Resonant scalar wave quartets

Summary: Four positive-frequency spatial waves permit reciprocal scalar norm transfer; the exact coherent contraction and its sharp bound depend on complete polarization vectors, with a right-angle cancellation restricted to one polarization per chiral band.

Positive incoming frequencies do not forbid all reciprocal transfer

The local scalar equation is

$$i\partial_t\Psi=-i\alpha\cdot\nabla\Psi+g(\Psi^\dagger\beta\Psi)\beta\Psi. \tag{1}$$

For two opposite complete kinetic chiralities, write $\Psi=(\Psi_u,\Psi_v)^T$ and $\zeta=\Psi_u^\dagger\Psi_v$ in the block convention $\beta=\left(\begin{smallmatrix}0&I\\I&0\end{smallmatrix}\right)$. The full scalar vertex gives

$$\sigma=2\operatorname{Re}\zeta, \qquad \dot Q_u=2g\operatorname{Im}\int\zeta^2dx, \qquad \dot Q_v=-\dot Q_u. \tag{2}$$

Strictly positive free frequency removes a particular conjugate contribution for the one-axis traveling pulse class. That argument uses its momentum support. It does not establish a universal prohibition for several incoming directions.

Here all four incoming waves have the same positive free frequency $k_0>0$. Their momenta are $p,-p$ in one chiral block and $q,-q$ in the other, with

$$|p|=|q|=k_0,\qquad p=k_0 e_z,\qquad q=k_0(\sin\theta\,e_x+\cos\theta\,e_z), \quad 0<\theta<\pi. \tag{3}$$

Thus the opposite pairs have equal total momentum and frequency. The physical coordinates are ordinary spatial directions of the same containing space. These plane waves are unbound controls; their amplitudes below are densities per spatial volume. A finite box can regulate a commensurate example without assigning physical compactification. The amplitudes are classical complex field coefficients. This calculation is not an antisymmetrized quantum-fermion scattering matrix or an observed particle reaction rate.

Retain every polarization in each occupied chiral band

Let the total spatial dimension be odd, $D\ge3$, and use $\alpha_j=\operatorname{diag}(\Gamma_j,-\Gamma_j)$. Choose a complete basis $W_+$ of the $+1$ eigenspace of $\Gamma_z$, and set $W_-=\Gamma_xW_+$. Each has

$$P=2^{(D-3)/2} \tag{4}$$

columns. Both are actual spinor frames, not additional spatial coordinates. Put $s=\sin(\theta/2)$ and $c=\cos(\theta/2)$. The two fields are

$$\begin{aligned} \Psi_u&=e^{-ik_0t} \left(W_+A_+e^{ip\cdot x}+W_-A_-e^{-ip\cdot x}\right),\\ \Psi_v&=e^{-ik_0t} \left[(-sW_++cW_-)B_+e^{iq\cdot x} +(cW_++sW_-)B_-e^{-iq\cdot x}\right], \end{aligned} \tag{5}$$

where all four $A_\pm,B_\pm\in\mathbb C^P$ are unrestricted complex polarization vectors. The lower block's minus sign in its kinetic operator is essential: the displayed frames at $q$ and $-q$ are its positive-frequency states. No negative-frequency incident wave has been inserted.

Define the complete overlap matrix

$$G_{st}=A_s^\dagger B_t,\qquad s,t\in\{+,-\}. \tag{6}$$

The cross scalar retains four spatial phases, with coefficients $-sG_{++},cG_{+-},cG_{-+},sG_{--}$ at momenta $q-p,-q-p,q+p,-q+p$, respectively. Under a common physical rotation, both momenta and all frames rotate together and leave this contraction unchanged.

The exact spatial quartic mean

Let brackets denote the spatial mean over these noncollinear plane phases, or over a compatible Fourier regulator. Put $a=s^2$, $b=c^2$. Orthogonality of distinct spatial Fourier phases gives

$$\begin{aligned} \langle|\zeta|^2\rangle &=a(|G_{++}|^2+|G_{--}|^2) +b(|G_{+-}|^2+|G_{-+}|^2),\\ \boxed{\langle\zeta^2\rangle=2\mathcal T,\qquad \mathcal T=-aG_{++}G_{--}+bG_{+-}G_{-+}.} \end{aligned} \tag{7}$$

The second line survives because opposite incoming pairs have the same total momentum and frequency. It is a coherent product, not a sum of squared channel counts. The full interaction energy density is

$$\boxed{\mathcal H= \frac g2\langle\sigma^2\rangle =g\left[a(|G_{++}|^2+|G_{--}|^2) +b(|G_{+-}|^2+|G_{-+}|^2)+2\operatorname{Re}\mathcal T\right].} \tag{8}$$

For $g\ge0$ this is nonnegative, as is clear from its original local square. The scalar mean itself vanishes at distinct $p,q$, but the local interaction and its quartic mean need not vanish. Replacing the current by its spatial mean before varying (1) would miss them.

