Positive-frequency scalar vertices

Summary: The full positive-frequency scalar Dirac vertex has an exact canonical singular strength, with a massless half-angle law, chirality-dependent zeros and reciprocal spatial interference that must be retained before channel counting.

Normalize the complete incoming bands

Use the scalar matrix $\beta$ appearing in the written Dirac source, on one flat containing spatial chart with $d\ge2$ ordinary directions and $\hbar=c=1$. Let

$$H(p)=\alpha\cdot p+m\beta,\qquad E_p=\sqrt{|p|^2+m^2},\qquad P_p=\frac{I+H(p)/E_p}{2}, \tag{1}$$

where $m\ge0$ is a specified mass coefficient of this free-band comparison. At $m=0$, exclude the single zero-momentum point when defining the positive band. The minimal full complex module has dimension $n=2^{\lfloor(d+1)/2\rfloor}$.

Choose any isometric positive-frequency frame $U_p$, $U_p^\dagger U_p=I_{n/2}$ and $U_pU_p^\dagger=P_p$. The complete scalar vertex between momenta $q$ and $p$ is

$$\boxed{C(p,q)=U_p^\dagger\beta U_q.} \tag{2}$$

This is the canonical positive-norm convention used by the massive current projection. Changing to a different external-wave normalization would change its factors and the associated kinetic metric together. The input and output spaces retain every positive-frequency polarization, not a single selected spin direction.

Exact singular strength

The Clifford relations give

$$\beta P_p\beta= \frac12\left[I+\frac{-\alpha\cdot p+m\beta}{E_p}\right]. \tag{3}$$

For $A=-\alpha\cdot p+m\beta$, its anticommutator with $H(q)$ is $\{A,H(q)\}=2(m^2-p\cdot q)I$. Compressing this identity to the $+E_q$ eigenspace yields

$$P_q A P_q=\frac{m^2-p\cdot q}{E_q}P_q. \tag{4}$$

Consequently

$$\boxed{C(p,q)^\dagger C(p,q)=\tau(p,q)I_{n/2},\qquad \tau(p,q)=\frac12\left[1+\frac{m^2-p\cdot q}{E_pE_q}\right].} \tag{5}$$

The reciprocal identity is $CC^\dagger=\tau I_{n/2}$, and $C(q,p)=C(p,q)^\dagger$. Every singular value is exactly $\sqrt{\tau}$. For $\tau>0$ the full vertex is a scaled unitary map between the two canonical bands. Its unitary part still contains polarization and phase information.

For $m>0$ and finite momenta, $\tau>0$. At $p=q$, (2) reduces to $(m/E_p)I$, recovering $\tau=m^2/E_p^2$. The forward scalar-density factor is a particular momentum configuration, not a general nonforward interaction strength.

The massless angular law

For $m=0$, let $\theta$ be the angle between the two nonzero spatial momenta. Then

$$\boxed{\tau(p,q)=\frac{1-\widehat p\cdot\widehat q}{2} =\sin^2\frac\theta2.} \tag{6}$$

Exactly co-propagating bands have $C=0$, even for different momentum magnitudes. Opposite directions have unit singular strength. At every nonzero angle, a nonzero input polarization has a nonzero response somewhere in the complete output band.

This does not make every individual vertex nonzero. For unit polarizations $a,b$,

$$|a^\dagger C(p,q)b|^2\le\tau(p,q), \tag{7}$$

and the chosen two patterns can be orthogonal under that map. A particular observed channel also involves its actual spatial inclusion and readout. Equation (6) alone neither identifies an electromagnetic channel nor predicts a scattering rate.

Under a common physical rotation, transform both momenta and spinor frames by the covering spin rotation. The scalar vertex is unchanged in the transported frames because the spin rotation commutes with $\beta$. Independent changes of polarization bases instead give $C\mapsto V_p^\dagger C V_q$ and preserve (5).

Chirality gives a channel zero, not a depth-five exception

At odd total spatial dimension, the kinetic involution commutes with every massless $P_p$ and anticommutes with $\beta$. Its positive-frequency eigenspaces each have dimension $n/4$. Writing the restricted involutions as $C_p^{\rm kin}$ and $C_q^{\rm kin}$ gives

$$C_p^{\rm kin}C(p,q)=-C(p,q)C_q^{\rm kin}. \tag{8}$$

Thus the scalar vertex vanishes between two equal kinetic chiralities for every pair of momenta. Between opposite chiralities, its singular strength is still $\sqrt\tau$. The complete vertex remains consistent with (5): its nonzero response is in the opposite output block.

This occurs at every odd total spatial dimension, including three, five, seven and nine. It supplies no unique depth-five selection and no electric-charge assignment. The neutrino control continues to require a derived neutral electromagnetic channel and surviving weak vertices. Both kinetic signs refer to components of the same field; their particle-antiparticle meaning remains open.

If a field also propagates through containing spatial directions, use the complete containing kinetic operator. A lower odd-dimensional involution anticommutes with every additional normal $\alpha_j$. The lower class can therefore mix under normal motion. The corresponding protection question is treated explicitly in spatially invariant scalar-null fields.

