Reciprocal scalar envelopes

Summary: Readjusting both nested spatial Dirac profiles derives a reciprocal scalar coupling matrix and distinct common and relative amplitude responses from one interaction coefficient.

Couple both envelopes through their actual spatial profiles

The nested Dirac pair determines both spatial profiles from one scalar moment interaction. The scalar envelope reduction determines how one profile responds when its amplitude changes. Combining these constructions gives the mutual amplitude response without introducing a new coefficient for each pair.

Use nested spatial sectors $Q_p\subset Q_{p+q}$ and one ordinary free coordinate in a sector containing both. All these directions belong to the same infinite physical manifold. The two envelope amplitudes describe orthogonal projected channels of $\Psi_\infty$ with the kinetic measures specified in the nested-pair construction; they are not independent interaction particles or further physical dimensions.

Keep the two centers and rest polarizations fixed in this scalar envelope reduction. The scalar stationary profiles retain that polarization degeneracy; changing spatial shapes and orientations has the additional responses in Dynamical shape opening. The reduction here does not establish that those other motions decouple from a physical experiment.

Scalar densities change both source coefficients

Let $s_a,s_b>0$ be the two proper scalar envelope densities, and let $t_a,t_x,t_y$ be the unit-profile scalar moments per spatial axis:

$$\mathsf M_a=t_aI_p,\qquad \mathsf M_b=\begin{pmatrix}t_xI_p&0\\0&t_yI_q\end{pmatrix}. \tag{1}$$

These moments depend on the self-consistent source. The common interaction matrix is $s_aI\mathsf M_aI^T+s_b\mathsf M_b$, where $I$ is the spatial inclusion. For scalar source coefficients $A$ in the shared directions and $B$ in the remaining directions, the field equations require

$$\boxed{A=g_0(s_at_a+s_bt_x),\qquad B=g_0s_bt_y.} \tag{2}$$

The lower profile solves its full radial Dirac equation with source $A$. The higher profile solves the four-component spatial equation with sources $A,B$. Both must be recomputed when either scalar density changes.

For vanishing bare scalar coefficients, let $\rho=B/A$. At unit shared scale write the moments as $t_a^{(1)},t_x^{(1)}(\rho),t_y^{(1)}(\rho)$. The exact scale reduction is

$$\rho=\frac{t_y^{(1)}(\rho)} {(s_a/s_b)t_a^{(1)}+t_x^{(1)}(\rho)},\qquad A=\left[g_0\big(s_at_a^{(1)}+s_bt_x^{(1)}(\rho)\big)\right]^{3/5}. \tag{3}$$

This determines the relative deformation and common scale. The two positive spatial frequencies $M_a,M_b$ then scale as $A^{1/3}$.

The reduced energy determines the entire coupling matrix

Let $U(s_a,s_b)$ be the full spatial kinetic plus scalar interaction energy evaluated on these stationary profiles. Independent variation of each profile gives

$$\boxed{\frac{\partial U}{\partial s_a}=M_a,\qquad \frac{\partial U}{\partial s_b}=M_b, \qquad U=\frac56(s_aM_a+s_bM_b).} \tag{4}$$

The first two identities use stationarity of both spatial profiles; their implicit variations do not add an unaccounted first-order term to the energy derivative. The last identity is the full massless-background spatial dilation law.

Define the scalar coupling matrix

$$\boxed{G_{ij}=\frac{\partial M_i}{\partial s_j} =\frac{\partial^2U}{\partial s_i\partial s_j},\qquad G_{ij}=G_{ji}.} \tag{5}$$

This includes the deformation of both profiles. It is not just the bare moment overlap evaluated at fixed shapes. Homogeneity of (3)–(4) also gives

$$\boxed{\sum_jG_{ij}s_j=\frac{M_i}{5}.} \tag{6}$$

Thus the two self responses and cross response cannot be independently chosen. A common change of density constrains each whole row of the coupling matrix.

