Polarized pulse collisions

Summary: The derived local displacement action gives exact reciprocal pulse delays while retaining both vector waveforms, and its causal cubic equation identifies where the full spatial response enters.

Keep the scope of the effective collision equation

The spatial displacement action integrates a specified stationary component field over its two actual bound coordinates. For a displacement $X(t,y)\in\mathbb R^2$ it gives

$$\mathscr L=-\mathcal T\sqrt{D},\qquad D=(1-|\dot X|^2)(1+|X_y|^2)+(\dot X\cdot X_y)^2. \tag{1}$$

The coefficient $\mathcal T$ is the same spatial integral that normalizes the wave energy. The coordinate $y$ is an existing propagation direction, and $X$ is an actual position of the bound profile.

Equation (1) is exact for constant gradients of the completed spatial field. Treating it as a local action for varying gradients defines its leading derivative description. The finite collisions solved below are exact solutions of that local action. They are not asserted to solve the complete finite-width component field: the retarded spatial response supplies its frequency-dependent deformation and outgoing modes.

Write reciprocal evolution without a fitted collision rule

Let $v=\dot X$, $w=X_y$ and $p=\mathcal T^{-1}\partial\mathscr L/\partial v$. The dimensionless Hamiltonian density is

$$h=\sqrt{1+|w|^2+|p|^2+(p\cdot w)^2}. \tag{2}$$

The complete local evolution is

$$\boxed{ \dot X=\frac{p+(p\cdot w)w}{h},\qquad \dot p=\partial_y\frac{w+(p\cdot w)p}{h},\qquad w=X_y.} \tag{3}$$

Its energy and normal momentum above the resting background are $E=\mathcal T\int(h-1)\,dy$ and $P_y=-\mathcal T\int p\cdot w\,dy$. Both vector polarizations, including their relative phases, remain in (3).

Solve a collision by actual spacetime coordinates

Take two arbitrary smooth real vector waveforms $A(u),B(v)$ with square-integrable derivatives. Define

$$a=A',\quad b=B',\qquad I_A(u)=\int_{-\infty}^{u}|a(s)|^2ds,\qquad I_B(v)=\int_{-\infty}^{v}|b(s)|^2ds. \tag{4}$$

An exact parametrized solution of (1) is

$$\boxed{ \begin{aligned} t&=\frac{u+v+I_A(u)+I_B(v)}2,\\ y&=\frac{v-u+I_A(u)-I_B(v)}2,\\ X&=A(u)+B(v). \end{aligned}} \tag{5}$$

The two tangent vectors of this sheet in the containing spacetime are

$$Y_u=\left(\frac{1+|a|^2}2,\frac{-1+|a|^2}2,a\right), \qquad Y_v=\left(\frac{1+|b|^2}2,\frac{1-|b|^2}2,b\right). \tag{6}$$

Both are null for the metric $dt^2-dy^2-dX^2$, and $\partial_u\partial_vY=0$. Moreover, $2Y_u\cdot Y_v=1-2a\cdot b+|a|^2|b|^2$. These identities are the conformal-coordinate minimal-area equation and constraints, proving (5).

A sufficient regularity condition is $|a(u)|^2|b(v)|^2<1$ everywhere. Indeed, the physical null-coordinate map is

$$t-y=u+I_B(v),\qquad t+y=v+I_A(u), \tag{7}$$

with Jacobian $1-|a|^2|b|^2>0$. It therefore describes a regular graph in $(t,y)$ for sufficiently small bounded slopes. The formula does not promise a single graph after that condition fails.

Extract the two outgoing delays

Before overlap, each pulse has its original free waveform. After the opposite pulse has passed, (7) gives

$$\boxed{ X_A^{\rm out}=A(t-y-\Delta_A),\quad \Delta_A=\frac{E_B}{\mathcal T}; \qquad X_B^{\rm out}=B(t+y-\Delta_B),\quad \Delta_B=\frac{E_A}{\mathcal T}.} \tag{8}$$

Here $E_A=\mathcal T\int|A'|^2du$ and $E_B=\mathcal T\int|B'|^2dv$ are the incoming pulse energies. Positive shifts in the arguments' arrival times are delays. Every spatial component and phase of each waveform is retained.

