Local polarized Dirac pulse coupling

Summary: The local scalar Dirac term gives exact polarized pulse equations, a complete spatial mismatch criterion and reciprocal weak-scattering maps with both ordinary and phase-conjugate contributions.

A direct local same-field calculation

Consider the written scalar Dirac contribution on a specified flat ordinary spatial region, with zero bare mass:

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

The Hermitian matrices satisfy $\{\alpha_i,\alpha_j\}=2\delta_{ij}I$, $\{\alpha_i,\beta\}=0$, $\beta^2=I$. The coefficient $g$ is the one scalar coefficient in this equation; no separate one-way or pair coefficient is inserted. Other contributions to a completed master action are not supplied by (1).

Choose a physical unit direction $e$ and let $x=e\cdot\mathbf x$. Fields homogeneous in all transverse ordinary directions form an exact class of (1). This is a plane pulse control with finite norm per unit transverse volume, not a transversely bound particle. It introduces no compactification and does not fix the global geometry of any IDWT sector. The bound-profile polarization waves have different profile and mass assumptions.

Let $E_+$ be an orthonormal frame of the complete $+1$ eigenspace of $\alpha(e)$, of complex dimension $s$, and set $E_-=\beta E_+$. Every spinor in the chosen Clifford module can be written

$$\Psi=E_+R+E_-L,\qquad R,L\in\mathbb C^s. \tag{2}$$

The vectors $R(t,x)$ and $L(t,x)$ retain every traveling polarization and its spatial phase. They are two parts of the same field, not mediator particles or extra spatial dimensions. Their positive kinetic norm is additive because the two frames are orthogonal. Their scalar current is the cross term

$$\boxed{\sigma=\Psi^\dagger\beta\Psi =R^\dagger L+L^\dagger R=2\operatorname{Re}(R^\dagger L).} \tag{3}$$

It is not the sum of two nonzero individual scalar sources. Substitution in the full equation gives exactly

$$\boxed{i(\partial_t+\partial_x)R=g\sigma L,\qquad i(\partial_t-\partial_x)L=g\sigma R.} \tag{4}$$

Both responses follow from the same local interaction. A common physical rotation acts on the direction, coordinates, Clifford frame and field together and preserves (3)–(4).

Causal support and conserved quantities

Pure one-way pulses $R=A(x-t),L=0$, or $R=0,L=B(x+t)$, solve (4) for arbitrary smooth complex vector profiles. No fixed polarization or carrier frequency is required for that statement.

In the coupled equation, $R=0$ makes its source zero on that characteristic, and similarly for $L$. While a regular solution exists, each pulse therefore stays within the characteristic support of its own incoming profile. The interaction acts only where those supports cross. This domain of dependence follows from a local equation, addressing the issue exposed by the instantaneous scalar test.

There is a matrix continuity identity:

$$\boxed{\partial_t(RR^\dagger+LL^\dagger) +\partial_x(RR^\dagger-LL^\dagger)=0.} \tag{5}$$

The two nonlinear matrix sources cancel. Thus

$$M_R+M_L=\int(RR^\dagger+LL^\dagger)dx \tag{6}$$

is conserved, including its trace, the total positive norm. This matrix identity reflects the common $U(s)$ symmetry of the transversely homogeneous class. It is not a derived physical gauge group; general transverse derivatives do not have that entire symmetry. The directional norms $N_R=\operatorname{Tr}M_R$ and $N_L$ can change oppositely. They have not been assigned electric charges.

The local action also conserves energy and momentum:

$$E=\int\left[-iR^\dagger R_x+iL^\dagger L_x+ \frac g2\sigma^2\right]dx,\qquad P=\int-i(R^\dagger R_x+L^\dagger L_x)dx. \tag{7}$$

Boundary terms vanish for decaying or supported profiles. For separated incoming and outgoing pulses the scalar term vanishes. Their directional kinetic energies then satisfy $E_R=(E+P)/2$ and $E_L=(E-P)/2$, so each agrees before and after the complete encounter. The classical Dirac kinetic energy has both signs; this observation does not turn arbitrary real pulses into positive-energy particle states.

Exact mismatch of complete spatial patterns

The free crossing $R=A(\xi)$, $L=B(\eta)$, $\xi=x-t$, $\eta=x+t$, is an exact nonlinear solution if and only if

$$\boxed{\operatorname{Re}\big[A(\xi)^\dagger B(\eta)\big]=0} \tag{8}$$

for every pair of occupied characteristics that meets, apart from the trivial case $g=0$. This criterion follows directly from both equations in (4). Realifying the complex vectors makes it an ordinary orthogonality condition between their full real spatial spans in $\mathbb R^{2s}$.

For $A=a f$, $B=b h$ with $a^\dagger b=0$, the interaction vanishes for every pair of complex envelopes $f,h$. Both pulses can occupy the same spatial region while the scalar source remains zero. If $a^\dagger b$ is purely imaginary and both envelopes are real, (8) also holds. That latter cancellation can be opened by spatial phase variation even when the intensity profiles are unchanged.

A fixed target vector does not represent a general texture. An incoming polarization must obey (8) against every direction and phase occupied by the target profile, not merely against its spatial average.

Complete scalar response shows why an instantaneously vanishing scalar is insufficient on its own: the source still has a responsive direction, and general spatial kinetics can activate it. Equation (8) uses actual traveling solutions and their full encounter.

