Moving nonlinear scalar planes and their response

Summary: The local scalar Dirac equation has continuous nonnull traveling-plane branches whose complete self-consistent response has a covariant closed form, with proper density, conjugate modes and the conserved norm treated explicitly.

Continuous branches of the written equation

Use the local scalar equation in ordinary flat spatial $\mathbb R^d$, $d\ge2$, with one time, $\hbar=c=1$ and the full minimal complex Dirac module:

$$i\partial_t\Psi=-i\alpha\cdot\nabla\Psi+ g(\Psi^\dagger\beta\Psi)\beta\Psi+J\beta\Psi. \tag{1}$$

The real scalar probe $J$ tests this same field. The plane backgrounds below are unbound controls with density per unit volume. Their effective scalar mass coefficient is not an identified particle mass, and no physical vacuum or global spatial topology has been selected.

At $J=0$ put $\Psi=e^{i\boldsymbol p\cdot x-i\Omega t}\phi$, $\rho=\phi^\dagger\phi>0$, $\sigma=\phi^\dagger\beta\phi$ and $M=g\sigma$. The stationary equation and its anticommutator with $\beta$ imply

$$(\alpha\cdot\boldsymbol p+M\beta)\phi=\Omega\phi, \quad \Omega^2=|\boldsymbol p|^2+M^2, \quad \Omega\sigma=M\rho. \tag{2}$$

For $M\ne0$, self-consistency therefore requires

$$\boxed{\Omega=g\rho,\qquad M=\pm\sqrt{g^2\rho^2-|\boldsymbol p|^2}.} \tag{3}$$

For $g>0$ these positive-frequency branches begin at $g\rho=|\boldsymbol p|$, the threshold of the scalar-null plane. The signs in (3) refer to the scalar density. No particle-antiparticle interpretation is assigned.

For example, let $p=|\boldsymbol p|>0$ and $\alpha(\boldsymbol p/p)w=w$. In the two-dimensional span of $w,\beta w$, write

$$\phi=\sqrt\rho\left(\cos\frac\theta2\,w+ \sin\frac\theta2\,\beta w\right),\quad \cos\theta=\frac p{g\rho},\quad \sin\theta=\frac M{g\rho}. \tag{4}$$

The fixed-norm energy density is $\mathcal E=\phi^\dagger\alpha\cdot\boldsymbol p\,\phi+g\sigma^2/2$. Writing $\nu=g\rho$ gives

$$\frac{\mathcal E_{\rm nonnull}}\rho= \frac\nu2+\frac{p^2}{2\nu},\qquad \frac{\mathcal E_{\rm nonnull}-\mathcal E_{\rm null}}\rho =\frac{(\nu-p)^2}{2\nu}. \tag{5}$$

This is a continuous density-dependent branch, with no discrete depth or mass selection.

Proper scalar density and the moving frame

Select $M>0$, $g>0$, and an arbitrary unit rest polarization $w_0$ with $\beta w_0=w_0$. Put $\Omega=\sqrt{M^2+|\boldsymbol p|^2}$ and $\rho=\Omega/g$. The explicit field is

$$\boxed{\phi= \sqrt{\frac{\rho(\Omega+M)}{2\Omega}} \left(I+\frac{\alpha\cdot\boldsymbol p}{\Omega+M}\right)w_0.} \tag{6}$$

This is the full positive-energy Dirac frame, with its mass coefficient now fixed by the same nonlinear source. Direct calculation gives

$$\phi^\dagger\phi=\frac\Omega g,\qquad \phi^\dagger\beta\phi=\frac M g,\qquad \phi^\dagger\alpha_j\phi=\frac{p_j}g. \tag{7}$$

Thus $\sigma=M/g$ is the proper rest density, whereas $\rho=\Omega/g$ is the positive density in the moving frame. Confusing the two changes the response coefficient.

