Causal scalar response of the local Dirac equation

Summary: The written local scalar Dirac term has an exact retarded response about a specified uniform background, with two signed propagation channels, a conserved zero mode and a controlled static limit.

A specified spatial background

Use the local scalar Dirac term, with $\hbar=c=1$ and a real scalar probe $J$:

$$i\partial_t\Psi=D\Psi+g(\Psi^\dagger\beta\Psi)\beta\Psi+J\beta\Psi. \tag{1}$$

The probe measures the response of this same field. It is not an additional interaction field, a mediator particle or an already-derived particle pair. The following background is a control calculation, not an identified physical vacuum or a bound particle.

Take the canonical Spin$^c$ bundle $\Lambda^{0,*}$ on the specified physical CP$m$ with Fubini–Study holomorphic sectional curvature $4/R^2$. Here $D=\sqrt2(\bar\partial+\bar\partial^*)$ and $\beta=(-1)^{\text{form degree}}$. The global geometry and physical radius $R$ remain assumptions. These are ordinary spatial directions; no direction is compactified.

The degree-zero section $\chi_0=1$ is parallel and has pointwise norm one. For physical density $\rho_0>0$, define

$$\Psi_0=\sqrt{\rho_0}\,e^{-i\varepsilon t}\chi_0, \qquad \varepsilon=g\rho_0, \qquad Q=\rho_0\operatorname{Vol}(\mathrm{CP}^m). \tag{2}$$

It solves (1) at $J=0$ exactly. The coefficient $g$ has units of frequency divided by density; $\varepsilon$ is a frequency, not a dimensionless particle coupling. Radius units replace it by $\varepsilon R$.

Complete scalar-generated spatial blocks

Let $A=-\Delta$ be the nonnegative scalar Laplacian. Choose a real orthonormal scalar eigenfunction $p$ with $Ap=\lambda^2p$, $\lambda>0$. The two spinors

$$e_p=p\chi_0,\qquad o_p=\frac{D(p\chi_0)}\lambda \tag{3}$$

are orthonormal, have opposite chirality, and satisfy $De_p=\lambda o_p$, $Do_p=\lambda e_p$. The identity $\bar\partial^2=0$ excludes a two-form remainder in the second equation. Include every scalar eigenfunction and its full angular multiplicity, not just functions of one projective coordinate.

Write $\Psi=e^{-i\varepsilon t}(\sqrt{\rho_0}\chi_0+\eta)$, $\eta=u e_p+h o_p$, and $r=(\operatorname{Re}u,\operatorname{Re}h)^{\mathsf T}$, $s=(\operatorname{Im}u,\operatorname{Im}h)^{\mathsf T}$. The complete ordinary and conjugate derivative of (1) gives

$$\dot r=L_-s,\qquad \dot s=-L_+r-\sqrt{\rho_0}J_p\binom10, \tag{4}$$

$$L_-=\begin{pmatrix}0&\lambda\\\lambda&-2\varepsilon\end{pmatrix}, \qquad L_+=\begin{pmatrix}2\varepsilon&\lambda\\\lambda&-2\varepsilon\end{pmatrix}. \tag{5}$$

The measured linear scalar density is $\delta\sigma_p=2\sqrt{\rho_0}\,r_1$. The orthogonal spinor complement contains no degree-zero component. It is invariant under $D$ and $\beta$; its response operator is simply the self-adjoint $D+\varepsilon\beta-\varepsilon$. Consequently (4) includes every mode that a linear scalar probe can excite at this uniform background. The complement still exists and must be retained when a different source or nonlinear perturbation excites it.

Retarded response and both signs

For time dependence $e^{-izt}$ with $\operatorname{Im}z>0$, solve (4). Since

$$L_-L_+= \begin{pmatrix} \lambda^2&-2\varepsilon\lambda\\ 0&\lambda^2+4\varepsilon^2 \end{pmatrix},$$

the exact scalar susceptibility is

$$\boxed{\chi_\lambda(z)=\frac{\delta\sigma_p(z)}{J_p(z)} =\frac{-4\varepsilon\rho_0\lambda^2} {(\lambda^2-z^2)(\lambda^2+4\varepsilon^2-z^2)}.} \tag{6}$$

Put $\nu=\sqrt{\lambda^2+4\varepsilon^2}$. For $\varepsilon\ne0$,

$$\chi_\lambda(z)=-\frac{\rho_0\lambda^2}{\varepsilon} \left[\frac1{\lambda^2-z^2}-\frac1{\nu^2-z^2}\right]. \tag{7}$$

The retarded time kernel therefore is

$$\boxed{\chi_\lambda^{\rm ret}(t) =-\theta(t)\frac{\rho_0\lambda^2}{\varepsilon} \left[\frac{\sin\lambda t}{\lambda}-\frac{\sin\nu t}{\nu}\right].} \tag{8}$$

Both terms arise from modes of the original Dirac field. Their opposite residues are essential. For $\varepsilon=0$ the linear scalar-density response vanishes; the full spinor still responds in other components.

The spatially constant mode also has $\chi_0(z)=0$ identically. A constant scalar probe changes the phase of (2), while its positive norm stays fixed. Varying the background density while holding its carrier frequency fixed is a different static problem. It must not replace this zero mode by a compressibility inferred from (6) after canceling $\lambda^2$.