Both complete amplitude systems follow from one energy

Use the canonical mean kinetic term and subtract the common carrier $k_0$. Define real matrices

$$w=\begin{pmatrix}a&b\\b&a\end{pmatrix}, \qquad \kappa=\begin{pmatrix}-a&b\\b&-a\end{pmatrix}. \tag{9}$$

The reciprocal gradients of (8) are

$$\boxed{ \begin{aligned} i\dot A_s&=g\sum_t\left[ w_{st}B_tG_{st}^*+\kappa_{st}B_tG_{-s,-t}\right],\\ i\dot B_t&=g\sum_s\left[ w_{st}A_sG_{st}+\kappa_{st}A_sG_{-s,-t}^*\right]. \end{aligned}} \tag{10}$$

In particular, with $Q_u=\|A_+\|^2+\|A_-\|^2$ and similarly $Q_v$,

$$\boxed{\dot Q_u=4g\operatorname{Im}\mathcal T, \qquad \dot Q_v=-4g\operatorname{Im}\mathcal T.} \tag{11}$$

These are norm densities for the plane-wave preparation. The imbalance $\|A_+\|^2-\|A_-\|^2$ is conserved, as is its $B$ counterpart; the exchanged pairs carry zero net spatial momentum. The canonical total norm and the same energy (8) are conserved. Full complex polarization vectors continue evolving in (10).

The exact local equation also generates spatial momenta $\pm p\pm2q$ and $\pm q\pm2p$. At the initial common source frequency $k_0$, they are off the positive free-frequency shell for noncollinear $p,q$. Thus (10) is the first-order resonant projection, or the corresponding weak-interaction amplitude system after excluding those detuned modes. It is not an exact four-mode invariant subspace of (1). The smallest positive detuning is $\Delta=k_0[\sqrt{5-4|\cos\theta|}-1]$; the weak-mode separation requires $|g|(Q_u+Q_v)\ll\Delta$. As $\theta$ approaches a collinear endpoint, the detuning can vanish; the distinct-mode reduction then needs reconsideration.

A one-polarization cancellation and its precise limit

For $P=1$ the two products in $\mathcal T$ contain the same four scalar amplitudes. Therefore

$$\mathcal T=\cos\theta\,A_+^*A_-^*B_+B_-. \tag{12}$$

The transfer contribution vanishes at $\theta=\pi/2$ for all phases. The ordinary terms in (8) remain present. This is a zero in the pair norm-transfer channel, not zero interaction.

For $P\ge2$, the products in (7) are independent. At right angles,

$$\boxed{\langle\zeta^2\rangle=-\det G.} \tag{13}$$

For example, at $Q_u=Q_v=1$, take

$$A_+=\frac{e_1}{\sqrt2},\quad A_-=\frac{e_2}{\sqrt2},\quad B_+=e^{i\phi}\frac{e_1}{\sqrt2},\quad B_-=e^{i\phi}\frac{e_2}{\sqrt2}. \tag{14}$$

Equations (11) and (13) give

$$\boxed{\dot Q_u=-\frac g2\sin(2\phi).} \tag{15}$$

Changing only $\phi$ from $\pi/4$ to $-\pi/4$ reverses the transfer at the same positive coupling. The complete massless chiral band at $D=5$ already permits (14), as do higher odd dimensions. The cancellation at $D=3$ here uses its single positive-frequency polarization within each selected kinetic chirality; it is not a statement that physical three-dimensional particles have only one polarization.

A sharp normalized bound

For arbitrary complex vectors in the stated preparation,

$$\boxed{ |\dot Q_u|\le |g|Q_uQ_v \begin{cases} |\cos\theta|,&P=1,\\[2pt] \dfrac{1+|\cos\theta|}{2},&P\ge2. \end{cases}} \tag{16}$$

Both bounds are attained. For $P=1$, use $|A_+A_-|\le Q_u/2$ and $|B_+B_-|\le Q_v/2$ in (12), then choose the product's phase to be imaginary.

For $P\ge2$, factor out the four vector norms. For unit vectors $u,v$ in the $B$ slots, the remaining bilinear contraction is governed by

$$M=b\,vu^T-a\,uv^T. \tag{17}$$

Its largest singular value is at most $\max(a,b)$. To see this, a common unitary change of basis and a phase put $u=e_1$, $v=re_1+\sqrt{1-r^2}e_2$, $0\le r\le1$. The nonzero block is

$$M=\begin{pmatrix}(b-a)r&-a\sqrt{1-r^2}\\ b\sqrt{1-r^2}&0\end{pmatrix}. \tag{18}$$

Assume $b\ge a$; the other case exchanges them. For $M^\dagger M$, its trace is $a^2+b^2-2ab r^2\le2b^2$, and

$$\det(b^2I-M^\dagger M) =b^2a r^2[2(b-a)+a r^2]\ge0. \tag{19}$$

Both eigenvalues are therefore at most $b^2$. It follows that $|\mathcal T|\le\max(a,b)Q_uQ_v/4$, giving (16). Orthogonal $B$ polarizations and appropriately matched or swapped $A$ polarizations attain the largest singular value; balanced norms and the appropriate overall phase attain the transfer bound.

The maximum stops increasing once two polarization directions are available. Additional component counts do not multiply (16). Which bound a physical spatial pattern approaches depends on its occupied vectors, phases and geometry.

Verification and physical scope

scalar_quartet_checks.py runs 2,158 checks. It constructs the full Clifford waves at $D=3,5,7,9$, builds every Fourier coefficient of the local scalar and its nonlinear source, and compares the resulting energy and both resonant gradients with (7)–(10). It retains the other generated spatial modes and checks their detuning. Independent energy variations, complete vector evolution, norm transfer and preparations attaining both bounds check scalar_wave_quartets.py.

This result demonstrates a permitted phase-sensitive channel of the specified scalar term. It does not assign electric charge, select bound particle geometries or derive observed weak rates. In particular, the neutrino control cannot be satisfied by asserting that five-dimensional polarization necessarily suppresses all interaction: the same full space permits the nonzero example (15).

Localized scalar wave encounters evolve spatially finite preparations with the complete nonlinear equation, including the momenta omitted by the resonant projection. They retain phase-sensitive reciprocal transfer and an exact continuum reflection law.