Rotation-neutral spatial modes

Summary: A common rotation of two spatial planes has an exact neutral spinor subspace, while three equal planes have none; full spatial motion and dynamical selection determine whether this can become a particle coupling rule.

A concrete cancellation between spatial polarizations

An interaction can vanish because a spatial pattern is unchanged by the particular geometric operation to which it couples. This is stronger than zero overlap of two unperturbed sources. The calculation below gives an explicit example involving two oppositely polarized rotation planes.

All coordinates are ordinary macroscopic spatial coordinates in the one nested manifold. Spinor components describe fields on those coordinates. The calculation uses a flat spatial chart and a specified rotation probe; it does not identify a spinor parameter space with the particle's spatial submanifold. Nor does it identify the probe with electromagnetism in advance.

The construction first works separately at each containing depth. Combining these probes into one action on the entire nested chain requires compatible embeddings and their variations. A common formula for several local probes is not by itself one globally defined conserved charge.

Rotations of nested spatial sectors derives the fixed-chain constraint and the moving-embedding terms required to pursue this candidate through the whole hierarchy.

Spin action of the specified spatial rotation

In $d$ Euclidean spatial directions let the Hermitian Clifford matrices obey

$$\{\gamma_a,\gamma_b\}=2\delta_{ab}I.$$

Choose $m=\lfloor d/2\rfloor$ mutually orthogonal, oriented spatial planes $(e_1,e_2),\ldots,(e_{2m-1},e_{2m})$, and define

$$H_j=-i\gamma_{2j-1}\gamma_{2j},\qquad H_j^2=I,\qquad [H_j,H_k]=0. \tag{1}$$

These are spatial rotation generators, not independent interaction fields. The spin action of rotation by the same angle in every chosen plane is

$$U_{\rm spin}(\theta)=e^{-i\theta Q_d}, \qquad Q_d=\frac12\sum_{j=1}^mH_j. \tag{2}$$

The factor one-half is the spin lift: a $2\pi$ rotation in one plane acts as $-I$. This convention can also be constructed with exterior creation and annihilation operators; see Woit's spinor representation notes. It is not an electromagnetic normalization.

In a simultaneous eigenbasis write $H_jz_s=s_jz_s$, $s_j\in\{+1,-1\}$. Then the weights are exactly

$$Q_dz_s=q_s z_s,\qquad q_s=\frac12\sum_js_j. \tag{3}$$

For the full spin module all sign choices occur. At even depth a chosen half-spin module restricts their product $\prod_js_j$ to one sign. The choice of half-spin module is an additional physical input.

The table counts field components, not extra spatial dimensions. Zero is possible when $m$ is even; the full neutral multiplicity is then $\binom m{m/2}$. In a half-spin module the neutral weights all have the same chirality $(-1)^{m/2}$, so the other half-spin module has none. Thus this is a plane-parity rule, not a unique characterization of depth five.

Five-dimensional cancellation as an operator identity

For $d=5$, equation (2) is $Q_5=(H_1+H_2)/2$. Its neutral projector follows from the Clifford algebra:

$$\boxed{P_0=\frac{I-H_1H_2}{2},\qquad P_0^2=P_0,\qquad \operatorname{rank}P_0=2,\qquad Q_5P_0=0.} \tag{4}$$

The two planes have opposite spin rotation weights on this subspace. Either weight can be nonzero, and each plane separately acts nontrivially. Their sum annihilates the complete state. Consequently

$$e^{-i\theta Q_5}P_0=P_0 \quad\text{for every real }\theta. \tag{5}$$

For example use

$$\gamma=(\sigma_1\otimes I,\sigma_2\otimes I, \sigma_3\otimes\sigma_1,\sigma_3\otimes\sigma_2, \sigma_3\otimes\sigma_3).$$

Then $H_1=\sigma_3\otimes I$, $H_2=I\otimes\sigma_3$, and $z=(0,1,0,0)^T$ has weights $(+1,-1)$. Both rotations are ordinary spatial rotations. The fifth spatial direction is still present in the field's profile; it is the null direction of this particular skew probe. No direction is reclassified as a different kind of space.