The spin boost

$$S(\boldsymbol p)=\sqrt{\frac{\Omega+M}{2M}} \left(I+\frac{\alpha\cdot\boldsymbol p}{\Omega+M}\right) \tag{8}$$

satisfies $S^\dagger\beta S=\beta$ and $\phi=S\sqrt{M/g}\,w_0$. Its current velocity is $\boldsymbol p/\Omega$. It is a change of spacetime frame, not a new spatial dimension or a compactification.

Derive the full feedback response

In the carrier frame let $v=\beta\phi$, and retain $\zeta_q=(\eta_q,\overline{\eta_{-q}})^{\mathsf T}$. Define

$$\begin{aligned} H_{\rm f}(q)&= \operatorname{diag}\!\left( \alpha\cdot(\boldsymbol p+q)+M\beta-\Omega,\, -\overline{\alpha\cdot(\boldsymbol p-q)+M\beta-\Omega}\right),\\ f&=\binom v{-\bar v},\qquad r=(v^\dagger,v^{\mathsf T}),\qquad \mathcal H(q)=H_{\rm f}(q)+gfr. \end{aligned} \tag{9}$$

Here $fJ$ is the probe force and $r\zeta=\delta\sigma$ its scalar readout. The rank-one update in doubled variables contains both ordinary and conjugate variations; it is not a prescribed occupied-ray mass contact.

For frequency $z$, introduce the Minkowski notation

$$K=(z,q),\quad K^2=z^2-|q|^2,\qquad P=(\Omega,\boldsymbol p),\quad P\cdot K=\Omega z-\boldsymbol p\cdot q,\quad P^2=M^2. \tag{10}$$

Inverting the two free Dirac blocks, using (7), gives

$$\begin{aligned} \chi_{\rm f} &=r(z-H_{\rm f})^{-1}f\\ &=\frac1g\left[ \frac{P\cdot K+2M^2}{K^2+2P\cdot K} -\frac{P\cdot K-2M^2}{K^2-2P\cdot K}\right]\\ &=-\frac4g \frac{(P\cdot K)^2-M^2K^2}{(K^2)^2-4(P\cdot K)^2}. \end{aligned} \tag{11}$$

For example, the upper numerator follows from $v^\dagger v=\Omega/g$, $v^\dagger\alpha_jv=-p_j/g$, and $v^\dagger\beta v=M/g$. Self-consistency says $\delta\sigma=\chi_{\rm f}(J+g\delta\sigma)$. The exact response of the full linearized equation is therefore

$$\boxed{\chi(z,q)=\frac{\chi_{\rm f}}{1-g\chi_{\rm f}} =-\frac4g\, \frac{(P\cdot K)^2-M^2K^2}{K^2(K^2-4M^2)}.} \tag{12}$$

Equivalently, for $u=P/M$, $\chi=-4M\sigma[(u\cdot K)^2-K^2]/[K^2(K^2-4M^2)]$. The formula is independent of the chosen constant rest polarization for this scalar-to-scalar probe. The full spinor response, other channel projections and nonuniform polarization textures retain additional information.

Complete spectrum and the zero-mode qualification

The matrix determinant lemma applied to (9) gives

$$\boxed{\det(z-\mathcal H)= (K^2+2P\cdot K)^{n/2-1} (K^2-2P\cdot K)^{n/2-1} K^2(K^2-4M^2).} \tag{13}$$

Thus the four scalar-sector frequencies are

$$\pm|q|,\qquad \pm\sqrt{|q|^2+4M^2}, \tag{14}$$

and the remaining frequencies are

$$-\Omega\pm\sqrt{|\boldsymbol p+q|^2+M^2},\qquad \Omega\pm\sqrt{|\boldsymbol p-q|^2+M^2}, \tag{15}$$

each with multiplicity $n/2-1$. All are real. Their algebraic multiplicities and possible degeneracies are retained in (13).