A complete geometric scattering functional

For finite incoming profiles define both kinds of moments:

$$M_A=\int AA^\dagger d\xi,\quad C_A=\int AA^{\mathsf T}d\xi, \qquad M_B=\int BB^\dagger d\eta,\quad C_B=\int BB^{\mathsf T}d\eta. \tag{9}$$

$M$ is Hermitian and positive semidefinite. The complex symmetric $C$ retains phase information that $M$ omits. These are integrals of spatial profiles of the same field, not new dynamical fields.

For the complete free encounter, $dx\,dt=\tfrac12d\xi\,d\eta$. Expanding $(2\operatorname{Re}A^\dagger B)^2$ gives

$$\boxed{\mathcal C[A,B]=\int\sigma_{\rm free}^2dx\,dt =\operatorname{Tr}(M_AM_B) +\operatorname{Re}\operatorname{Tr}(C_A^\dagger C_B)\ge0.} \tag{10}$$

For profiles initially separated, integrate over their full future crossing; for decaying scattering profiles, use the complete time axis. The right-hand side is zero precisely for (8), subject to the stated support conditions. Discarding $C_A,C_B$ would misclassify some complete scalar-source zeros.

Reciprocal weak-scattering map

Integrate (4) along the right-moving characteristic, evaluating its source on the free profiles at first order in $g$. Since $d\eta=2dt$,

$$\boxed{A_{\rm out}=A-\frac{ig}{2} \left(M_BA+C_B\bar A\right)+O(g^2).} \tag{11}$$

Integration along the left-moving characteristic similarly gives

$$\boxed{B_{\rm out}=B-\frac{ig}{2} \left(M_AB+C_A\bar B\right)+O(g^2).} \tag{12}$$

For example, the term $(A^\dagger B)B$ integrates to $C_B\bar A$, whereas $(B^\dagger A)B$ integrates to $M_BA$. The first correction is therefore real-linear in the incident pattern; it cannot generally be replaced by a single complex-linear overlap.

The integrated interaction functional is $g\mathcal C/2$. Its derivatives with respect to $\bar A$ and $\bar B$ are exactly the two brackets in (11)–(12), multiplied by $g/2$. Thus one scalar action generates both scattering responses with the canonical kinetic normalization. Small $g$ here means the corresponding dimensionless products $|g|N_A,|g|N_B$ are small in the chosen units. For smooth finite profiles on a regular finite encounter, the Duhamel expansion supplies the displayed $O(g^2)$ remainder.

Taking the norm derivative yields

$$\boxed{\Delta N_A =g\operatorname{Im}\operatorname{Tr}(C_BC_A^\dagger)+O(g^2), \qquad \Delta N_B=-\Delta N_A+O(g^2).} \tag{13}$$

The ordinary Hermitian term alone does not transfer directional norm at this order. Phase-sensitive spatial coherence can do so. This is not a particle–antiparticle conversion claim.

Positive free frequencies sharpen the selection rule

On the full line, suppose the incoming profiles have strictly positive free kinetic frequencies:

$$A(\xi)=\frac1{\sqrt{2\pi}}\int_0^\infty a(k)e^{ik\xi}dk, \qquad B(\eta)=\frac1{\sqrt{2\pi}}\int_0^\infty b(q)e^{-iq\eta}dq. \tag{14}$$

Assume sufficiently decaying profiles and no zero-frequency distribution. Parseval's identity pairs opposite Fourier frequencies in $\int AA^T$, so $C_A=C_B=0$. Consequently

$$\mathcal C=\operatorname{Tr}(M_AM_B),\qquad \Delta A=-\frac{ig}{2}M_BA+O(g^2),\quad \Delta B=-\frac{ig}{2}M_AB+O(g^2). \tag{15}$$

Each leading correction is a unitary polarization rotation to first order. No independent coefficient or phase average was used to remove the conjugate term; its absence follows from the occupied spectrum.

The exact zero (8) also strengthens. The two-variable Fourier transform of $A^\dagger B$ has support in the negative-negative frequency quadrant; its conjugate has support in the positive-positive quadrant. These supports are disjoint. Their sum can vanish only if

$$\boxed{a(k)^\dagger b(q)=0\quad\text{for almost every }k,q>0.} \tag{16}$$

Equivalently, the complete complex polarization spans of the two spectra are orthogonal. A quarter-turn phase with real envelopes is not an additional strictly positive-frequency escape from (16). Exactly supported spatial pulses cannot also be strictly one-sided in Fourier frequency; these two controlled assumptions must remain distinct.

In particular,

$$\ker M_B=\{v:\ B(\eta)^\dagger v=0\text{ for almost every }\eta\}. \tag{17}$$

A target whose polarization texture spans the complete traveling space has positive definite $M_B$ and no constant dark polarization. Extra polarization dimensions allow possible orthogonal patterns; they do not ensure that the dynamics select them.

Scope and verification

local_polarized_collision.py implements the complete local equations, both moments and both weak maps. local_polarized_collision_checks.py verifies the full Clifford reconstruction and physical rotation covariance at several spatial dimensions, exact nonlinear source zeros, independent spacetime source integrals, complete complex action derivatives and quadratic convergence to the weak map.

The same checks test continuous Fourier integrals of positive-frequency packets and full nonlinear collisions. Their changing spatial compatibility is measured after both patterns have responded, including spatial textures in all four traveling polarizations of an eight-component module.

The result establishes local polarization-sensitive coupling in this specified term and spatial class. It does not identify photons, neutrinos or bound leptons, select different particle depths, supply transverse localization, or derive a common coefficient after distinct-depth reduction. Every tested depth admits nonzero scalar coupling as well as permitted mismatches; no special depth-five zero follows from component counting.