Conserved spatial charge

Summary: A local charge generator must preserve the complete spatial Dirac kinetics, distinguishing a conserved electric-charge candidate from a rotation-neutral polarization of one probe.

A neutral pattern must remain neutral under its field equation

A spatial polarization can be annihilated by a particular rotation operator. That supplies an exact compatibility condition for that probe. An electric-charge assignment requires more: its density and current must obey continuity under the complete spatial dynamics.

This matters for the conditional current used in Soft spatial current response. Its leading charged-particle response cannot be suppressed by simply freezing an unfavorable orientation. A proposed geometric origin of neutrality must specify the physical generator and preserve its kernel.

All derivatives below act on ordinary macroscopic spatial coordinates in the one nested manifold. A matrix acting on field components does not introduce further spatial dimensions.

Compute the full local continuity equation

Take

$$i\partial_t\Psi =-i\alpha_i\partial_i\Psi+V[\Psi]\Psi, \qquad V^\dagger=V, \tag{1}$$

with constant Hermitian spatial Dirac matrices $\alpha_i$. For a Hermitian, possibly position-dependent matrix $Q(x,t)$, define

$$n_Q=\Psi^\dagger Q\Psi,\qquad j_{Q,i}=\frac12\Psi^\dagger\{Q,\alpha_i\}\Psi. \tag{2}$$

These are candidate charge density and current, not an electric identification by definition. Differentiating (2) using (1) gives

$$\boxed{ \begin{aligned} \partial_t n_Q+\partial_i j_{Q,i} ={}&\Psi^\dagger \left[\partial_tQ+\frac12\{\alpha_i,\partial_iQ\} +i[V,Q]\right]\Psi\\ &+\frac12\left[ (\partial_i\Psi^\dagger)[Q,\alpha_i]\Psi -\Psi^\dagger[Q,\alpha_i]\partial_i\Psi \right]. \end{aligned}} \tag{3}$$

For example, the two derivative terms in $\partial_t n_Q$ are $-(\partial_i\Psi^\dagger)\alpha_iQ\Psi -\Psi^\dagger Q\alpha_i\partial_i\Psi$. Combining them with $\partial_i j_{Q,i}$ gives exactly the second line. The potential contributes $i\Psi^\dagger[V,Q]\Psi$. This proof includes arbitrary complex field values and derivatives.

For a constant $Q$ and a scalar Dirac potential $V=\beta\mathfrak m[\Psi]$, conservation for arbitrary local field data requires

$$\boxed{[Q,\alpha_i]=0\quad\hbox{for every retained direction}, \qquad [Q,\beta]=0.} \tag{4}$$

The second condition applies where the scalar potential is nonzero. The first follows because the first spatial derivatives at a point can be varied independently. It cannot be replaced by vanishing expectation of the second line for one stationary field.

Apply the complete kinetic representation

If $\alpha_1,\ldots,\alpha_d,\beta$ act irreducibly on the full complex field module, their common commutant consists of scalar matrices. One proof uses the associative matrix algebra they generate: irreducibility makes this the full matrix algebra, whose commuting matrices are $qI$. Consequently (4) gives

$$Q=qI. \tag{5}$$

This does not derive $q$. It says that a constant local phase-type charge on one such complete module is the same for all its spatial polarizations. The canonical normalization preserves that supplied charge.

Several equivalent copies instead permit $Q=I_{\rm Dirac}\otimes q_{\rm copies}$, provided the remaining dynamics preserves it. Inequivalent invariant blocks may also have different scalar charges. Assigning and occupying those blocks requires a physical derivation. Their labels are field components, not additional spatial dimensions.

Every retained propagation direction matters. For example, a containing representation of six bound coordinates and one existing normal coordinate has

$$\Xi=\beta i^3\alpha_1\cdots\alpha_6,\qquad [\Xi,\alpha_i]=[\Xi,\beta]=0,\qquad \{\Xi,\alpha_y\}=0. \tag{6}$$

The bound operator alone permits separate constants on the two $\Xi$ blocks. Its normal kinetic term connects them and removes that freedom as a conserved local charge of the complete equation. Normal Dirac propagation explicitly retains this normal partner. The example concerns the stated representation; it does not forbid distinct invariant particle families in a larger derived master field.