Compute the profile response with the full Dirac operator

Let $H(A,B)\phi=M\phi$, $\phi^T\phi=1$, be the real higher-profile matrix equation. If $B_j=\partial_jH$ is one scalar moment operator, differentiate the complete eigenproblem:

$$(H-M)\partial_j\phi =-(B_j-\phi^TB_j\phi)\phi,\qquad \phi^T\partial_j\phi=0. \tag{7}$$

Solving this constrained equation retains both signs of the Dirac spectrum. The frequency curvature is

$$C_{ij}=\partial_i\partial_jM =2\phi^TB_i\partial_j\phi. \tag{8}$$

It is the static full-source response, rather than a bare projection onto the occupied positive mode. The principle is the same as the dressed source calculation.

For the massless lower profile $t_a'=-2t_a/(3A)$. Define

$$\mathcal J=I-g_0 \begin{pmatrix} s_at_a'+s_bC_{AA}/p&s_bC_{AB}/p\\ s_bC_{BA}/q&s_bC_{BB}/q \end{pmatrix}. \tag{9}$$

When this Jacobian is invertible, differentiating (2) gives

$$\frac{\partial(A,B)}{\partial(s_a,s_b)} =g_0\mathcal J^{-1} \begin{pmatrix}t_a&t_x\\0&t_y\end{pmatrix}, \tag{10}$$

and therefore

$$\boxed{G= \begin{pmatrix}pt_a&0\\pt_x&qt_y\end{pmatrix} \frac{\partial(A,B)}{\partial(s_a,s_b)}.} \tag{11}$$

The asymmetric-looking factors in (9)–(11) account for the different numbers of spatial axes. Their product satisfies the reciprocal identity (5), which can also be checked by recomputing the profiles at independently varied $s_a,s_b$.

There is also a symmetric form which makes the source dependence explicit. Put $c=(A,B)^T$, $R=\operatorname{diag}(p,q)/g_0$, and collect the two frequency gradients in

$$F=(\nabla_cM_a,\nabla_cM_b) =\begin{pmatrix}pt_a&pt_x\\0&qt_y\end{pmatrix}. \tag{11a}$$

The scalar source equation is $Rc=Fs$. Define the stationary coefficient functional

$$\mathcal V(s,c)=s_aM_a(c)+s_bM_b(c)-\frac12c^TRc. \tag{11b}$$

Its derivative with respect to $c$ vanishes at the self-consistent source, and its value there is the complete reduced spatial energy $U(s)$. These coefficients describe the already specified spatial source; this rewrite introduces no independent interaction field.

If

$$\mathcal H_c=R-\sum_i s_i\nabla_c^2M_i \tag{11c}$$

is invertible, differentiation of the same stationary equation gives

$$\boxed{\frac{\partial c}{\partial s}=\mathcal H_c^{-1}F, \qquad G=F^T\mathcal H_c^{-1}F.} \tag{11d}$$

This proves reciprocity directly. If $\mathcal H_c$ is positive definite, $G$ is a Gram matrix of the two gradients with the weight $\mathcal H_c^{-1}$. It obeys $G_{ab}^2\leq G_{aa}G_{bb}$ and has full rank whenever the two gradients are independent. Both conditions hold for the tested nested pairs. A soft relative response can therefore arise when the two amplitudes change nearly the same common source. This is a consequence of the solved spatial profiles, not an assumed overlap or a fitted pair coefficient.

The massless source homogeneity supplies another check: $F^Tc=M/3$ and $\mathcal H_cc=(5/3)Rc$. Together with $Rc=Fs$, these give $Gs=M/5$ again. Positivity of this two-coefficient response matrix does not imply positivity of the full spatial field Hessian; the latter retains the radial and angular degrees of freedom.

One common-density response and a relative response

In the specified two-envelope action, let $D=\operatorname{diag}(M_a,M_b)$ and define

$$\mathcal A_{ij}=\sqrt{s_is_j}\,G_{ij},\qquad \mathcal C=D^{-1/2}\mathcal A D^{-1/2}. \tag{12}$$

The eigenvalues of $\mathcal C$ are the squared long-wavelength normal characteristic speeds of this reduced action. Equation (6) gives an exact eigenvector:

$$\boxed{\mathcal C \begin{pmatrix}\sqrt{s_aM_a}\\\sqrt{s_bM_b}\end{pmatrix} =\frac15 \begin{pmatrix}\sqrt{s_aM_a}\\\sqrt{s_bM_b}\end{pmatrix}.} \tag{13}$$

This mode changes the two densities together. The other eigenvector changes their relative amplitudes and has a geometry-dependent response. It is a collective mode of the pair, not the coupling of either isolated particle to every possible source.

