Transition-density coupling

Summary: The complete derivative of a bilocal density interaction pairs both objects' transition densities, giving reciprocal mode vertices and an exact signed quadrupole matching law from the existing spatial kernel.

The spatial patterns which must match

A coupling calculation compares the changes that two spatial patterns can undergo. For a density interaction the relevant objects are their transition densities, which can have signs and complex phases even when each original density is positive. Their pairing provides a precise version of shape matching with both objects polarized.

Use one field $\Psi(t,x)$ on a specified finite containing spatial region of the nested macroscopic manifold. The labels $x,y$ below are two points in that same physical space. Let $A_\alpha(x)$ be fixed Hermitian matrices on its field components, and define real scalar bilinears

$$\rho_\alpha(x)=\Psi(x)^\dagger A_\alpha(x)\Psi(x). \tag{1}$$

For a compatible Dirac scalar channel one may take $A_\alpha=\beta P_\alpha$ when $\beta P_\alpha$ is Hermitian. Positivity of the bilinear is not assumed. The channel labels do not supply spatial embeddings; the appropriate nested maps, transport and measures must be included in the kernel used for a physical application.

The same energy fixes both responses

Take a real exchange-symmetric spatial kernel $K_{\alpha\beta}(x,y)=K_{\beta\alpha}(y,x)$ and the energy term

$$E_{\rm int}[\Psi]=\frac12\sum_{\alpha,\beta} \iint\rho_\alpha(x)K_{\alpha\beta}(x,y)\rho_\beta(y) d\mu_xd\mu_y. \tag{2}$$

Define $U=K\rho$ with the indicated integrals. Its field source is

$$\big(\mathcal N[\Psi]\big)(x)=\sum_\alpha U_\alpha(x)A_\alpha(x)\Psi(x). \tag{3}$$

This is the Hamiltonian convention $\delta E=2\operatorname{Re}\langle\delta\Psi,\mathcal N\rangle$. The bilocal variation derives (3) directly from the two density factors in (2).

At a specified background $\phi$, let

$$(D\eta)_\alpha(x)=2\operatorname{Re} \big[\phi(x)^\dagger A_\alpha(x)\eta(x)\big], \qquad V_\phi=\sum_\alpha(K\rho[\phi])_\alpha A_\alpha. \tag{4}$$

$D\eta$ is the change in the source density. Differentiating the whole source, including that change, gives

$$\boxed{D\mathcal N[\phi]\eta =V_\phi\eta+\sum_\alpha A_\alpha\phi\,(KD\eta)_\alpha.} \tag{5}$$

The first term is multiplication by the existing potential. The second term is the response of the potential to the changing pattern. It is generally real-linear and depends on $\eta$ and its conjugate.

For two independent field variations, the complete mixed second derivative is

$$\boxed{\big(E_{\rm int}''[\phi]\big)(\eta,\zeta) =2\operatorname{Re}\langle\eta,V_\phi\zeta\rangle +\langle D\eta,KD\zeta\rangle_{\rm real}.} \tag{6}$$

Exchange symmetry of $K$ makes (6) symmetric in the two variations. This is reciprocity before any mode truncation or amplitude convention. With the real field pairing $\operatorname{Re}\langle\eta,\zeta\rangle$, the density-map adjoint satisfies $D^*f=2\sum_\alpha A_\alpha\phi f_\alpha$; thus (5) is also $V_\phi+\tfrac12D^*KD$.

A response with $D\eta=0$ has no density-feedback term, but it can still respond through $V_\phi$, kinetic cross terms or other action terms. In particular $D(i\phi)=0$ while $D\mathcal N[\phi]\,(i\phi)=i\mathcal N[\phi]$. The full phase zero mode requires an actual stationary solution and the kinetic part of its equation.

The complete coherent mode vertex

Let $\Phi_i(x)$ be modes normalized in the derived kinetic metric. If they are not orthonormal, apply the canonical normalization before interpreting amplitude magnitudes. Define transition densities

$$r_{ij}^{\alpha}(x)=\Phi_i(x)^\dagger A_\alpha(x)\Phi_j(x). \tag{7}$$

For $\Psi=\sum_i c_i\Phi_i$ the exact quartic term is

$$E_{\rm int}=\frac12\sum_{i,j,k,l} \bar c_i c_j\bar c_kc_l\,T_{ij;kl}, \qquad T_{ij;kl}=\sum_{\alpha,\beta}\iint r_{ij}^{\alpha}(x)K_{\alpha\beta}(x,y)r_{kl}^{\beta}(y) d\mu_xd\mu_y. \tag{8}$$

Both pairs of modes enter. Reversing a transition changes its density to its complex conjugate, giving

$$T_{ij;kl}=T_{kl;ij},\qquad \overline{T_{ij;kl}}=T_{ji;lk}. \tag{9}$$

This retains the spatial shapes and relative phases required by the one field. The diagonal densities alone do not determine (8). Likewise, frequency matching does not guarantee a nonzero spatial vertex. A rotating wave approximation can select resonant terms only after this full tensor and the temporal frequencies have been calculated.