In six dimensions, the sum of three signs is odd. Therefore, on either half-spin module and on the full module,

$$\boxed{\|Q_6z\|\geq\frac12\|z\|.} \tag{6}$$

The same bound holds for the five-plane operator at depth ten. This supplies an explicit geometric distinction between these chosen probes without setting a five-dimensional coupling coefficient to zero.

A geometric origin for the two equal plane rates

The equal rates can arise from a higher spatial complex structure. Let $J^2=-I$ on six ordinary spatial directions and choose a unit normal $n$ to a five-dimensional hyperplane $V$. With $P=I-nn^T$, put $B=(PJP)|_V$ and $u=Jn\in V$. For $v\in V$, direct multiplication gives

$$\boxed{B^2=-I_V+uu^T,\qquad Bu=0,\qquad |u|=1.} \tag{6a}$$

Thus the tangential compression of any orthogonal six-dimensional complex structure has two equal unit rotation planes and one null direction. It supplies precisely the equal-rate operator type used in (4), without adjusting the two rates independently. The hyperplane is an observer-relative or partner-relative subspace of the same physical space.

Compression has also removed a nonzero mixed term. With the ordered decomposition $V\oplus\operatorname{span}(n)$,

$$J=\begin{pmatrix}B&u\\-u^T&0\end{pmatrix}. \tag{6b}$$

Take $J$ to be the pure-ray moment in the chosen four-component Spin$(6)$ half-spin module, as in the existing chiral source construction. Denote the tangential bivector action by $S_5$ and the mixed action by $R$, so the full unscaled spin action is $S_6=S_5+R$. In an adapted pure-ray basis one has

$$S_5=\operatorname{diag}(2,0,0,-2),\qquad R=\operatorname{diag}(1,-1,-1,1),\qquad S_6=\operatorname{diag}(3,-1,-1,-1). \tag{6c}$$

These spectra and simultaneous relations are unchanged under the corresponding spatial spin rotations. The neutral projector of the tangential action obeys

$$P_0=I-\frac{S_5^2}{4},\qquad S_5P_0=0,\qquad RP_0=-P_0,\qquad S_6P_0=-P_0. \tag{6d}$$

The full spin generator is $S_6/2$. Consequently the tangential neutral mode still has a nonzero response to the mixed part of this higher source. This is useful for a mechanism with distinct geometric channels, but it requires the physical lower-channel projection to be derived. Simply dropping the mixed part would create a neutral coupling by omission.

The equal-plane source and its surviving complement are both fixed by the same six-dimensional geometry. Which component the photon current samples, and whether the mixed component acts directly or through other spatial modes, remain dynamical questions.

Spatial transport of a neutral doublet derives another response directly from this projector: variation of its physical null axis produces a nonzero traceless geometric curvature and a fixed positive contribution to the spatial kinetic energy.

Which higher depths guarantee this cancellation

The equal-rate guarantee in (6a) uses a codimension-one hyperplane. For a lower spatial subspace of general codimension, write an orthogonal complex structure on the containing space as

$$J=\begin{pmatrix}B&C\\-C^T&D\end{pmatrix},\qquad \boxed{B^2=-I+CC^T.} \tag{6e}$$

There can now be several independent mixed directions, and the nonzero rotation rates of $B$ need not agree. In particular, a direct ten-dimensional source compressed to five dimensions does not automatically inherit the two equal rates of the six-to-five case.

