Coherent spatial overlap

Summary: Exact complex Gaussian overlaps retain both objects' shapes, polarizations, and spatial phases, while normalized opposing lobes give a concrete low-momentum source cancellation.

Spatial source patterns, before averaging

After deriving a source or response operator, a local overlap may take the form

$$\mathcal A(k)=\int \langle F_a(x),F_b(x)\rangle e^{-ik\cdot x}\,d^Dx. \tag{1}$$

Both $F_a$ and $F_b$ are polarized spatial patterns. Their components must be expressed in compatible frames and normalized according to the action. Equation (1) adopts a local identity pairing; a derivative or nonlocal kernel must be retained when the action supplies one.

The coordinates belong to one common physical chart of the nested manifold. This benchmark uses finite test functions or wavepackets after the spatial mode maps have been specified. It does not replace a lower field's cylindrical continuation by a thin sheet, or identify wavepacket localization in a free direction with a new bound dimension.

For an actual nonlinear source, first calculate $F_b$ from the higher field as in descending photon source. Substituting a bare wavefunction or a positive density for a different source would change (1).

Exact overlap of two Gaussian amplitude profiles

Let $A,B$ be real positive-definite $D\times D$ matrices. Choose

$$\psi_a(x)=\left(\frac{\det A}{\pi^D}\right)^{1/4} \exp\left[-\frac12(x-X_a)^TA(x-X_a)+ip_a\cdot x+i\phi_a\right],$$

and the analogous $\psi_b$ using $B,X_b,p_b,\phi_b$. Both have unit $L^2$ norm. If $F_a=\psi_a T_a$ and $F_b=\psi_b T_b$ with constant normalized tensors, the result is

$$\mathcal A(k)=\langle T_a,T_b\rangle\,I_{ab}(k). \tag{2}$$

Put

$$S=A+B,\qquad m=S^{-1}(AX_a+BX_b), \qquad q=p_b-p_a-k,\qquad \Delta X=X_a-X_b.$$

Completing the real square gives

$$(x-X_a)^TA(x-X_a)+(x-X_b)^TB(x-X_b) =(x-m)^TS(x-m)+D_0,$$

where

$$D_0=X_a^TAX_a+X_b^TBX_b-m^TSm =\Delta X^T(A^{-1}+B^{-1})^{-1}\Delta X.$$

The last identity holds without assuming that $A$ and $B$ commute. Diagonalizing $S$ and taking the characteristic function of the centered Gaussian now yields

$$\boxed{ I_{ab}(k)= \frac{2^{D/2}(\det A\det B)^{1/4}}{\sqrt{\det S}} \exp\left[-\frac12D_0-\frac12q^TS^{-1}q +iq\cdot m+i(\phi_b-\phi_a)\right].} \tag{3}$$

Thus spatial separation, shape covariance, phase-gradient mismatch, and tensor polarization all remain visible. At equal profiles and $k=0$, $I_{aa}=1$. Cauchy–Schwarz gives $|I_{ab}(k)|\leq1$ for every $k$, since the Fourier phase has unit magnitude.

The positive density overlap in Gaussian contact profiles is a different quantity: it has different normalization and no relative wave phase. It cannot stand in for (3) when interference is relevant.

Two structured objects

For finite patterns

$$F_a=\sum_i a_i\psi_{ai}T_{ai}, \qquad F_b=\sum_j b_j\psi_{bj}T_{bj},$$

linearity gives the exact amplitude

$$\boxed{\mathcal A(k)= \sum_{i,j}\overline a_i b_j \langle T_{ai},T_{bj}\rangle I_{ai,bj}(k).} \tag{4}$$

Both objects' orientations and phases occur in every term. Individual nonzero overlaps can cancel in the sum. The full Gram matrix of the component profiles determines the normalization of $a$ and $b$; the lobes are not assumed orthogonal merely because their centers differ. If physical dynamics makes some outputs distinguishable, sum their intensities only after separating those outputs.

A normalized shape with a vanishing long-wavelength source

For an explicit finite source use the real normalized amplitude

$$\psi_\sigma(x)=(\pi\sigma^2)^{-D/4} e^{-|x|^2/(2\sigma^2)},\qquad \sigma>0.$$

For a fixed nonzero displacement $L$, define opposing lobes

$$f_-(x)= \frac{\psi_\sigma(x-L/2)-\psi_\sigma(x+L/2)} {\sqrt{2\left(1-e^{-|L|^2/(4\sigma^2)}\right)}}. \tag{5}$$

Their overlap is $e^{-|L|^2/(4\sigma^2)}$, so $\|f_-\|_2=1$. With Fourier convention $\widetilde f(k)=\int f(x)e^{-ik\cdot x}d^Dx$,

$$\boxed{ \widetilde f_-(k)= \frac{-2i\sin(k\cdot L/2)} {\sqrt{2\left(1-e^{-|L|^2/(4\sigma^2)}\right)}} \widetilde\psi_\sigma(k),} \tag{6}$$

where

$$\widetilde\psi_\sigma(k) =2^{D/2}\pi^{D/4}\sigma^{D/2} e^{-\sigma^2|k|^2/2}.$$

The zero at $k=0$ comes from spatial cancellation while the source has unit norm. For fixed $L,\sigma$ and $|k\cdot L|\ll1$, its squared Fourier amplitude begins at order $(k\cdot L)^2$. Directions with $k\cdot L=0$ remain exactly dark for this source. This is a shape-induced selection within the declared linear source map, not a small coefficient inserted into that map.

For a tensor source $f_-(x)T_b$, additionally contract its tensor with the other object's polarized test pattern. The node in (6) and the tensor pairing are separate factors only for this separated preparation.

General moment cancellation

Suppose an integrable scalar or tensor source $s(x)$ has finite absolute moment of order $m$ and

$$\int x^\alpha s(x)d^Dx=0 \quad\text{for every multi-index }|\alpha|<m.$$

Subtract the degree-$m-1$ Taylor polynomial of $e^{-ik\cdot x}$ under the integral. Its real-variable integral remainder is bounded by $|k\cdot x|^m/m!$, which proves

$$\boxed{ \|\widetilde s(k)\| \leq\frac{|k|^m}{m!} \int |x|^m\|s(x)\|d^Dx.} \tag{7}$$

This gives a calculable low-momentum suppression from vanishing source moments. The moments must vanish for the source produced by the actual field equation. Squaring an odd field can remove its odd parity, so an odd bare shape alone does not establish the moment conditions.

The same construction is available at many depths. A distinctive particle prediction requires the field dynamics to select the relevant shape, scale, and source symmetry, and requires the coupling to allowed response tangents to be checked as well. A zero source does not by itself exclude a nonzero susceptibility.

Verification

polarization_coupling_checks.py checks the displayed identities with independent spatial quadrature, energy variations, or full finite-system evolution as appropriate. These checks verify the declared mathematics and benchmarks, not empirical couplings.

Related

• Bilateral polarization coupling — retains both tensor states and distinguishes source values from mixed response derivatives.
• Descending photon source — defines which higher-field source is projected.
• Polarized response spectrum — combines spatial matrix elements with frequency and exposure.
• Gaussian contact profiles — contrasts density contact with the complex amplitude overlap derived here.