Causal limit of an instantaneous scalar moment interaction

Summary: Smooth spatially separated initial data show that the instantaneous scalar moment equation changes a local current before causal contact, so its physical use requires a controlled retarded reduction.

A local observable tests the domain of dependence

$$i\partial_t\Psi=(D+V[\Psi]\beta)\Psi, \qquad V_\Psi(X)=\kappa\int K(X,Y)\sigma(Y)\,d\nu(Y), \quad \sigma=\Psi^\dagger\beta\Psi, \tag{1}$$

where $K(X,Y)=\operatorname{Tr}(P_XP_Y)$ and $d\nu$ is unit FS measure. The coefficient incorporates the conversion from the chosen physical measure. The potential uses spatially separated sources at the same time. Norm conservation, reciprocal variation and global existence do not settle whether this equation propagates causally.

At a target point $X$, take a unit spatial direction $e$ and its Hermitian Dirac velocity matrix $\alpha(e)$. Its local current is $j_e=\Psi^\dagger\alpha(e)\Psi$. Consider two smooth initial fields that agree on an open neighborhood of $X$, including all spinor components, and differ only in separated regions. Their local Dirac derivatives at $X$ agree. Equation (1) gives

$$\boxed{\Delta\dot j_e(X) =i\Delta V(X)\Psi(X)^\dagger[\beta,\alpha(e)]\Psi(X).} \tag{2}$$

This follows by differentiating both field factors in $j_e$ and using $\Delta\dot\Psi=-i\Delta V\beta\Psi$. The commutator is generally nonzero because $\beta$ anticommutes with the spatial velocity matrices. A state with both chirality components can therefore have a different local current derivative immediately, even though all its initial data near the target are identical.

For a local relativistic initial-value equation, changes outside a target's past domain of dependence cannot do this. Equation (2) identifies a specific failure of that property in the standalone instantaneous scalar model. No instantaneous potential plus compensating local constraint has been derived for (1); such a structure cannot be presumed.

Smooth projective example with unchanged total norm

Use an ordinary CP1 spatial chart $u$, with

$$P(u)=\frac1{1+|u|^2} \begin{pmatrix}1&\bar u\\u&|u|^2\end{pmatrix}, \qquad d\nu=\frac{d^2u}{\pi(1+|u|^2)^2}. \tag{3}$$

This is a conditional projective geometry of radius one, not a compactified direction. The target is $u=0$. Let $b_c(u)$ be a normalized nonnegative smooth bump supported on the disk $|u-c|<0.07$, and define

$$M_c=\int P(u)b_c(u)d\nu(u),\qquad\operatorname{Tr}M_c=1. \tag{4}$$

Move a scalar density $a=0.1$ from the disk centered at $c_2=1.2i$ to the disk centered at $c_1=0.6$. Both disks are disjoint from the target. This can be realized by smooth even-chirality field densities on top of a common positive background, with the same spinor in a neighborhood of the target. The positive total norm and total scalar source are unchanged. No distributional point source or change at the target is required.

At $\kappa=1$ the target potential difference is

$$\Delta V(0)=0.1\operatorname{Tr}[P(0)(M_{c_1}-M_{c_2})] \simeq0.03253166758. \tag{5}$$

Choose a local Clifford frame with $\beta=\sigma_z$, $\alpha(e)=\sigma_x$, and the common target spinor $\Psi(0)=(1,i)^{\mathsf T}/\sqrt2$. Then

$$\boxed{\Delta\dot j_e(0)=-2\Delta V(0) \simeq-0.06506333516.} \tag{6}$$

The minimum distances of the two source disks from the target are $\arctan(0.53)\simeq0.48736$ and $\arctan(1.13)\simeq0.84636$. Equation (6) holds at the initial time, before either distance can be traversed at speed one. Restoring $R$ multiplies these distances by $R$; it does not remove the immediate response.

Consequence for the candidate interaction

An effective instantaneous approximation could still be useful in a regime derived from local same-field dynamics. It must specify the retained spatial modes, response frequencies, initial conditions and error from neglecting propagation. The local Dirac retarded response supplies one explicit calculation in which both propagation channels follow from the written scalar term and signals have the required delayed support. Its static approximation has a frequency bound. That calculation does not derive the projective moment action or identify a physical background.

Verification

local_dirac_response_checks.py computes both normalized smooth source moments with independent quadrature refinement, verifies the unchanged total source, and differentiates the local current independently of (2). Its separate full nonlinear local Dirac propagation control has zero receiver response before causal arrival. The contrast tests the equations' domain of dependence, without assigning electric charge, weak interaction rates or an antiparticle interpretation.