This is a spectral result, not a proof of bounded linear evolution. At $q=0$ the phase vector and fixed-momentum norm derivative satisfy

$$\zeta_{\rm ph}=\binom{i\phi}{-i\bar\phi},\qquad \zeta_\rho=\frac g{2M}\binom{\beta\phi}{\overline{\beta\phi}}, \qquad \mathcal H(0)\zeta_{\rm ph}=0,\quad \mathcal H(0)\zeta_\rho=-ig\zeta_{\rm ph}. \tag{16}$$

This is the norm-frequency Jordan pair. Its secular phase change must be treated before making any stability claim, and real eigenvalues alone do not establish nonlinear stability. Tiny numerical splittings of this exact zero block are checked against (16), rather than interpreted as a measured growth rate.

Retarded and static meanings

The stable spectral poles in (12) use the retarded boundary value $z=\omega+i0$. Their two wave factors are the same ones in the uniform rest response. The numerator is a local differential operator, so the response preserves the causal support of those wave propagators. The first-order Dirac system fixes the compatible initial data; rewriting (12) as a fourth-order scalar equation does not add independent initial data. Both response channels are modes of the original field, not mediator particles.

At nonzero static momentum,

$$\boxed{\chi(0,q)= -\frac4g\, \frac{M^2+(\boldsymbol p\cdot q)^2/|q|^2}{|q|^2+4M^2}.} \tag{17}$$

At exactly zero momentum the continuous frequency expression is

$$\chi(z,0)=-\frac{4|\boldsymbol p|^2}{g(z^2-4M^2)},\qquad \chi(0,0)=\frac{|\boldsymbol p|^2}{gM^2}. \tag{18}$$

At rest, a spatially constant scalar probe changes only phase and this response is zero. Equation (17) has a different small-momentum limit. The frequency and momentum limits do not commute. An undamped initial-value experiment can also retain oscillatory transients; its late-time field need not settle to a static branch.

There is an independent nonlinear check of (18). Hold $\rho$ and $\boldsymbol p$ fixed and add a constant small $J$. The total mass coefficient $\bar M$ of a nearby positive-frequency stationary plane satisfies

$$\bar M=g\rho\frac{\bar M} {\sqrt{|\boldsymbol p|^2+\bar M^2}}+J,\qquad \sigma(J)=\frac{\bar M-J}g. \tag{19}$$

Differentiate at the unforced solution $\bar M=M$, $\Omega=g\rho$:

$$\frac{d\bar M}{dJ}=\frac{\Omega^2}{M^2},\qquad \boxed{\frac{d\sigma}{dJ}=\frac{|\boldsymbol p|^2}{gM^2}.} \tag{20}$$

This fixes the norm throughout, instead of changing the background density to keep an arbitrarily fixed carrier.

Connection to the null threshold

For fixed $p=|\boldsymbol p|>0$ and $g>0$, let $M\to0^+$. Then $\Omega\to p$, $\rho\to p/g$, and (12) tends to

$$\chi(z,q)\longrightarrow -\frac{4p^2}{g}\, \frac{(z-q_\parallel)^2}{(z^2-|q|^2)^2}. \tag{21}$$

This is exactly the scalar-null response at $\nu=p$. The branch connection follows from one local equation with one coefficient. It does not turn the continuous control family into a mass quantization or an empirical coupling law.

Verification

scalar_plane_response.py implements the complete Fourier symbol and exact responses. scalar_plane_response_checks.py runs 2,594 checks across the linked plane and kinetic-null calculations. For (6)–(21) it checks arbitrary complex rest polarizations in spatial dimensions two, three, five, six and ten, diverse momenta, the full nonlinear source, current and spin boost.

Direct full-spinor resolvents and determinant ratios check the feedback derivation independently of its scalar formula. Both field and momentum are rotated together to test physical rotation covariance. The exact zero-mode vectors are checked separately from the nonzero spectrum. Independently solved nonlinear stationary equations with positive and negative probes verify (20) at fixed norm. The approach to (21) has quadratic error in $M$ away from its poles.