Changing compatibility in a local Dirac collision

Summary: Independent full spatial evolutions of the local scalar Dirac equation show reciprocal pulse deformations that increase or decrease normalized scalar-pattern compatibility while preserving the combined polarization matrix.

An autonomous local encounter

Use the exact plane pulse equations derived from the local scalar Dirac term:

$$i(\partial_t+\partial_x)R=g\sigma L,\qquad i(\partial_t-\partial_x)L=g\sigma R, \qquad\sigma=2\operatorname{Re}R^\dagger L. \tag{1}$$

Both complex spatial patterns evolve; there is no prescribed target, externally forced probe or instantaneous spatial convolution. The scalar cross term of the one field is retained at each spatial point. This control uses zero bare mass and fields homogeneous in transverse ordinary spatial directions. It is not a transversely bound particle pair or a derived relation between different depths.

Prepare both spatial polarization textures

Let

$$b(s)=\begin{cases}\exp[1-(1-s^2)^{-1}],&|s|<1,\\0,&|s|\ge1, \end{cases} \quad F(s)=\frac{b(s/1.5)}{\sqrt{\int b(u/1.5)^2du}}, \tag{2}$$

and define a real unitary rotation in a selected two-polarization plane,

$$U(\theta)=\begin{pmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{pmatrix}. \tag{3}$$

The initial pulses are

$$\begin{aligned} R(0,x)&=F(x+3)e^{3i(x+3)}U(\theta_R(x+3))a,\\ L(0,x)&=F(x-3)e^{-3i(x-3)}U(\theta_L(x-3))b_\phi, \end{aligned} \tag{4}$$

$$a=\binom10,\quad b_\phi=\binom{0.7e^{i\phi}}{\sqrt{0.51}e^{-0.4i}}, \quad\theta_R(s)=0.35\sin(1.3s),\quad \theta_L(s)=0.25\cos(0.8s). \tag{5}$$

Each initial positive norm is one. The two pulses start on disjoint intervals centered at $-3$ and $3$. Their characteristic supports cross only during $1.5<t<4.5$ and are separated again at $t=6$. The coefficient is fixed at $g=1$ for both preparations below.

These supported pulses have nonzero carriers but are not strictly positive-frequency states: exact spatial support and exact one-sided Fourier support cannot both hold. The positive-frequency selection theorem in the analytic pulse note is established separately. The pulses in (4) have not been identified with physical incoming particles.

Measure the changed patterns after both responses

At a given time, form

$$M_R=\int RR^\dagger dx,\quad C_R=\int RR^{\mathsf T}dx, \qquad M_L=\int LL^\dagger dx,\quad C_L=\int LL^{\mathsf T}dx.$$

The scalar-pattern measure is

$$\boxed{\mathcal C=\operatorname{Tr}(M_RM_L) +\operatorname{Re}\operatorname{Tr}(C_R^\dagger C_L), \qquad \widehat{\mathcal C}=\frac{\mathcal C}{N_RN_L}.} \tag{6}$$

For a free crossing of the specified two profiles, $\mathcal C$ equals the complete spacetime integral of $\sigma^2$. Evaluating it on outgoing profiles asks how those changed patterns would engage in another complete crossing. It is not their instantaneous local contact after they have separated, and it is not a measured scattering rate. Independent translations of the profiles leave (6) unchanged.

The normalized quantity removes changes of their two positive norms. Both are relevant because (1) permits directional norm transfer, while preserving the total. Pointwise Cauchy–Schwarz gives $0\le\widehat{\mathcal C}\le2$.

The first weak-scattering map is generated by $g\mathcal C/2$ itself. Its Hamiltonian variation therefore gives $\mathcal C_{\rm out}=\mathcal C_{\rm in}+O(g^2)$. That leading approximation alone cannot establish the changing-match effect. The following results come from the complete nonlinear equations.

