Spatial and moving-cavity response

Summary: Exact oscillator matrix elements connect cumulative state counts to polarized spatial response and energy-transfer distributions, while moving scalar and Dirac modes show why a frozen cavity overlap is only one part of coupling.

A spatial operator with a specified meaning

Let $P_{N,d}$ project onto the normalized isotropic oscillator modes of total degree at most $N$ in $d$ ordinary spatial coordinates. Their common oscillator length is $L$ and their count is $D_N=\binom{N+d}{d}$. The operator

$$A(k)=e^{ik\cdot x},\qquad \eta=\frac{|k|^2L^2}{2}, \tag{1}$$

is a Fourier component of a declared scalar density probe. It is unitary on the complete spatial Hilbert space. Its compressed matrix can lose weight to modes outside a selected band.

These are conditional spatial controls. Positive density is not automatically electric charge or a Dirac scalar. A band trace represents specified equal weights; a coherent field instead requires its actual complex coefficients. The calculation neither changes the particle depth assignments nor adds compactified coordinates.

Hockey-Stick gives an exact form factor

For one normalized oscillator mode,

$$\langle a|e^{ikx}|a\rangle =e^{-\eta/2}L_a(\eta), \tag{2}$$

where $L_a$ is a Laguerre polynomial. Align the first coordinate with $k$. The other $d-1$ coordinates contribute their oscillator multiplicities. The generating function for the shell traces is

$$e^{-\eta/2}(1-t)^{-d} \exp\left(-\frac{\eta t}{1-t}\right). \tag{3}$$

Multiplication by $(1-t)^{-1}$ performs the cumulative Hockey-Stick sum. Comparison with the generalized Laguerre generating function gives

$$\begin{aligned} F_{N,d}(k)&=\frac{\operatorname{Tr}(P_{N,d}A(k))}{D_N}\\ &=e^{-\eta/2}\frac{L_N^{(d)}(\eta)}{D_N}. \end{aligned} \tag{4}$$

In particular $F_{N,d}(0)=1$. Its small-$k$ expansion gives

$$\langle |x|^2\rangle_N =dL^2\left(\frac{N}{d+1}+\frac12\right). \tag{5}$$

This explicitly separates a common well length from the spatial extent of different modes. Equation (4) is a normalized trace amplitude. Squaring it is not generally the probability of remaining in a band: amplitudes may cancel in a trace, while transitions to different final modes are distinct.

Complete transition weights

For $\ell=\min(a,b)$ and $h=\max(a,b)$, the one-coordinate displacement matrix $A_{ba}=\langle b|A(k)|a\rangle$, with $k$ chosen positive along that coordinate, is

$$\begin{aligned} A_{ba}&=e^{-\eta/2}\sqrt{\frac{\ell!}{h!}}\\ &\quad\times(i\sqrt\eta)^{h-\ell}L_\ell^{(h-\ell)}(\eta). \end{aligned} \tag{6}$$

Let $r$ be the total degree in the spectator coordinates. Their shell multiplicity is $g_{d-1}(r)=\binom{r+d-2}{d-2}$ for $d\ge2$. For $d=1$, only $r=0$ contributes, with weight one. The probability weight captured in a final cumulative band $M$ is

$$\begin{aligned} I_{N\to M}(\eta) ={}&\frac1{D_N}\sum_{r=0}^{\min(N,M)}g_{d-1}(r)\\ &\times\sum_{a=0}^{N-r}\sum_{b=0}^{M-r} |A_{ba}|^2. \end{aligned} \tag{7}$$

This retains every allowed final mode. It obeys

$$\begin{aligned} D_NI_{N\to M}&=D_MI_{M\to N},\\ \lim_{M\to\infty}I_{N\to M}&=1. \end{aligned} \tag{8}$$

The first identity follows from the symmetric squared entries in (6); the second follows from completeness and unitarity. It is a reciprocity identity for this operator, not a complete two-particle scattering law. Occupation blocking, an energy budget or a specified time-dependent probe would impose additional physical conditions.

