Radial dynamics of two-frequency fields

Summary: An exact map from complex radial motion to the full two-frequency spatial field separates number norm from kinetic weight and transfers radial dynamics, source response and instability without a dimension-specific symmetry assumption.

Extend the stationary construction to radial motion

The two-frequency current construction uses two stationary components of one spatial field. Its radial dynamics can also be derived exactly when the profile changes with time. The map below is defined on actual radial functions over $d$ ordinary macroscopic spatial directions, with one existing containing normal direction used for the partner matrix $\alpha_y$. All sectors remain submanifolds of the one nested physical manifold.

Choose fixed unit rest polarizations $w_1,w_2$ satisfying $\beta w_j=w_j$. They need not coincide. For arbitrary complex radial functions $u=(F,G)$, define

$$\Phi_w[u]\,(x)=\frac{F(r)w+iG(r)\alpha(\widehat x)w} {\sqrt{\Omega_{d-1}}},\qquad \|u\|_r^2=\int_0^\infty(|F|^2+|G|^2)r^{d-1}dr. \tag{1}$$

Here $\Omega_{d-1}$ is the area of the unit sphere. The functions obey regularity at the origin and the decay needed for the stated integrals and integrations by parts. For constants $a,b$, put

$$\boxed{\Psi[u]=a\Phi_{w_1}[u]+b\alpha_y\Phi_{w_2}[u^*], \quad n=|a|^2+|b|^2,\quad s=|a|^2-|b|^2>0.} \tag{2}$$

Conjugation in $u^*$ acts on the radial functions $F,G$. It does not conjugate the fixed spatial Clifford matrices or the chosen rest polarization. Equation (2) defines a real-linear map into the full field. It does not assume a global antilinear symmetry on arbitrary spatial fields. When a two-frequency spatial symmetry exists derives the separate matrix and scalar-interference requirements for such a global map, including its compatibility with normal propagation.

The scalar source remains radial pointwise

The Clifford relations give

$$\Phi_{w_1}[u]^\dagger\beta\alpha_y\Phi_{w_2}[u^*]=0. \tag{3}$$

The diagonal terms connect opposite $\beta$ eigenspaces and vanish. The two radial cross terms cancel because their coefficient is $F^*G^*-G^*F^*=0$ and $\beta\alpha_y$ commutes with $\alpha(\widehat x)$. This holds for complex $F,G$ and both independent rest polarizations. Consequently

$$\boxed{\Psi^\dagger\beta\Psi =\frac{s}{\Omega_{d-1}}(|F|^2-|G|^2).} \tag{4}$$

The scalar center stays at the origin whenever its defining scalar normalization is nonzero. Its second-moment tensor is $M_\sigma=(sT/d)I_d$, where

$$T[u]=\int_0^\infty r^2(|F|^2-|G|^2)r^{d-1}dr. \tag{5}$$

Thus the common scalar moment interaction $g_0$ gives $\mathfrak m(r)=m_0+(g_0s/d)T[u]r^2$.

A nonlinear radial equation generates the full spatial solution

The real radial Dirac differential operator is

$$H_r[u]= \begin{pmatrix} \mathfrak m(r)&\partial_r+(d-1)/r\\ -\partial_r&-\mathfrak m(r) \end{pmatrix}. \tag{6}$$

It is self-adjoint on the appropriate radial domain with measure $r^{d-1}dr$. Suppose

$$\boxed{i\partial_tu=H_r[u]u.} \tag{7}$$

The field $\Phi_w[u]$ then satisfies the complete bound-coordinate Dirac equation. Since (6) has real coefficients, the conjugate radial functions satisfy $i\partial_tu^*=-H_r[u]u^*$. Anticommutation of the full bound operator with $\alpha_y$ supplies that same negative sign for the second term of (2). Both terms therefore solve the full spatial equation with precisely the scalar source in (4).

This proves an exact invariant radial family, uniform in the normal coordinate $y$. A varying $y$-dependent amplitude or an angular shape perturbation requires the additional spatial dynamics. The complete periodic response retains those possibilities rather than assuming this radial family exhausts the field.

For a stationary real profile $u_0$, $u(t)=u_0e^{-iMt}$ makes (2) the original two-frequency solution. Time-dependent complex profiles in (7) also give full nonlinear spatial solutions; the construction is not confined to harmonic motion.

Number norm and kinetic normalization have different weights

The positive and normal-partner radial frames are orthogonal after integration over the actual bound spatial directions. For variations $\delta u,\delta v$, equation (2) gives

$$\boxed{ \begin{aligned} \|\Psi[u]\|^2&=n\|u\|_r^2,\\ \operatorname{Re}\langle\delta\Psi[u],\delta\Psi[v]\rangle &=n\operatorname{Re}\langle\delta u,\delta v\rangle_r,\\ \operatorname{Im}\langle\delta\Psi[u],\delta\Psi[v]\rangle &=s\operatorname{Im}\langle\delta u,\delta v\rangle_r. \end{aligned}} \tag{8}$$

The second component is conjugated, so it reverses the imaginary part of the radial inner product. At $n=1$, the map preserves the real perturbation norm exactly. Its symplectic form, and hence the first-order kinetic term, instead has weight $s$.

