Rotation-neutral Dirac fields in physical space

Summary: A full radial Dirac field can preserve an exactly neutral total spatial rotation through compensating orbital and spin motion, even though a projector on neutral spin components discards part of the field.

Keep the position dependence of the polarization

The neutral-doublet calculation shows that projecting the first-order spatial Dirac operator onto neutral spin components removes some momentum responses. Full spatial modes provide a constructive way to retain those components. The conserved rotation acts on position and spin together.

Work on a flat $d$-dimensional spatial model of a sector in the one nested macroscopic manifold. All $d$ directions are ordinary spatial dimensions. The finite spinor components below describe the same field on those coordinates. Choosing rotation planes specifies a probe or a physical preparation; it does not divide a particle's dimensions into different kinds.

This note derives neutrality under that specified spatial rotation. Its identification with an electromagnetic interaction remains a separate current calculation. No mediator particle or additional physical dimension is introduced.

The full Dirac representation

Let Hermitian matrices satisfy

$$\{\alpha_i,\alpha_j\}=2\delta_{ij}I, \qquad \{\alpha_i,\beta\}=0, \qquad\beta^2=I. \tag{1}$$

Let $V$ be an isometric basis of the positive rest eigenspace, $\beta V=V$, $V^\dagger V=I$. The map $\alpha(v)V$ lies in the opposite $\beta$ eigenspace for every spatial vector $v$. The two eigenspaces are field components, not parts of physical space.

The full central spatial Hamiltonian is

$$H=-i\alpha_i\partial_i+\beta M(r)+V_0(r)I, \qquad r=|x|. \tag{2}$$

Its radial coefficients may be specified backgrounds or come from a self-consistent source. A derivation must say which applies and keep their response when they are dynamical.

For $z^\dagger z=1$, consider the spatial field

$$\boxed{\Psi(x)=\frac{1}{\sqrt{\Omega_{d-1}}} \left[f(r)Vz+i g(r)\alpha(\widehat x)Vz\right],} \qquad \widehat x=x/r, \tag{3}$$

where $\Omega_{d-1}$ is the area of the unit $(d-1)$-sphere and $f,g$ are real radial functions. Regularity requires $g(r)=O(r)$ at the origin. The norm and scalar source are

$$\|\Psi\|^2=\int_0^\infty(f^2+g^2)r^{d-1}dr, \qquad \overline\Psi\Psi=\frac{f^2-g^2}{\Omega_{d-1}}. \tag{4}$$

The lower component has a direction-dependent polarization. Removing it would change both the field equation and the scalar source.

Substitution into $H\Psi=E\Psi$ gives the full radial equations

$$\boxed{(E-M-V_0)f=g'+\frac{d-1}{r}g, \qquad (E+M-V_0)g=-f'.} \tag{5}$$

They follow from $\alpha(\widehat x)^2=I$ and $\alpha_i\partial_i\alpha(\widehat x)=(d-1)I/r$. No nonrelativistic approximation has been made.

Self-consistent scalar Dirac moment modes solves these radial equations with the potential generated by the existing nonlocal scalar source, and checks its complete radial response.

Orbital motion preserves the same total rotation weight

Choose $m=\lfloor d/2\rfloor$ ordinary orthogonal rotation planes and set

$$S=-\frac i2\sum_{j=1}^{m}\alpha_{2j-1}\alpha_{2j}, \qquad L=-i\sum_{j=1}^{m} \left(x_{2j-1}\partial_{2j}-x_{2j}\partial_{2j-1}\right), \qquad\mathscr Q=L+S. \tag{6}$$

The Dirac matrices rotate with their spatial arguments. Explicitly,

$$[S,\alpha(x)]=-L\alpha(x),\qquad [\mathscr Q,\alpha(\widehat x)]=0. \tag{7}$$

Every radial function is annihilated by $L$. If the rest polarization has $S Vz=qVz$, equation (3) therefore obeys

$$\boxed{\mathscr Q\Psi=q\Psi.} \tag{8}$$

This is an operator identity for the entire position-dependent field. For a physical rotation $R(\theta)$ in the chosen planes, its finite version is

$$e^{-i\theta S}\Psi\big(R(\theta)^{-1}x\big) =e^{-iq\theta}\Psi(x). \tag{9}$$

The kinetic operator and both central terms in (2) commute with $\mathscr Q$. A rotation-equivariant nonlinear scalar source also preserves an initially fixed rotation sector, subject to the adopted domains and uniqueness of evolution. Retaining the lower component is fully consistent with that conservation.