At $g_0=1$, $s_a=s_b=1$, the five/six-dimensional pair gives

$$(M_a,M_b)\simeq(1.944521645,\,2.156533481),$$

$$G\simeq \begin{pmatrix} 0.200082767&0.188821562\\ 0.188821562&0.242485134 \end{pmatrix}. \tag{14}$$

This is full rank. Its normalized squared speeds are approximately $0.015337715$ and $0.2$. The following comparison uses the same $g_0=1$ and densities for every row:

The relative response is softer because the two amplitudes can partly offset each other's shared scalar source. Its coefficient is determined by the solved profiles and nesting. This comparison does not assign the relative mode to a neutrino, a photon, or a measured weak-interaction rate.

Check the complete normal fluctuation problem

For a common normal Fourier wavenumber $k$, the full ordinary/conjugate matrix of the two-envelope model is

$$\mathcal B(k)= \begin{pmatrix} \mathcal A&kI&\mathcal A&0\\ kI&-2D&0&0\\ -\mathcal A&0&-\mathcal A&kI\\ 0&0&kI&2D \end{pmatrix}. \tag{15}$$

Eliminating the other components gives the Hermitian quadratic pencil for $\lambda=\omega^2$:

$$\boxed{\left[(\lambda-k^2)^2I-4\lambda D^2 +4k^2D^{1/2}\mathcal A D^{1/2}\right]r=0.} \tag{16}$$

This also supplies a useful stability condition for any number of envelopes. Suppose $0\preceq\mathcal A\preceq D$, with $D>0$. For a normalized possibly complex vector $r$, put $a=r^\dagger D^2r>0$ and $b=r^\dagger D^{1/2}\mathcal A D^{1/2}r$, so $0\le b\le a$. Multiplying (16) by $r^\dagger$ gives

$$\lambda^2-2(k^2+2a)\lambda+k^4+4k^2b=0. \tag{17}$$

Its discriminant is $16[a^2+k^2(a-b)]\geq0$, and its roots are nonnegative. Hence the reduced normal frequencies are real. The tested pairs satisfy the stronger bound $0\prec\mathcal A\preceq D/5$. Negative scalar compressibility or excessive curvature need not satisfy this result; the diagnostic retains growing modes in both corresponding controls.

This is a condition excluding exponential growth in the reduced linear spectrum. At $k=0$, the upper-amplitude block $\left(\begin{smallmatrix}\mathcal A&\mathcal A\\-\mathcal A&-\mathcal A\end{smallmatrix}\right)$ squares to zero. When $\mathcal A\ne0$, its generalized zero directions describe phase drift caused by changing the conserved background densities. Real frequencies alone do not exclude that linear-in-time drift; a bounded orbital-stability statement must fix the appropriate densities and account for phase symmetry.

The scalar bound profiles can also have the separate radial instabilities documented elsewhere. Fixed centers and polarizations omit further collective motions. The calculation provides a reciprocal amplitude interaction and its reduced spectral criterion; full spatial particle stability requires the other response channels too.

Verification

coupled_scalar_envelope_checks.py solves the actual nested profiles at increased radial resolution and computes their constrained source derivatives. Independent changes of each scalar density verify the frequencies, full spatial energy and coupling matrix. Unequal densities and changed common coefficients test the homogeneity relations without fitting each pair separately.

The complete eight-component fluctuation matrix is checked against its long-wave limits and the independent pencil (16). Direct complex field variations recompute both spatial profiles at every perturbed scalar density and reproduce the ordinary/conjugate source matrix. Conditions remain active with Python optimization enabled.

Related

• Self-consistent nested Dirac pair — supplies the spatial equations and both readjusting profiles.
• Scalar envelope of a bound Dirac field — supplies the normalized normal-coordinate action and scalar density convention.
• Dynamical opening of spatial shape channels — retains the additional polarization and spatial shape responses.