Both strengthening and weakening survive independent checks

The refined spatial solution gives:

The other norm is $N_L=2-N_R$. Dividing by $N_RN_L$ preserves both conclusions: normalized compatibility increases by more than $0.012$ in the first preparation and decreases by more than $0.003$ in the second. The difference is not an imposed alignment rule or merely the change in the amount of field in either outgoing pulse.

The full combined polarization matrix $M_R+M_L$ remains conserved. This stronger identity checks all complex polarization components, not only their summed norm. The local interaction source is zero before the supports meet and after they separate.

The comparison uses exact characteristic transport and exact local nonlinear reaction. For a reaction step of duration $h$, $\sigma$ is constant during that substep and

$$\boxed{R'=\cos(g\sigma h)R-i\sin(g\sigma h)L,\qquad L'=\cos(g\sigma h)L-i\sin(g\sigma h)R.} \tag{7}$$

Half a reaction, one characteristic shift, and half a reaction give a second-order spatial and temporal scheme with $\Delta t=\Delta x$. The numerical interval $[-16,16)$ has an unreached buffer throughout the run. Circular array indexing is a storage method, not an assumption that a physical direction is compactified.

An independent solver uses Fourier spatial derivatives and adaptive continuous-time DOP853 integration of the full complex first-order equation. It does not use (7), the weak-scattering map or the compatibility formula to evolve the fields. Its outgoing values are $0.578944463$ and $0.336433102$ respectively.

The values approach the independent solution at the expected second order. The maximum intermediate energy errors decrease from $1.65\times10^{-4}$ to $1.06\times10^{-5}$ in the first case, and from $1.03\times10^{-4}$ to $6.43\times10^{-6}$ in the second. The combined norm and matrix identities remain accurate to numerical roundoff. Once the fields are separated, both directional kinetic energies agree with their incoming values, as required by total energy and momentum.

Full traveling multiplet and exact controls

The two-plane data in (4) provide transparent comparison parameters; they are not asserted to represent every polarization of a six-dimensional field. A further calculation occupies all four traveling polarizations of a chosen eight-component spatial Clifford module. Both profiles have complex, spatially varying polarization and the combined matrix has four strictly positive eigenvalues,

$$(0.00087970,\ 0.02007333,\ 0.54629695,\ 1.43275002). \tag{8}$$

The complete four-component evolution conserves that matrix and is covariant under a common change of traveling polarization basis. Its scalar-pattern measure changes from $0.211053195$ to $0.210230023$. The full physical rotation test separately rotates the propagation direction and Clifford matrices; the larger basis symmetry of the transversely homogeneous class is not labeled a physical gauge symmetry.

Exact mismatched crossings provide another control. Orthogonal complex polarizations with arbitrary carriers evolve as the two free pulses through their entire overlap, with zero scalar source. Real-envelope quarter-turn phase preparations also give the exact zero predicted by the local equation; adding a spatial phase opens that latter cancellation. The same spatial depths permit a nonzero source when their polarizations are changed.

The weak-scattering test independently reduces $g$ from $0.04$ to $0.02$ to $0.01$. Its combined full-field error against the first-order map is $5.61\times10^{-4}$, $1.39\times10^{-4}$, and $3.47\times10^{-5}$, respectively, confirming the predicted quadratic remainder.

Interpretation

This calculation supplies a concrete mechanism for reciprocal change of spatial compatibility within the written local scalar equation. It also supplies exact polarization-sensitive noninteractions under that same rule. Both results retain the complete scalar cross term, physical propagation, complex phases and the response of both patterns.

Its limits remain substantial: the pulses are unbound in transverse space, their geometry is flat, their physical particle identity is unspecified, and distinct-depth embeddings and observed electric and weak vertices have not been derived. Successful conservation and convergence establish the conditional field calculation, not experimental support for IDWT.

Verification

local_polarized_collision.py contains the complete local evolution and moment functions. local_polarized_collision_checks.py contains the independent full spatial solver, refinement comparisons, complete Clifford and polarization tests, exact mismatch controls, continuous positive-frequency integrals and both complex weak-scattering responses. All checks remain active under Python optimization.