Assemble the actual spatial current before projecting

A positive-frequency free packet has the form

$$\Psi(x,t)=\int\frac{d^dp}{(2\pi)^{d/2}}\, U_p a(p)e^{ip\cdot x-iE_pt},\qquad \|\Psi\|_{L^2}^2=\int\|a(p)\|^2d^dp. \tag{9}$$

Its scalar current is the full coherent contraction

$$\boxed{\sigma(x,t)= \iint\frac{d^dp\,d^dq}{(2\pi)^d}\, a(p)^\dagger C(p,q)a(q) e^{i(q-p)\cdot x-i(E_q-E_p)t}.} \tag{10}$$

At $m=0$ every monochromatic forward scalar is zero, but the off-diagonal momentum pairs in (10) can remain nonzero. A spatial packet cannot generally be replaced by an incoherent sum of those forward scalars. A packet confined to one complete odd-dimensional kinetic chirality does have zero scalar, because every matrix element in (10) then vanishes.

For two such packets of opposite kinetic chirality, denote them by $\psi_+$ and $\psi_-$ and put $c(x,t)=\psi_+^\dagger\beta\psi_-$. Their individual scalar self-currents vanish, while the total scalar is

$$\sigma=2\operatorname{Re}c. \tag{11}$$

For the local equation $i\partial_t\Psi=D\Psi+g\sigma\beta\Psi$, the exact two rows are

$$\boxed{ i\partial_t\psi_+=D\psi_++g\sigma\beta\psi_-, \qquad i\partial_t\psi_-=D\psi_-+g\sigma\beta\psi_+. } \tag{12}$$

Both patterns respond to the same local scalar. For sufficiently decaying fields, their positive norms satisfy

$$\boxed{\frac{dQ_+}{dt}=2g\operatorname{Im}\int c^2dx,\qquad \frac{dQ_-}{dt}=-\frac{dQ_+}{dt}.} \tag{13}$$

The derivative follows from $2\operatorname{Re}\langle\psi_+,-ig\sigma\beta\psi_-\rangle$. Its sign depends on the full complex cross pattern. Individual norms need not be conserved even though their sum is. Equation (12) retains all generated frequency components; positive frequency is an initial preparation, not a projection imposed at every nonlinear time step.

These full spatial equations generalize the local cross-source structure of polarized traveling pulses. They do not assume automatic alignment, a fixed target or a mediator particle.

What counting can and cannot multiply

For a fixed unit incoming polarization $b$, summing over a complete orthonormal output band gives

$$\sum_{j=1}^{n/2}|e_j^\dagger Cb|^2=\tau. \tag{14}$$

Summing additionally over all orthonormal inputs gives $\operatorname{Tr}(C^\dagger C)=(n/2)\tau$, while averaging over those inputs gives $\tau$ again. The number of available components is not an extra enhancement of one normalized incident field.

Similarly, a cumulative spatial-mode count can organize the available channels, but their actual current contraction, kinetic metric and spectral weights remain necessary. The oscillator source marginals show how Hockey-Stick sums enter a specified projection after those ingredients are supplied. They do not replace (10) by a depth-only coupling coefficient.

Verification

positive_scalar_vertex_checks.py checks full positive-frequency eigenspaces, all singular values, reciprocity, projector identities and physical rotations in every spatial dimension from two through ten. Independent spatial superpositions verify the complete current, including diagonal massive terms and complex interference. General angular superpositions check the exact chiral source zeros and both reciprocal nonlinear source rows.

Containing-direction tests retain every additional Clifford matrix and distinguish lower-class leakage from a scalar zero protected by a complete containing symmetry. These results verify the specified scalar operator and incoming preparations; a physical particle-channel embedding, flux normalization and observed rate remain separate calculations.

scalar_wave_packets.py implements (12) on a full three-dimensional Cartesian grid with all four complex components. Its incoming Fourier Gaussian is projected onto the complete positive-frequency band once, then evolved with the exact local reaction and exact free Fourier propagator in a symmetric time splitting. The numerical periodic box is a regulator for localized ordinary spatial fields, not a physical compactification.

scalar_packet_collision_checks.py checks that solver against an independently assembled four-component nonlinear differential equation. The default 101 checks include energy variation, reciprocal norm transfer, positive-frequency preparation and second-order time convergence. Its optional larger packet runs record spatial and temporal refinement data; a finite-grid collision is a conditional unbound-field benchmark, not an identified particle pair.

Axial spatial Dirac packets retain transverse localization and complete complex polarization vectors in an exact angular family at odd total spatial dimension. Their radial and axial equation comes from the same local scalar, with both patterns evolving and the full angular spinor dependence reconstructed explicitly.

Resonant scalar wave quartets provide a multi-directional positive-frequency preparation with reciprocal chiral norm transfer. The exact coherent contraction has a sharp normalized bound. A right-angle cancellation for one polarization per selected chiral band is removed by allowed complete higher-dimensional vectors; counting the components cannot determine that contraction.

Frequency accounting for scalar Born response uses both complete vertices to control the negative-output source. A resonant cancellation for positive incoming scattering data coexists with nonzero finite-time forcing from a positive initial preparation.