Test the balanced five-dimensional rotation

The rotation-neutral construction uses

$$Q_{\rm rot}=\frac12(B_{12}+B_{34}),\qquad B_{ij}=-i\alpha_i\alpha_j,\qquad P_0=I-Q_{\rm rot}^2. \tag{7}$$

On the positive-$\beta$ rest polarization module, $P_0$ has rank two. Every $w$ in that doublet satisfies $Q_{\rm rot}w=0$. However,

$$[Q_{\rm rot},\alpha_1]=i\alpha_2,\qquad [Q_{\rm rot},\alpha_5]=0. \tag{8}$$

Thus this spin-only rotation generator does not satisfy (4). The same issue appears as a missing transverse kinetic complement in Neutral doublet transport.

An exact spatial plane-wave evolution makes the distinction explicit. Choose normalized $\beta w=w$, $Q_{\rm rot}w=0$, and initial field $e^{ipx_1}w$. Under the full free Dirac operator,

$$\Psi(t,x)=e^{ipx_1} \left[\cos(Et)-i\frac{p\alpha_1+m\beta}{E}\sin(Et)\right]w, \qquad E=\sqrt{p^2+m^2}. \tag{9}$$

Applying (8) gives

$$\boxed{ \|Q_{\rm rot}\Psi\|^2 =\frac{p^2}{p^2+m^2}\sin^2(Et).} \tag{10}$$

The mean is $\Psi^\dagger Q_{\rm rot}\Psi=-\|Q_{\rm rot}\Psi\|^2 w^\dagger B_{12}w$. It remains zero for an equal-weight preparation of the two opposite $B_{12}$ eigenpolarizations, despite the nonzero response norm. General complex preparations can change both the mean and the norm. Propagation along $x_5$ instead preserves the kernel because the second commutator in (8) vanishes. A favorable propagation direction is therefore a conditional geometric property, not a conserved electric-neutral particle assignment.

Equation (9) is a full spatial Dirac initial-value example, not an identified neutrino mode. It tests the proposed matrix generator on allowed spatial data.

Full spatial rotations have an orbital term

The actual rotation generator is

$$\mathcal L_{ij} =-i(x_i\partial_j-x_j\partial_i) -\frac i2\alpha_i\alpha_j. \tag{11}$$

For a radial scalar source, $[\mathcal L_{ij},H_b]=0$. The orbital commutator cancels the spin commutator with the spatial kinetic operator. This is why the full rotation-neutral Dirac field can remain neutral under a chosen total rotation while its spin components alone do not satisfy (4).

That conserved quantity is an angular momentum of the actual spatial field. It is a first-order differential generator, so it lies outside the constant zero-order phase-charge hypothesis of (4)–(5).

Promoting such a spatial rotation to a local coupling requires its coordinate and derivative terms too. Formally replacing a derivative by $\partial_\mu+iA_\mu\mathcal L$ changes the principal spatial operator through the orbital part of $\mathcal L$. The action must then derive the corresponding frame dynamics, connection source, conservation law and photon modes. Retaining only its spin matrix would discard the very term which conserved the rotation charge.

The useful distinction is therefore between a specified probe's geometric nullspace and a conserved physical source. Both can be geometrical. Their identification must follow from the complete same-field action and its occupied spatial solutions.

Particle and antiparticle identification remains open

V2 has not yet supplied a physical account of antiparticles or specified what changes in a particle's spatial pattern to produce its antiparticle. The conserved-generator conditions above constrain a candidate charge assignment; they do not identify that physical transformation. Assigning opposite charge labels or using opposite-frequency solutions still requires a derivation connecting those mathematical descriptions to the occupied particle and antiparticle fields.

Verification

spatial_charge_generator_checks.py checks (3) against direct field time derivatives for arbitrary complex spatial data. Independent commutant calculations include all Dirac matrices through depth ten and verify the effect of the normal derivative. Exact matrix evolution checks (9)–(10) for arbitrary complex doublet polarizations, momenta and masses. Direct spatial polynomial derivatives verify the orbital-plus-spin kinetic cancellation in (11). All checks remain active under Python optimization and reports use timestamped filenames.