These delays are reciprocal and involve the same tension. They hold for linear, elliptical and circular patterns within (1). For this isotropic local action, rotating one linear polarization does not change the delay at fixed pulse energy. That is a derived property of the specified geometry, rather than an imposed compatibility factor.

For example, $A(u)=\epsilon L e^{-u^2/(2L^2)}e_1$ has $E_A/\mathcal T=\epsilon^2L\sqrt{\pi}/2$. A second Gaussian pulse, in any real unit direction, gives the corresponding delay from its own energy.

Restore the causal spatial response at cubic order

For a weak displacement, define its trace-free source matrix

$$S=\dot X\dot X^T-X_yX_y^T -\frac12\left(|\dot X|^2-|X_y|^2\right)I_2. \tag{9}$$

Let $\mathcal F_{\rm ret}$ act on each matrix component by the Fourier multiplier $F(\lambda\sqrt{(\Omega+i0)^2-q^2})$ derived from the full spatial Hessian. With no independent incoming complementary deformation, its eliminated field gives the cubic response equation

$$\boxed{ \Box X_i+ \partial_\mu\!\left[ (\mathcal F_{\rm ret}S)_{ij}\partial^\mu X_j \right]=0 \quad\hbox{through cubic order in the weak field}.} \tag{10}$$

Here $\mu$ runs over $t,y$. To see its normalization, write the two real quadrupole components as $S_1=S_{11}$ and $S_2=S_{12}$. The full deformation has radial variables $h_A$, source $fS_A$ and quadratic kinetic factor $\mathcal T\lambda^2/4$. Solving their retarded equations gives $\tfrac12\langle f,h_A\rangle_w=\mathcal F_{\rm ret}S_A$. Variation of their original coupling to $S_A(X)$ produces $\mathcal T(\mathcal F_{\rm ret}S)_{ij}\partial^\mu X_j$. An independent full-profile check uses $\eta_h=\tfrac12(Sx)_i\partial_i n_0\,h(r)$ and gives

$$Z\int\left[ \partial_i n_0\cdot\partial_j\eta_h+ \partial_j n_0\cdot\partial_i\eta_h \right]d^2x =\mathcal T F_h S_{ij},\qquad F_h=4\int_0^\infty\frac{r^3h(r)}{(1+r^2)^3}\,dr, \tag{10a}$$

where the radius in $F_h$ is scaled by $\lambda$. This is the change in the actual translation kinetic matrix. Substituting each retarded radial response yields the same force coefficient. This derives (10) from the retained and complementary field equations; one does not vary a dissipative retarded kernel as an ordinary real energy.

Setting $F=1$ gives the cubic expansion of (3). For $X=A(t-y)+B(t+y)$, its leading correction is

$$\delta X=-A'(u)I_B(v)-B'(v)I_A(u), \tag{11}$$

which is exactly the first correction obtained by inverting (7). The full $F$ adds finite-width dispersion and radiation. Polarized spatial radiation calculates that outgoing energy from the same spatial source.

The component action and its identification with the master field remain conditional. Neither the exact local delays nor the causal cubic equation assign electric charges or measured elementary-particle interaction rates.

Verification

polarized_pulse_collision_checks.py checks the actual spacetime tangents and mixed derivatives of (5). A separate normalized variation of the full spatial profile checks (10a). A separate Fourier-space time evolution of the nonlinear Hamiltonian equations (3) checks outgoing vector profiles, both delays, energy and normal momentum for parallel, perpendicular and elliptical inputs. Spatial and time refinement distinguish integration error from a polarization effect. Reports use timestamps and checks survive Python optimization.