Two-frequency spatial Dirac currents

Summary: Opposite-frequency components of the same spatial Dirac field give exact current oscillations at twice the bound frequency, with a common scalar source and explicit normalization, size and response constraints.

Construct a spatial clock from the bound field itself

A stationary field's common phase cancels from its bilinear observables. Nevertheless, the scalar Dirac model admits exact fields whose spatial currents oscillate at a frequency fixed by their bound profile. This provides a concrete construction to examine when asking whether slow particle oscillations can weaken coupling.

Use ordinary bound coordinates $x\in\mathbb R^d$ and one existing normal coordinate $y$ in a containing sector of the same nested physical manifold. The two components below belong to one field. No interaction particle or extra internal spatial direction is introduced.

The specific source model used here is local in time and $y$, with a centered spatial moment over the bound coordinates:

$$\begin{gathered} \sigma=\Psi^\dagger\beta\Psi,\qquad Q_\sigma=\int\sigma\,dx\ne0,\qquad X_\sigma=Q_\sigma^{-1}\int x\sigma\,dx,\\ M_\sigma=\int(x-X_\sigma)(x-X_\sigma)^T\sigma\,dx,\\ i\partial_t\Psi=\left[-i\alpha_i\partial_i-i\alpha_y\partial_y +\beta\left(m_0+g_0(x-X_\sigma)^TM_\sigma(x-X_\sigma)\right)\right]\Psi. \end{gathered}$$

The Hermitian matrices obey the ordinary spatial Clifford relations; $\beta$ anticommutes with each $\alpha$. With $X_\sigma=0$ and a radial source, the bound operator is $H_b=-i\alpha_i\partial_i+\beta(m_0+a_{\rm source}r^2)$. Only boosts in the $(t,y)$ plane are used below; full spacetime Lorentz covariance of this bound-coordinate kernel is not assumed.

For a real radial bound profile, the normal-partner construction supplies

$$H_b\phi=M\phi,\qquad H_b(\alpha_y\phi)=-M\alpha_y\phi, \qquad \int\phi^\dagger\phi\,dx=1. \tag{1}$$

The two fields are orthogonal. The stronger pointwise scalar identity is

$$\phi^\dagger\beta\alpha_y\phi=0, \qquad (\alpha_y\phi)^\dagger\beta(\alpha_y\phi) =-\phi^\dagger\beta\phi. \tag{2}$$

It follows from the full upper and lower radial components, as established for all selected rest polarizations in Spatial Dirac polarization waves.

An exact nonlinear two-frequency family

Choose complex constants $a,b$ and define

$$\boxed{\Psi(t,x)=a\phi(x)e^{-iMt} +b\alpha_y\phi(x)e^{iMt}.} \tag{3}$$

Set $n=|a|^2+|b|^2$ and $s=|a|^2-|b|^2$. Equation (2) gives

$$\boxed{\Psi^\dagger\beta\Psi=s\,\phi^\dagger\beta\phi, \qquad \int\Psi^\dagger\Psi\,dx=n.} \tag{4}$$

Solve the bound profile with this same scalar weight $s$. Its scalar center and every scalar binding moment are then constant. Each frequency component solves the same frozen bound equation, and their sum preserves exactly the source which defined that equation. Thus (3) solves the full nonlinear scalar moment equation, uniformly in $y$.

Two-frequency solitary waves also occur in nonlinear Dirac models with local scalar interaction; see Boussaïd and Comech. Equations (1)–(4) prove the present construction directly for the specified spatial moment source. Stability results for a different source model are not being assumed here.