Let $H_{r,0}$ be (6) with the bare scalar coefficient $m_0$ and no self-interaction. The full energy restricted to (2) is

$$\boxed{\mathcal E_{\rm full} =s\langle u,H_{r,0}u\rangle_r+ \frac{g_0s^2}{2d}T[u]^2 =s\,\mathcal E_{r,g_0s}[u].} \tag{9}$$

The kinetic term has the same overall factor $s$. Therefore the complete restricted action is $s$ times the radial action with coefficient $g_0s$, and variation gives (7) for fixed $a,b$. The amplitudes and their relative polarizations remain additional possible variations outside this restricted calculation.

For a radial scalar moment, a tangent source variation has the form $\delta(sT)=2s\operatorname{Re}\langle u,Q\delta u\rangle_r$, where $Q$ is the signed radial second-moment operator. In the canonical coordinate $\xi=\sqrt{s}\,\delta u$, this becomes

$$\boxed{\delta(sT)=2\sqrt{s}\operatorname{Re}\langle u,Q\xi\rangle_r.} \tag{10}$$

Both the changing profile and its kinetic weight must enter the normalized coupling vertex. Unit total number norm alone does not determine the normalization of every collective coordinate.

Radial instability is inherited

Let $u_0e^{-iMt}$ be a radial stationary solution at coefficient $g_0s$. A growing perturbation of its radial equation maps through (2) into a perturbation of the full two-frequency field. At $n=1$, equation (8) preserves its norm at every time. A radial exponential growth rate is therefore preserved by the map, including an oscillating growing mode. The negative-frequency component cannot remove that instability.

This statement is conditional on the radial instability being present. It does not assert that every scalar Dirac source is unstable. The zero-bare-mass moment branches already have such modes. Direct nonlinear evolution of the five- and six-dimensional radial equations confirms their transfer through (2), while both the full number norm and energy remain conserved.

A phase rotation of the stationary background should be removed when comparing a nonlinear perturbation with a tangent mode. If its real linearization has eigenvalue $\lambda$ with nonzero imaginary part, the physical radial tangent evolves as $\operatorname{Re}(e^{\lambda t}v)$ in real coordinates. Retaining only $e^{\operatorname{Re}\lambda t}\operatorname{Re}v$ would miss its oscillation and give an incorrect verification.

The current retains both polarizations and the radial phase

Integrating the normal number current gives

$$\boxed{\int j_y\,dx =2\operatorname{Re}\left[ \bar a b\,\langle w_1,w_2\rangle \int_0^\infty(F^{*2}+G^{*2})r^{d-1}dr\right].} \tag{11}$$

For a real unit stationary profile this reduces to a $2M$ oscillation, with an additional factor from the overlap of the two rest polarizations. Orthogonal rest polarizations can remove this integrated normal-current oscillation. That is a statement about one integrated current; local bound-direction currents and other response channels still need to be calculated.

The current in (11) is the spatial number current of the single Dirac field. Identifying its projection with a physical photon source remains a separate dynamical step.

A readjusted stationary current clock

The frequency of a nearby stationary branch changes when its scalar source changes. Add an isotropic coefficient $f_{\rm ext}$ multiplying $r^2/\sqrt d$. On the zero-bare-mass branch, write the total quadratic coefficient as $A$ to distinguish it from the amplitude $a$ in (2). Self-consistency and spatial scaling give

$$A=\frac{f_{\rm ext}}{\sqrt d}+\frac{g_0s}{d}T(A),\qquad T(A)=\frac{M(A)}{3A},\qquad M(A)\propto A^{1/3}. \tag{12}$$

At zero external coefficient,

$$\boxed{\frac{dA}{df_{\rm ext}}=\frac{3}{5\sqrt d},\qquad \frac{dM}{df_{\rm ext}}=\frac{M}{5A\sqrt d}.} \tag{13}$$

The frozen-profile scalar matrix element is $T/\sqrt d=M/(3A\sqrt d)$. Profile readjustment reduces that derivative by a factor $3/5$ in this specific homogeneous source model. Independent stationary solves verify (13). Following the derivative as an actual slow dynamical evolution would also require stability; a stationary branch derivative does not supply it.

Verification

radial_bifrequency_checks.py uses arbitrary complex radial profiles and independent rest polarizations. Full spatial quadrature checks the scalar density, center, moments, number current, tangent metric and kinetic form. Independent finite derivatives in every bound coordinate verify (7) as an actual spatial field equation.

A nonlinear radial trajectory and its reconstructed two-component field check the source equation, conserved norm and energy, and the complete oscillating growth mode. Separate stationary profile solves check the current-clock derivative. Reports preserve previous runs with timestamped filenames and all checks remain active under Python optimization.

Related

• Two-frequency spatial Dirac currents — supplies the stationary family extended here.
• Periodic spatial source response — retains normal propagation and angular motions outside the radial family.
• Self-consistent scalar Dirac moment modes — supplies the radial equation, scaling and stability calculation.
• Canonical coupling normalization — explains why a source coefficient must be combined with its kinetic normalization.

Complete radial polarization fields allows independent complex vector profiles in both radial components. That separate family closes under the local scalar equation in even and odd dimensions and retains the full tangential current. It does not assume the two components share one scalar radial profile or require a partner normal matrix.