The five-dimensional neutral modes remain available

The rest weights are the spin lifts of the chosen spatial rotations. At depth five the four-component rest module has weights $(-1,0,0,1)$, so two independent choices in (3) give $\mathscr Q\Psi=0$. Those fields can have nonzero density throughout all five spatial coordinates.

At depth six the rest module is a selected four-component half-spin representation. Three equal rotation planes give half-integer weights with no zero. The five-plane depth-ten construction likewise has no zero weight. With ordinary single-valued spatial wavefunctions, the full $2\pi$ rotation acts as $-I$ when the number of planes is odd; orbital integer weights cannot turn it into a zero eigenvalue. The rotation-weight note gives the representation assumptions and the other depths, including the even-plane cases which also admit zero weights.

The comparison concerns the same type of geometric operation. It is not a proof that this operation is the photon current or that all five-dimensional particles occupy its neutral modes. Neutral and nonneutral radial polarizations in (3) have the same scalar density and radial equations. Their occupation still requires a dynamical explanation.

What a spin-only projection would discard

Let

$$F=\int f^2r^{d-1}dr,\qquad G=\int g^2r^{d-1}dr, \qquad F+G=1.$$

Spherical averaging and the Clifford identity $\sum_i\alpha_iS\alpha_i=(d-4)S$ give

$$\langle S\rangle=q\left(1-\frac{4G}{d}\right), \qquad \langle L\rangle=\frac{4Gq}{d}, \qquad\langle\mathscr Q\rangle=q. \tag{10}$$

For a five-dimensional zero-weight state, the lower component also obeys

$$\boxed{\langle S^2\rangle=\langle L^2\rangle=\frac45G, \qquad\mathscr Q\Psi=0.} \tag{11}$$

The spin and orbital actions cancel on the full field, although each can have a nonzero norm. One way to obtain (11) is to use $S^2=I/2+$ a Clifford grade-four term and $\sum_i\alpha_iT_4\alpha_i=(d-8)T_4$. On a rest state with $S^2Vz=0$, the lower angular average gives $4/5$.

A projector onto $\ker S$ alone discards a fraction $4G/5$ of the complete field norm: the lower component in four of the five spatial directions has nonzero spin weight, compensated by its orbital dependence. That discarded norm is not a violation of the conserved total neutrality in (8). Equations (7)–(11) specify the distinction exactly.

A neutral total rotation does not imply zero local current

Define the rest polarization bivector by

$$J_{ij}=z^\dagger V^\dagger(-i\alpha_i\alpha_j)Vz, \quad i<j, \qquad J_{ji}=-J_{ij}. \tag{12}$$

Direct substitution of (3) into the Dirac velocity current yields

$$\boxed{j_i(x)=\Psi^\dagger\alpha_i\Psi =-\frac{2f(r)g(r)}{\Omega_{d-1}}J_{ij}\widehat x_j.} \tag{13}$$

The current is tangent to the radial spheres. Its divergence vanishes: $J$ is skew, $x^TJx=0$, and $\operatorname{tr}J=0$. These are literal spatial circulation planes of the field. The unpaired direction of a five-dimensional rotation can be a null direction of its current tensor while remaining a direction in which the field has spatial extent.

For a specified charged minimal action, the electric current would include its supplied charge factor. Equation (13) itself is the velocity current; zero weight under $\mathscr Q$ does not make this current vanish. The physical source must be obtained from the same action and the appropriate photon projection. Dressed source vertices shows how to retain the required current if modes are subsequently eliminated.

Compatibility with the entire nested geometry

The rotation used here can move lower members of a nested chain. Rotations of nested sectors distinguishes preserving each fixed member from transporting the whole chain while preserving its nesting. A physical application must include those embedding motions and both interacting objects' polarizations. Equations (8)–(13) retain full spatial motion but do not select a universal embedding or assign a charge generator to every particle.

Normal propagation of spatial Dirac modes retains the containing-space Dirac partner and derives exact free-direction boosts of this full spatial field, including its scalar source and current.

Verification

spatial_dirac_mode_checks.py builds full Clifford matrices independently at depths three, five, six and ten. It tests finite rotations of position-dependent fields, spatial finite differences of $L$, the complete current, and positive spherical cubature for the spin variance and discarded fraction. Its independent radial checks also retain both equations in (5).

Related

• Rotation-neutral spatial modes — establishes the equal-plane weights and source-compression conditions.
• Spatial transport of a neutral doublet — identifies the first-order components whose orbital completion is retained here.
• Dressed spatial source vertices — carries the complete field response through a subsequent mode reduction.