Resolve the response by energy as well as by band

Take oscillator energies $\hbar\omega(k+d/2)$. A transition from degree $a$ to $b$ in the active coordinate changes the energy by $\hbar\omega j$, where $j=b-a$ can be positive, zero or negative. Resolve the sum (7) by this signed difference and let $w_N(j)$ denote the resulting complete weight, with the final cutoff removed.

Put $u(t)=4\eta\sinh^2(t/2)$. The exact one-coordinate generating function is

$$\begin{aligned} G_a(t)&=\sum_b|A_{ba}|^2e^{t(b-a)}\\ &=e^{\eta(e^t-1)}L_a(-u(t)). \end{aligned} \tag{9}$$

The same spectator and cumulative sums give the full moment-generating function

$$\begin{aligned} \mathcal G_N(t)&=\sum_jw_N(j)e^{tj}\\ &=e^{\eta(e^t-1)} \frac{L_N^{(d)}(-u(t))}{D_N}. \end{aligned} \tag{10}$$

At $t=0$, this is one. Its first two cumulants are

$$\begin{aligned} \langle j\rangle&=\eta,\\ \operatorname{Var}(j)&=\eta\left(1+\frac{2N}{d+1}\right). \end{aligned} \tag{11}$$

There is an independent spatial check. Conjugating the oscillator Hamiltonian by $A(k)$ adds $\hbar k\cdot p/m_{\rm kin}$ and $\hbar^2|k|^2/(2m_{\rm kin})$. The band has zero mean momentum, and its directional momentum variance gives the second line of (11). Thus the generating function and the actual kinetic operator give the same two sum rules.

The inclusive weight remains one, but it need not lie in the frequencies available to a particular encounter. For example, every $t<0$ gives the rigorous bound

$$\Pr(j\le J)\le e^{-tJ}\mathcal G_N(t). \tag{12}$$

This follows by applying the elementary exponential Markov inequality to $e^{tj}$. The program evaluates the logarithm before optimizing $t$, so an exponentially small bound is not confused with floating-point zero. Neither (11) nor (12) determines an IDWT encounter's available energy; that requires its actual field dynamics and conservation equations.

Complex polarization and a nested spatial probe

An isotropic trace removes orientation information. A normalized complex polarization $v\in\mathbb C^d$ instead defines the degree-$N$ mode

$$|N,v\rangle=\frac{(v\cdot a^\dagger)^N}{\sqrt{N!}}|0\rangle. \tag{13}$$

Its actual spatial density form factor is

$$F_{N,v}(k)=e^{-\eta/2} L_N\!\left(\frac{L^2|v^\dagger k|^2}{2}\right). \tag{14}$$

Relative complex phases enter $|v^\dagger k|^2$. Circular and linear polarizations give different directional responses. The diagnostic compares (14) with complete complex mode vectors and separately integrates a general complex superposition in physical coordinates.

Let $E$ be the real isometric inclusion of a lower spatial sector into a containing sector. For a declared product of two density profiles, a shared wavevector $k$ samples the higher profile at $Ek$. Its Fourier pairing contains both factors

$$F_{N_a,v_a}(k)\,F_{N_b,v_b}(-Ek). \tag{15}$$

A common physical rotation of $v_b$ and $E$ leaves the result unchanged. Both polarizations are retained. A rotation of only one object is a different preparation. The normal marginal must be formed from the bilinear current before constructing this pairing. Equation (15) remains a specified profile overlap until both fields and their reciprocal motion are derived from an interaction action.