This can be checked by an explicit family with every containing direction still equivalent in kind. Let $J_2=\left(\begin{smallmatrix}0&1\\-1&0\end{smallmatrix}\right)$, $B=\operatorname{diag}(\lambda_1J_2,\lambda_2J_2,0)$, and $0\leq\lambda_2\leq\lambda_1\leq1$. Then

$$C=(I+B^2)^{1/2},\qquad J_{10}=\begin{pmatrix}B&C\\-C&-B\end{pmatrix} \quad\Longrightarrow\quad J_{10}^2=-I_{10},\quad J_{10}^T=-J_{10}. \tag{6f}$$

The blocks describe the relation to a chosen five-dimensional partner, not an inherent division of a ten-dimensional particle. They can be rotated together with the entire nested chain. The compressed spin action has the four weights

$$q=\frac{\pm\lambda_1\pm\lambda_2}{2},\qquad \min_{\|z\|=1}\|Q_5z\|=\frac{|\lambda_1-\lambda_2|}{2}. \tag{6g}$$

For example, $(\lambda_1,\lambda_2)=(0.9,0.4)$ gives weights $\pm0.65,\pm0.25$ and no neutral vector. Equal rates do give the neutral doublet, with the completely zero source as a separate degenerate case. Thus extending the mechanism through the whole hierarchy requires the actual source map and compatible geometric preparations. A higher depth alone is not sufficient. Direct ten-to-five response and a projection through an independently selected six-dimensional complex structure are different physical assumptions and must follow from the same field action.

Spatial motion must be included

A field transforms through its position dependence as well as its spinor components. Define the ordinary orbital operators

$$L_j=-i\left(x_{2j-1}\partial_{2j}-x_{2j}\partial_{2j-1}\right), \qquad \mathscr Q_d=\sum_jL_j+Q_d. \tag{7}$$

For a radial envelope $f(|x|)$, every $L_jf=0$. Hence $\Phi(x)=f(|x|)z$, $z\in\operatorname{ran}P_0$, is an actual five-dimensional spatial field satisfying $\mathscr Q_5\Phi=0$. Its density can have nonzero extent in all five coordinates. This is a neutral spatial mode of the specified rotation, not a particle collapsed onto fewer dimensions.

More general angular profiles carry integer weights under the simultaneous orbital rotation and must be tested with (7). With the usual single-valued spatial functions, an odd number of equal planes gives

$$e^{-2\pi i\mathscr Q_d}=(-1)^mI=-I. \tag{8}$$

A nonzero vector annihilated by $\mathscr Q_d$ would instead be fixed by this finite rotation. Thus there is no such vector for odd $m$. Additional bundle twists, gauge phases, boundaries, or a different rotation action can change this premise and require their own calculation.

When neutrality survives dynamics

Let $\mathcal F[\Phi]$ be the full evolution vector field of the one master field. Suppose solutions are unique on the adopted domain and it is equivariant under this rotation:

$$\mathcal F[U(\theta)\Phi]=U(\theta)\mathcal F[\Phi]. \tag{9}$$

If $U(\theta)\Phi_0=\Phi_0$ for all $\theta$, both the evolved solution and its rotation solve the same initial-value problem. Uniqueness therefore preserves the neutral subspace. For linear dynamics this reduces to $[H,\mathscr Q_d]=0$ with compatible operator domains.

A reciprocal mode interaction $a\langle\Phi,\mathscr Q_d\Phi\rangle$ then has both its source and its matter variation zero on a neutral field:

$$\partial_aE_{\rm int}=\langle\Phi,\mathscr Q_d\Phi\rangle=0, \qquad \frac{\delta E_{\rm int}}{\delta\Phi^*}=a\mathscr Q_d\Phi=0. \tag{10}$$

Here $a$ is a specified perturbation amplitude of that same field. A full spatially varying connection requires a local covariant action and its complete current, not just the uniform-mode expression (10).

Zero expectation alone would not suffice. The normalized spinor $(1,0,0,1)^T/\sqrt2$ has $\langle Q_5\rangle=0$ but $\|Q_5z\|=1$: it combines two charged weights and responds to (2).

Other couplings and imperfect cancellation

An operator $R=-i\gamma_1\gamma_5$ acts nontrivially on the neutral example above: $\|Rz\|=1$ and $Q_5Rz\ne0$. Thus neutrality under $Q_5$ is compatible with transitions under another spatial operation. This is the kind of channel distinction needed in a neutrino calculation; it does not identify $R$ with an observed weak vertex or calculate a scattering rate.