The current and number-density pattern really move

Write the reference bilinears as $\rho_0=\phi^\dagger\phi$, $j_{i0}=\phi^\dagger\alpha_i\phi$ and $j_{y0}=\phi^\dagger\alpha_y\phi$. Define

$$z(t)=\bar a b e^{2iMt},\qquad v_i(x)=\phi^\dagger(-i\alpha_i\alpha_y)\phi\in\mathbb R.$$

Direct multiplication of the complete spatial field gives

$$\boxed{ \begin{aligned} \rho(t,x)&=n\rho_0(x)+2\operatorname{Re}z(t)\,j_{y0}(x),\\ j_y(t,x)&=n j_{y0}(x)+2\operatorname{Re}z(t)\,\rho_0(x),\\ j_i(t,x)&=s j_{i0}(x)-2\operatorname{Im}z(t)\,v_i(x). \end{aligned}} \tag{5}$$

These are currents over literal spatial coordinates. Their identification with a particular electromagnetic source still requires the physical current map. The scalar binding density in (4) does not move with them.

For $n=1$, the integrated normal current has oscillation amplitude $2|ab|=\sqrt{1-s^2}$. It can remain large when the bound frequency is small. The bound number-density centroid provides another check. For a radial rest polarization $w$, put

$$J_{iy}=w^\dagger(-i\alpha_i\alpha_y)w,\qquad F_0=\frac{1-2G/d}{2M}, \quad G=\int g(r)^2r^{d-1}dr. \tag{6}$$

Then

$$\boxed{X_{\rho,i}(t)=2\operatorname{Re}z(t)F_0J_{iy}, \qquad \dot X_{\rho,i}=-4M\operatorname{Im}z(t)F_0J_{iy}.} \tag{7}$$

Independent integration of $j_i$ yields precisely the second expression, checking spatial continuity. The scalar center stays fixed. For the tested pure odd-dimensional preparations, $\sum_iJ_{iy}^2=1$; the selected six-dimensional rest chirality instead has $J_{iy}=0$.

A slower clock increases the excursion scale $F_0\propto1/M$ at fixed radial shape. The corresponding speed amplitude is $\sqrt{1-s^2}(1-2G/d)$ for the unit odd-dimensional circulation. It stays finite: low frequency alone does not imply a small moving current. The five-dimensional field still occupies all five bound spatial directions.

Keep the canonical norm and the common coefficient fixed

Set $n=1$, $0<s\le1$, and choose

$$|a|^2=\frac{1+s}{2},\qquad |b|^2=\frac{1-s}{2}. \tag{8}$$

For the zero-bare-mass scalar moment branch, let $M_*$ and $\phi_*$ denote the profile at $s=1$ with a fixed coefficient $g_0$. The exact self-consistent scaling is

$$\boxed{\lambda=s^{1/5},\qquad \phi_s(x)=\lambda^{d/2}\phi_*(\lambda x),\qquad M(s)=\lambda M_*,\qquad a_{\rm source}(s)=\lambda^3a_{\rm source}(1).} \tag{9}$$

Both frequency components use this readjusted profile. The full field's source satisfies $a_{\rm source}=g_0sT/d$, where $T=\int r^2\phi_s^\dagger\beta\phi_s\,dx$. Under dilation, $T\mapsto\lambda^{-2}T$, so self-consistency gives $\lambda^3=s\lambda^{-2}$ and hence $\lambda=s^{1/5}$. The full field's canonical norm remains one; $s$ measures their signed scalar balance. The total scalar second moment scales as $s\lambda^{-2}=s^{3/5}$, while the squared number-density radius scales as $s^{-2/5}$. Reducing the scalar balance therefore makes the current clock slower and the bound field broader using the same $g_0$.

For $d=5$, $g_0=6.5$ and zero bare mass, independent spatial solves give:

These are model units and a continuous family, not a neutrino mass table. The same construction works at the tested depths three and six. The action has not selected a particular $s$ or a uniquely five-dimensional preparation. As $s\to0$, the bound profile spreads without limit, so the endpoint is not a finite localized zero-frequency particle.

The conserved full field energy per chosen normal normalization interval is the bound kinetic expectation plus $\mathcal E_{\rm int}=g_0\operatorname{tr}(M_\sigma^2)/2$. For the zero-bare-mass family it is

$$\boxed{\mathcal E(s) =sM(s)-\mathcal E_{\rm int}(s) =\frac56sM(s)=\frac56M_*s^{6/5}.} \tag{10}$$

The first equality retains the opposite signs of the two frequencies; the last uses the spatial dilation law. A periodic field containing both signs is not a single stationary energy eigenmode. Neither $M$ nor (10) can be assigned a measured neutrino mass without deriving that particle identification and its propagation normalization.

