation symmetry and corresponding momentum. The interval and radial scale are numerical choices requiring convergence checks; they do not specify a physical compactification or the geometry of the observed three-dimensional space.

Verification and limits

axial_scalar_checks.py reconstructs the complete Cartesian spinor and differentiates it in every ordinary direction at $D=3,5,7,9$. It checks (6)–(8), physical rotation covariance, the normalized radial basis and its independent derivatives, full complex source variation, exact quartic quadrature, conserved norm, axial momentum and the matrix (11).

Independent full matrix exponentials verify the free propagator and complete positive-frequency projection implemented by axial_scalar_packets.py. The diagnostic runs 192 checks, including full nonlinear evolution of complex spatial textures and a five-dimensional localized comparison with both zero and nonzero scalar sources. Spatial, radial and temporal refinement remain necessary for any particular collision result.

The family provides transverse localization and several spatial polarizations with a tractable full-source equation. It supplies no bound-particle depth assignment or electric and weak channel embedding. With dimensionless $Q$ and $\hbar=c=1$, $[g]=\mathrm{length}^{D-1}$. Choosing equal numerical $g$ values at different $D$ does not by itself define one physically normalized interaction across nested sectors.

Changing compatibility of localized spatial packets uses this full angular family to compare both outgoing complex patterns after removing their known free evolution. A precisely specified future encounter has either increased or decreased scalar contact for different five-dimensional preparations under the same local dynamics.

A separate spherical vector family works in both odd and even dimensions, including six and ten. Its full radial polarization vectors retain tangential current while scalar density is radial. It has one radial center and does not extend the present axial encounter ansatz to a general even-dimensional collision.