Spatial causal support

With zero initial response, (6) is equivalently

$$\boxed{(\partial_t^2+A)(\partial_t^2+A+4\varepsilon^2)\delta\sigma =-4\varepsilon\rho_0 A J.} \tag{9}$$

For a smooth source switched on after the initial time, set

$$S_0(t)=\frac{\sin(t\sqrt A)}{\sqrt A},\qquad S_\varepsilon(t)= \frac{\sin(t\sqrt{A+4\varepsilon^2})}{\sqrt{A+4\varepsilon^2}}.$$

The zero eigenvalue in $S_0$ uses its continuous limit $t$. Then

$$\delta\sigma(t)=-\frac{\rho_0}{\varepsilon}A \int_0^t[S_0(t-s)-S_\varepsilon(t-s)]J(s)\,ds. \tag{10}$$

Each wave operator propagates at speed at most one on the static product spacetime with spatial metric $g_R$. The local differential operator $A$ does not enlarge support. Thus (10) vanishes outside the source's causal future. The general finite-propagation statement for symmetric hyperbolic systems is given in Corollary 5.4 of Bär, Green-hyperbolic operators on globally hyperbolic spacetimes. Here it also applies directly to the real-linear first-order Dirac system. The fourth-order expression (9) does not authorize extra initial data: its compatible data come from that first-order system.

Retarded elimination is performed in the initial-value equations. An ordinary real quadratic functional $\int J(t)G_{\rm ret}(t-s)J(s)$ symmetrizes its kernel under variation and does not reproduce a purely retarded force. This distinction is demonstrated explicitly by Galley, Tsang and Stein.

Uniform spectral stability and its limits

The local blocks have frequencies $\pm\lambda$ and $\pm\nu$, all real. Together with the self-adjoint complement, this excludes exponential linear eigenmodes for every spatial polarization at the uniform solution. The constant phase and norm-frequency pair supplies the usual zero Jordan block; fixing the norm and quotienting the overall phase removes it. This is a spectral statement, not a nonlinear stability theorem.

More generally, replace the local scalar convolution by a real isotropic kernel normalized to preserve constants. If its multiplier on the chosen scalar harmonic is $r_\lambda\in[0,1]$, only the upper-left entry of $L_+$ changes, to $2\varepsilon r_\lambda$. The squared frequencies are

$$\boxed{\omega_\pm^2=\lambda^2+2\varepsilon^2 \pm2|\varepsilon|\sqrt{\varepsilon^2+(1-r_\lambda)\lambda^2}.} \tag{11}$$

Their sum is positive and their product is $\lambda^2(\lambda^2+4\varepsilon^2r_\lambda)>0$. Hence both are positive for every $\lambda>0$. This extends the uniform spectral result to those specified kernels; it does not give an instantaneous nonlocal kernel causal support.

Even in the local case the quadratic energy is indefinite. For $\varepsilon>0$, real-position eigenvectors can be chosen as $r_+=(1,0)$ and $r_-=(-\lambda/(2\varepsilon),1)$, with

$$r_+^{\mathsf T}L_+r_+=2\varepsilon>0,\qquad r_-^{\mathsf T}L_+r_-=-\frac{\nu^2}{2\varepsilon}<0. \tag{12}$$

Real frequencies alone therefore do not permit replacing (7) by a positive passive spectral measure. Nor do these two signs identify particles and antiparticles.

Controlled static approximation

For a nonconstant harmonic,

$$\boxed{\chi_\lambda(0)=-\frac{4\varepsilon\rho_0} {\lambda^2+4\varepsilon^2}.} \tag{13}$$

If all retained temporal frequencies satisfy $|\omega|\le\eta\lambda_{\min}$, $0<\eta<1$, then

$$\left|\frac{\chi_\lambda(\omega)}{\chi_\lambda(0)}-1\right| \le\frac1{(1-\eta^2)^2}-1. \tag{14}$$

Moving flat backgrounds

Moving nonlinear scalar planes derive the covariant flat-space continuation with the proper rest density and both conjugate modes retained. Its scalar response reduces to (6) at rest, while a spatially constant probe has a different response when the background carries momentum. The scalar-null plane supplies the continuous zero-scalar threshold, including its instability boundary. These are flat unbound controls, distinct from a global CP$m$ assignment.

Independent spatial verification

local_dirac_response.py implements (4), (6), (8) and a flat spatial propagation control. local_dirac_response_checks.py compares the blocks with the complete spatial Dirac Hessian, and compares independent first-order time evolution with the retarded convolution.

The flat control uses $D=-i\sigma_x\partial_x$, $\beta=\sigma_z$, $\rho_0=1$, $\varepsilon=0.7$, and $J=a\,b((x+4)/0.75)b(t-1)$, where $b(s)=\exp[1-1/(1-s^2)]$ for $|s|<1$ and zero otherwise. The receiver at $x=4$ cannot respond before $t=7.25$. Exact characteristic shifts and local reactions evolve both complex spinor components. A numerical buffer remains unreached; its periodic array storage is not a physical compactification.

The field error decreases by a factor approximately four at each refinement. The nonlinear equation conserves its positive norm to $3\times10^{-14}$ in these runs and approaches the linear solution at the expected first order in probe amplitude. These checks establish the declared response calculation, not the measured coupling of a particle. The instantaneous scalar comparison tests the contrasting domain of dependence of the moment action.