A small mean scalar source does not fix its response

The shrinking scalar moment in (9) reduces its diagonal scalar cross energy against a fixed partner in the specified moment kernel. The other partner must readjust at finite coupling, and the full variation still includes changes of $s$, shape and polarization.

For example, parametrize (8) by $a=\cos\theta$, $|b|=\sin\theta$, with $0\le\theta<\pi/4$, so $s=\cos2\theta$. Differentiating the readjusted energy gives

$$\frac{d\mathcal E}{d\theta} =-2M(s)\sqrt{1-s^2},\qquad \boxed{\frac{d^2\mathcal E}{d\theta^2} =\frac{4M(s)}{5s}(1-6s^2).} \tag{11}$$

For small positive $s$, the second derivative grows as $s^{-4/5}$. Thus a small mean binding source can coexist with a large response to changing the relative components. This curvature is not a stability proof for the periodic field.

Its linearized source also has explicit time dependence. If the spatial moment columns are $C(t)=aC_+e^{-iMt}+bC_-e^{iMt}$, the ordinary and conjugate feedback blocks contain respectively

$$C(t)\kappa C(t)^\dagger, \qquad C(t)\kappa C(t)^T. \tag{12}$$

Here $\kappa$ is the constant moment curvature at the unchanged scalar source, including the scalar-center response where required. The blocks in (12) contain frequencies $0,\pm2M$. A perturbation can therefore drive sidebands separated by $2M$. Its response requires a calculation with periodic coefficients, or an equivalent Floquet expansion; substituting $2M$ into a stationary single-frequency susceptibility does not supply it. Periodic spatial source response derives that complete equation. Some scalar readouts satisfy an exact finite spectral cancellation at rest, while normal propagation reopens their sidebands. The full shape response provides the relevant spatial source columns, with the time dependence retained here.

Motion changes the source phase seen by the partner

The normal boost symmetry gives another exact solution:

$$\Psi_\eta(t,x,y) =e^{\eta\alpha_y/2}\Psi_0(\tau,x),\qquad \tau=\cosh\eta\,t-\sinh\eta\,y. \tag{13}$$

Its scalar source remains the same pointwise. The current's oscillating phase is $2M\tau$, with frequency $2M\cosh\eta$ and normal wavenumber $2M\sinh\eta$ in these coordinates. Along a comoving trajectory it accumulates at $2M$ per proper time. At a fixed position the uniform boosted pattern instead oscillates at $2M\cosh\eta$.

A finite encounter also includes the spatial envelope and overlap of both objects. Their motion cannot be represented by inserting the rest clock alone into a target response. The general source-frequency accounting is in Source frequencies in spatial coupling.

Radial dynamics of two-frequency fields extends this stationary construction to arbitrary complex radial motion. It proves the distinct number and kinetic weights and transfers an existing radial instability into the full two-frequency field.

Verification

source_frequency_checks.py recomputes the scalar profile at fixed common coefficient and unit total norm for several scalar balances and depths. Full spatial quadrature checks the density, moments, currents, centroid and conserved energy. Independent finite derivatives in every bound coordinate, time and the normal direction verify both unboosted and boosted nonlinear solutions.

Independent temporal Fourier transforms recover the twice-frequency current and the periodic source-feedback blocks. Energy differences verify (11). The construction supplies an exact spatial clock and its limitations; full stability, the occupied neutrino geometry and physical photon coupling remain to be established.

Related

• Normal propagation of spatial Dirac modes — supplies the partner, boost and current-moment identities.
• Spatial Dirac polarization waves — supplies the complete spatial polarization frame.
• Source frequencies in spatial coupling — explains the clock sampled by a physical output map.