Equal plane rates matter. If

$$Q_5(\epsilon)=\frac12\big[(1+\epsilon)H_1+(1-\epsilon)H_2\big],$$

then $Q_5(\epsilon)z=\epsilon z$ for the $(+1,-1)$ example. The squared coupling action is $\epsilon^2$. A dynamical symmetry selecting equal rates could protect the cancellation; choosing them by hand leaves it conditional. Under a common physical rotation of the probe and the field, the annihilation condition is unchanged. Rotating just one changes their relative preparation and must be evaluated afresh.

What a physical mechanism still has to select

There is an exact self-bound spatial example in the existing positive scalar moment model, extended to a multicomponent field with $\rho=\Phi^\dagger\Phi$. In units with kinetic coefficient one, take

$$E[\Phi]=\int\|\nabla\Phi\|^2dx+\frac g2\operatorname{tr}M^2, \qquad M=\int(x-X)(x-X)^T\rho(x)dx,\qquad \int\rho=1. \tag{11}$$

The centre $X=\int x\rho$ is varied with the field, as in centroid-relative self-binding. For any constant normalized spinor $z$,

$$\Phi_z(x,t)=e^{-id\omega t} \left(\frac\omega\pi\right)^{d/4}e^{-\omega|x-X|^2/2}z, \qquad \omega^3=\frac g2 \tag{12}$$

is an exact resting solution. Its moment is $M=I/(2\omega)$, so the nonlinear potential $g(x-X)^TM(x-X)$ is $\omega^2|x-X|^2$ and cancels the quadratic part of the Gaussian Laplacian.

These profiles minimize (11) at fixed norm. Indeed, integration by parts in the square $\int\|\nabla\Phi+\tfrac12M^{-1}(x-X)\Phi\|^2dx\geq0$ gives

$$E[\Phi]\geq\frac14\operatorname{tr}M^{-1} +\frac g2\operatorname{tr}M^2\geq\frac{3d\omega}{4}. \tag{13}$$

For each positive eigenvalue $s$ of $M$, the minimized expression is $1/(4s)+gs^2/2$, whose minimum has $gs^3=1/4$. Equality in the first square requires a Gaussian times a constant spinor, and equality in the eigenvalue bound requires all $s=1/(2\omega)$.

Taking $d=5$ and $z\in\operatorname{ran}P_0$ therefore supplies a normalized neutral spatial solution with nonzero extent in all five directions. But (12) also supplies equally minimal solutions in the two nonzero rotation-weight subspaces. Their energy and shape width are identical. The scalar density kernel has no preference between them. This is an auxiliary spatial field solution, not the actual electron orbit or a derived Dirac particle solution.

The derivation supplies actual neutral modes and a definite six-dimensional comparison. It does not force a five-dimensional excitation to occupy the neutral subspace: its two nonzero weights also exist. A spin-independent radial self-binding energy gives the same energy to all constant normalized spinors, as the full spatial example now demonstrates. An initial constant spinor polarization is preserved by this scalar evolution, but there is no attraction of an arbitrary polarization toward the neutral subspace.

Applying this result to electromagnetic neutrality requires deriving the probe from the one field's action, including the orbital contribution, photon current, kinetic normalization, physical boundary conditions, and the selected particle solutions. Consistency along the nested spatial chain is part of that derivation. The existence of a neutral eigenspace is a concrete mechanism candidate, not a completed charge assignment.

Verification

spatial_coupling_mechanism_checks.py constructs Clifford matrices independently of the sign-counting formula, checks the full spectra, the neutral projector, finite rotations, a surviving other channel, the unequal-rate leakage law, and explicit ten-to-five compressions with both equal and unequal rates. Spatial Gaussian quadrature also checks the self-bound solution, its energy and the neutral/nonneutral degeneracy in (11)–(13).

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