A moving final cavity has a different spectral projector

The moving-cavity calculation gives an exact identity for quadratic kinetics:

$$\begin{aligned} H_v&=H_0-v\cdot p,\\ U_v&=e^{im_{\rm kin}v\cdot x/\hbar},\\ P_v&=U_vP_0U_v^\dagger. \end{aligned} \tag{16}$$

For $k=m_{\rm kin}(v_f-v_i)/\hbar$, the spatial phase $A(k)$ maps the initial moving band isometrically onto the corresponding final moving band. In consistently related bases,

$$\langle U_{v_f}u_b|A(k)|U_{v_i}u_a\rangle =\delta_{ab}. \tag{17}$$

Projection back into an unchanged cavity is a different operation. In the fifteen-state quasiperiodically perturbed oscillator used for the inertia control, a boost of $1.4$ in its nonrelativistic units leaves fixed-band weights $0.65253335$ and $0.65394785$ for the two well phases. The corresponding moving-band weights are one to within $3\times10^{-14}$.

The prescribed well and its velocity are part of this calculation. The identity does not derive a physical collision that accelerates the well, or remove the need to include its source and energy. It shows why a frozen profile overlap alone cannot decide whether a moving particle pattern can persist after receiving momentum.

Full first-order Dirac current overlaps

For the exact Dirac control

$$H=\sigma_xp+b\sigma_yx+m_0\sigma_z, \tag{18}$$

use the complete normalized moving spinors derived in the inertia note. With $\hbar=c=1$, the current matrix element is

$$\begin{aligned} J_\Gamma(k,v)={}&\int dx\,\phi_{n,v}^\dagger(x)\\ &\qquad\times\Gamma\phi_{n,0}(x)e^{ikx}. \end{aligned} \tag{19}$$

The vertices $I,\sigma_x,\sigma_y,\sigma_z$ give the density, spatial, pseudoscalar and scalar channels of this two-component comparison. Both spinor components, the boost phase and the changed width remain inside the integral.

At $b=0.8$, $m_0=0$, $n=1$, $v=0.8$, choose the phase-matched wavevector $k=E'\gamma v=1.30639453$. This matches the spinor phase; it is not automatically the total conserved momentum transfer of a collision. The calculated density amplitudes give

For the moving mode, the spatial-current amplitude is $0.43306907$, the scalar amplitude is $0.01397361$, and the pseudoscalar amplitude is $0.00698680i$. They are distinct channels. Independent coordinate quadrature at orders 80 and 160 verifies the complete complex integrals. This comparison contains an external well; it is not a closed reciprocal IDWT encounter or a prediction of a physical interaction rate.

The large inverse-scale ratio is a conditional suppression test

The existing mass-scale benchmarks suggest $L_5/L_6\simeq3.6\times10^7$ if the geometric conversion factors in the common-size relation are equal. For the illustrative probe $|k|L_6=10^{-3}$, (4) gives $F_e=0.9999988929$ and $F_\mu=0.9999973214$ at the unchanged degree assignments. The same probe has $\eta=6.48\times10^8$ in the depth-five oscillator comparison.

At depth-five degrees $9,14,21$, the logarithms of the absolute fixed-band trace amplitudes are approximately $-3.24\times10^8$. The finite-band capture weights are also exponentially small. Equation (12) gives a similarly small bound if the allowed increase in oscillator energy is only ten quanta. All these statements retain the prescribed stationary well and specified energy window.

This establishes strong suppression for those operator channels. It does not yet explain rare neutrino interactions: the oscillator frequency in the energy window, the physical current, allowed final states, moving confinement and both objects' response must be supplied by the same-field action. In particular, neither a trace amplitude nor a positive density marginal establishes electric neutrality. The broader spatial contact calculation also keeps the physical measure and its units explicit.

Reproduction

cavity_form_factors.py implements the exact matrix elements and generating functions. cavity_form_factor_checks.py checks direct finite matrices, independent complex coordinate integration, reciprocal band weights, energy moment sum rules, the complete transfer generating function, rotated polarized profiles, moving bands and Dirac currents. The numerical controls retain the fixed particle indices; their success establishes the stated mathematics under the stated operators.