Spatially invariant scalar-null fields

Summary: The minimal massless spatial Dirac module admits complete kinetic scalar-null classes at every odd spatial depth, while even depths have no such constant linear class; this parity result does not select a particle or a physical charge.

State the dynamical question

The complete source derivative distinguishes a zero background scalar from a zero response to every perturbation. A further question is whether spatial motion preserves a proposed family of zero-source fields.

Consider the specified local equation in $d$ ordinary flat spatial directions, with $\hbar=c=1$:

$$i\partial_t\Psi=-i\sum_{j=1}^d\alpha_j\partial_j\Psi +g(\Psi^\dagger\beta\Psi)\beta\Psi. \tag{1}$$

The Hermitian matrices $\alpha_j,\beta$ square to one and anticommute pairwise. Use one minimal irreducible complex module of their full Clifford algebra, of dimension $n=2^{\lfloor(d+1)/2\rfloor}$. Components of this module are field amplitudes over ordinary space. The calculation introduces neither compactification nor a choice of global physical topology.

We classify constant complex-linear subspaces $E\subset\mathbb C^n$ with two properties: all spatial derivatives preserve fields valued in $E$, and every such field has zero scalar source. For the orthogonal projector $P_E$, these conditions are precisely

$$[P_E,\alpha_j]=0\quad(1\le j\le d),\qquad P_E\beta P_E=0. \tag{2}$$

Hermiticity makes an invariant subspace reducing, giving the commutator condition. The scalar condition follows from complex polarization of $u^\dagger\beta u=0$ for all $u\in E$. It retains every relative phase; testing just a few real vectors would be insufficient.

Odd spatial depth

For odd $d$, define

$$C=i^{(d-1)/2}\alpha_1\cdots\alpha_d,\qquad P_\pm=\frac{I\pm C}{2}. \tag{3}$$

Moving a matrix through the product gives

$$C^\dagger=C,\quad C^2=I,\quad [C,\alpha_j]=0,\quad \{C,\beta\}=0. \tag{4}$$

Thus $\beta$ exchanges the two eigenspaces, each of complex dimension $n/2$, and

$$\boxed{P_\pm\beta P_\pm=0.} \tag{5}$$

For any sufficiently regular initial field with $P_\pm\Psi=\Psi$, the free massless Dirac evolution stays in that same space. Equation (5) makes its nonlinear scalar term vanish pointwise for the entire evolution. It is therefore also an exact solution of (1). Spatially varying complex polarization and all momentum directions are allowed. The positive-frequency spectral projection of $\alpha\cdot p$ commutes with $C$, so positive-frequency wavepackets can be prepared in these classes as well.

If two full fields belong to the same class, their sum does too:

$$(\Psi_1+\Psi_2)^\dagger\beta(\Psi_1+\Psi_2)=0. \tag{6}$$

Their local scalar interaction vanishes even where their spatial profiles overlap. Opposite classes have a possible cross scalar $2\operatorname{Re}(\Psi_+^\dagger\beta\Psi_-)$, determined by their actual profiles and phases. This is a dynamical sufficient condition for a source zero, complementing the traveling-pulse conditions in local polarized coupling.

Completeness and the even-depth result

The usual complex Clifford classification states that an even number of generators gives one full matrix algebra, while an odd number gives two simple blocks. See the complex case in Etingof's Mathematical Ideas and Notions of Quantum Field Theory, section 11.3. Here its application concerns the kinetic generators alone, inside the larger module that also contains $\beta$.

For even $d$, the $2^d=n^2$ products of distinct $\alpha_j$ form a basis of $\operatorname{Mat}_n(\mathbb C)$. Consequently a projector commuting with them is either $0$ or $I$. The latter fails $P\beta P=0$. There is no nonzero class satisfying (2).

For odd $d$, the kinetic algebra acts irreducibly on each $C$ eigenspace, with commutant exactly $\operatorname{span}_{\mathbb C}\{I,C\}$. Its orthogonal projectors are $0,I,P_+,P_-$; only $P_+$ and $P_-$ are nonzero and scalar-null. Thus (3) supplies the complete answer under the stated module and constant-subspace assumptions.