Exact factorization for the spatial dot-square kernel

For two prescribed spatial charts and their rectangular identification $B:\mathbb R^e\to\mathbb R^d$, adopt the scalar term $K(x,y)=g(x^TBy)^2$. Define complex transition moments

$$M_{ij}^{(a)}=\int xx^T\bar\Phi_i^{(a)}\Phi_j^{(a)}\,dx, \qquad M_{kl}^{(b)}=\int yy^T\bar\Phi_k^{(b)}\Phi_l^{(b)}\,dy.$$

Expansion of the two dot products gives

$$\boxed{T_{ij;kl}=g\operatorname{tr} \left(M_{ij}^{(a)}B M_{kl}^{(b)}B^T\right).} \tag{10}$$

There is no extra complex conjugate in this bilinear expression; the reversal identities are already in (9). The geometric action supplies the identification $B$, and common physical rotations transform all three tensor factors consistently. Centre motion or a position-dependent contact factor must also be varied when they belong to the adopted energy.

The source channel has at most $d(d+1)/2$ independent symmetric moment components at fixed depth. These are descriptions of spatial shape, not additional spatial axes. Higher spatial structure can enter other terms of the action even when it is absent from this particular kernel.

A normalized shape-matching example

Use ordinary flat Gaussian modes with width parameter $\omega>0$,

$$\phi_0(x)=\left(\frac\omega\pi\right)^{d/4} e^{-\omega|x|^2/2}.$$

For a real symmetric matrix $A$ with $\operatorname{tr}A^2=1$, define the polarized quadratic shape mode

$$\phi_A(x)=\sqrt2\,\omega \left(x^TAx-\frac{\operatorname{tr}A}{2\omega}\right)\phi_0(x). \tag{11}$$

Gaussian second and fourth moments give

$$\langle\phi_0,\phi_A\rangle=0,\qquad \|\phi_A\|=1,\qquad \int xx^T\phi_0\phi_A\,dx=\frac{A}{\sqrt2\,\omega}. \tag{12}$$

For example the covariance of $x^TAx$ under $|\phi_0|^2$ is $\operatorname{tr}A^2/(2\omega^2)$, establishing the normalization. The covariance of $xx^T$ with $x^TAx$ is $A/(2\omega^2)$, establishing the transition moment.

Two such patterns, with shape tensors $A,C$ and widths $\omega_a,\omega_b$, therefore have the exact vertex

$$T_{0A;0C}=\frac{g}{2\omega_a\omega_b} \operatorname{tr}(A B C B^T). \tag{13}$$

For two real fluctuation amplitudes $q_a,q_b$ about the respective backgrounds, the mixed second derivative of the cross energy is four times (13): each density variation contributes a factor two. Equation (13) itself is the normalized quartic mode vertex, not that Hessian.

In a shared two-plane take $B=I$ and

$$A=\frac1{\sqrt2}\begin{pmatrix}1&0\\0&-1\end{pmatrix}, \qquad C=R(\theta)A R(\theta)^T.$$

Then

$$\boxed{T_{0A;0C}=\frac{g}{2\omega_a\omega_b}\cos(2\theta).} \tag{14}$$

The patterns have the same nonzero spatial support and norm, yet this transition is exactly blocked at $\theta=\pi/4$. At $\theta=\pi/2$ the amplitude reverses sign. Rotating both objects by the same angle changes neither result. Squaring the amplitude gives a $\cos^2(2\theta)$ angular factor for an otherwise specified transition probability.

There is also an orientation-independent mismatch: a traceless shape $A$ has zero pairing with the breathing shape $C=I/\sqrt d$ for every relative rotation in this isotropic kernel. Other compatible shape transitions can remain nonzero. These are explicit geometric selection rules obtained from the existing interaction, without assigning a separate zero to a selected particle species.

Scope of the selection rule

For harmonic-oscillator basis functions, multiplication by $x_ix_j$ changes total oscillator number by $0$ or $\pm2$. Thus (10) connects only the corresponding mode pairs on each side. This is a rule for the transition density and its moment; it is not a claim that an odd spatial field has zero self-energy. The distinction is derived in angular kernel parity.

Equations (11)–(14) are a conditional flat-profile example. They do not replace the actual Coulomb-plus-precession electron orbit or specify a neutrino profile. Their role is to demonstrate, using an explicit kernel, how polarization becomes a higher-order spatial shape selection rule. The occupied particle solutions, photon map, accessible target motions, kinetic scales and scattering boundary conditions remain to be derived.

Verification

spatial_coupling_mechanism_checks.py compares (5)–(6) with finite differences of the full nonlocal complex-field energy using unequal spatial weights, indefinite bilinears and both positive and indefinite kernels. It reconstructs the complete quartic energy from (8), checks reciprocity, and independently evaluates (12)–(14) by Gaussian double quadrature with both shape orientations varied.

Related

• Projected bilocal variation — supplies the action and first variation extended here.
• Dirac source linearization — gives the local scalar analogue, including conjugate response blocks.
• Cross-sector moment kernel — derives the tensor identification and its covariance.
• Bilateral polarization coupling — motivates varying both spatial patterns.