An explicit matrix check of the commutant avoids choosing a polarization basis. The commuting conjugation projections give

$$\mathcal E(X)= \prod_{j=1}^d\frac{I+\operatorname{Ad}_{\alpha_j}}2(X) = \begin{cases} \dfrac{\operatorname{Tr}X}{n}I,&d\ {\rm even},\\[6pt] \dfrac{\operatorname{Tr}X}{n}I+ \dfrac{\operatorname{Tr}(CX)}n C,&d\ {\rm odd}. \end{cases} \tag{7}$$

Multiple module copies or additional independent action terms change the problem and can enlarge or break these invariant spaces. A position-dependent projector requires its own connection terms. Equation (7) does not classify those extensions, nonlinear families, or all individual zero-source solutions at even depth.

Propagation through containing spatial directions

The dimension $d$ in (1) counts all derivatives retained in that equation. Applying it to a lower particle while omitting its permitted normal propagation is a further restriction. Use the containing Clifford module when that propagation is present, as in normal Dirac propagation.

For an odd set of lower directions, form $C_d$ from their $\alpha_j$. Every additional normal kinetic matrix $\alpha_a$ anticommutes with it:

$$\{C_d,\alpha_a\}=0,\qquad P_\pm\alpha_aP_\pm=0. \tag{8}$$

The extra derivative maps the lower class into its complementary class. For a normal momentum $k$, its full symbol obeys

$$A_N(k)=\sum_a k_a\alpha_a,\qquad A_N(k)^2=|k|^2I. \tag{9}$$

If an initial field lies in $P_+$ and evolves under this normal kinetic term alone, the fraction in $P_-$ is $\sin^2(|k|t)$. Projecting back onto $P_+$ at every step would delete the actual propagation.

More generally, a smooth ambient field constrained to the fixed lower class for all times would have to satisfy $D_N\Psi=0$. For flat normal $\mathbb R^q$, (9) shows that a square-integrable normal profile satisfying this condition is zero: its Fourier transform vanishes at every nonzero $k$, and an $L^2$ function cannot be supported at the single point $k=0$. This concerns the stated smooth ambient extension and fixed projector. It does not exclude a field defined on a lower spatial submanifold, other confinement dynamics, or a complete containing-space symmetry.

In particular, an odd-dimensional containing space has its own kinetic involution. That complete class can preserve scalar nullity while the lower class mixes. A full positive-frequency plane also has zero scalar at every spatial depth. Lower-class mixing alone is therefore not a proof of nonzero scalar interaction. The positive-frequency vertex calculation tests both the complete angular current and these containing-direction qualifications.

Physical scope

This is a parity result; depths three, five, seven and nine share it. It does not single out the depth-five assignment. Within either class the written scalar term supplies no self-binding, so the theorem does not construct a localized particle.

Exact invariance also differs from stability against perturbations outside the class. The complete null-plane response has a density-dependent instability even though its background scalar vanishes identically.

The operator $C$ anticommutes with $\beta$. It therefore fails the general scalar-potential charge condition in conserved spatial charge. Its sign has not been identified with electric charge or an observed weak representation. The neutrino control still requires electric neutrality and nonzero weak vertices. The two signs are not assigned particle and antiparticle meanings.

This kinetic involution also differs from the even-bound-depth $\Xi=\beta\Gamma_b$ in collective translation waves, which commutes with $\beta$, and from the rest Spin(6) polarization in the massive current projection. They solve different spatial representation problems.

Verification

scalar_plane_response_checks.py checks the involution, every kinetic commutator, scalar-null complex superpositions, arbitrary spatial derivatives, and the full commutant projection for every depth from one through ten. scalar_plane_response.py provides the shared Clifford involution and complete plane-response operators. The results test the specified equation and representation, not the physical realization of a particle sector.