Periodic spatial source response

Summary: The full response of a two-frequency spatial Dirac field reduces to a frequency-indexed moment equation that retains scalar-center motion, normal propagation and the conditions for source-sideband cancellation.

Follow the complete periodic background

The two-frequency spatial field has an unchanging scalar binding source and a changing number-current pattern. Its linearized field equation nevertheless has periodic coefficients. The response can be derived without assigning its bound frequency to an external forcing frequency or discarding one of its two components.

All coordinates describe ordinary macroscopic spatial directions of the one nested manifold. The bound coordinates are $x\in\mathbb R^d$ and $y$ is one existing containing direction. The field-component indices, source moments and temporal harmonics below are mathematical descriptions of that same field; they add no physical spatial dimensions or particles. The present normal momentum is along the direction used to construct the opposite-frequency partner.

Let $H_b=-i\alpha_i\partial_i+\beta(m_0+a_{\rm source}|x|^2)$ be the self-consistent bound operator, with $\{H_b,\alpha_y\}=0$. Choose normalized radial fields with possibly different rest polarizations,

$$H_b\phi_1=M\phi_1,\qquad H_b\phi_2=M\phi_2,\qquad \Psi_0(t)=a\phi_1e^{-iMt}+b\alpha_y\phi_2e^{iMt}. \tag{1}$$

Their complete radial frame has the pointwise scalar identities $\phi_j^\dagger\beta\phi_j=\sigma_0$ and $\phi_1^\dagger\beta\alpha_y\phi_2=0$. Thus the binding density is $s\sigma_0$, where $s=|a|^2-|b|^2>0$. Both profiles must be solved at that scalar weight and the same interaction coefficient. Set $|a|^2+|b|^2=1$ for the unit canonical norm used in the examples.

Retain shape and scalar-center variation

Use Hermitian spatial source operators

$$O_A=\beta(x^TAx),\qquad O_i=\beta x_i, \tag{2}$$

where $A$ runs over a Frobenius-orthonormal basis of all symmetric spatial tensors, including breathing. Define $C_\sigma$ as the column operator whose columns are $O_A\psi_\sigma$, with $\psi_+=a\phi_1$ and $\psi_-=b\alpha_y\phi_2$. Then

$$C(t)=\sum_{\sigma=\pm1}C_\sigma e^{-i\sigma Mt}. \tag{3}$$

The real symmetric moment curvature is constant on this background. Its shape block is $g_0I$, and its dipole block is $-2a_{\rm source}I/Q_\sigma$, with $Q_\sigma=\int\Psi_0^\dagger\beta\Psi_0\,dx\ne0$. Translational response derives that second block by varying the scalar center. An unchanged background center does not justify omitting its perturbations.

For the complexified ordinary and conjugate perturbation $Z=(\eta,\eta^*)^T$, the full equation at normal momentum $q$ is

$$\boxed{ i\partial_tZ=\left[\mathcal B_0(q)+S(t)\kappa T(t)\right]Z +S(t)f(t),} \tag{4}$$

with

$$\begin{gathered} H_q=H_b+q\alpha_y,\qquad \mathcal B_0(q)=\begin{pmatrix}H_q&0\\0&-H_{-q}^*\end{pmatrix},\\ S(t)=\begin{pmatrix}C(t)\\-C(t)^*\end{pmatrix},\qquad T(t)=(C(t)^\dagger,C(t)^T),\qquad h=T(t)Z. \end{gathered} \tag{5}$$

The star means entrywise conjugation; the conjugate block retains the opposite normal momentum. These are laboratory field frequencies, with no stationary carrier subtracted from $H_q$. The additional source coefficients are $f$, while $h$ contains the induced moments. A physical real perturbation includes the conjugate Fourier component at $(-\omega,-q)$.

Equation (4) follows by the complete real-linear differentiation in Dirac source linearization. Its energy Hessian obeys $\Sigma\mathcal B=(\Sigma\mathcal B)^\dagger$, where $\Sigma=\operatorname{diag}(I,-I)$. This action identity does not make the energy positive. The field coefficients contain frequencies $0,\pm2M$ and have period $\pi/M$.

A monochromatic source produces three bare moment frequencies

First remove the feedback term $\kappa h$ and impose $f e^{i(qy-\omega t)}$. Multiplication by (3) drives ordinary fields at $\omega+\sigma M$ and conjugate fields at $\omega-\sigma M$. Reading those fields through $T(t)$ gives moment outputs at $\omega+\Delta M$, for $\Delta=0,\pm2$:

$$\boxed{ \begin{aligned} \Pi_\Delta(\omega,q) ={}&\sum_{\sigma-\rho=\Delta} C_\rho^\dagger(\omega+\sigma M-H_q)^{-1}C_\sigma\\ &-\sum_{\rho-\sigma=\Delta} C_\rho^T(\omega-\sigma M+H_{-q}^*)^{-1}C_\sigma^*. \end{aligned}} \tag{6}$$

The first sum has ordinary propagation; the second has the conjugate response with its required sign. For example:

Normal propagation can be evaluated with the full squared bound operator:

$$\begin{aligned} (\zeta-H_q)^{-1} &=(\zeta+H_b+q\alpha_y)(\zeta^2-H_b^2-q^2)^{-1},\\ (\zeta+H_{-q}^*)^{-1} &=(\zeta-H_b^*+q\alpha_y^*) (\zeta^2-(H_b^*)^2-q^2)^{-1}. \end{aligned} \tag{7}$$

The domain and pole qualifications are those of the full normal response. No radial or angular field components have been eliminated by simply neglecting them.

Repeated feedback couples the frequency sidebands

Let $h_j$ and $f_j$ denote moment amplitudes at $\omega+2jM$. Define an operator on their frequency-indexed sequence by

$$\mathbb P_{jk}(\omega,q)= \begin{cases} \Pi_{2(j-k)}(\omega+2kM,q),&|j-k|\le1,\\ 0,&|j-k|>1. \end{cases} \tag{8}$$

The complete source equation is

$$\boxed{h=\mathbb P(f+\mathbb K h),\qquad h=(I-\mathbb P\mathbb K)^{-1}\mathbb P f,\qquad \mathbb K_{jk}=\delta_{jk}\kappa.} \tag{9}$$

Its repeated feedback can populate more frequencies than the three in (6). A finite harmonic cutoff is an approximation to this sequence; convergence of a physical response or pole must be checked as that cutoff increases. At singular frequencies use the complete field operator or the appropriate limiting resolvent.

There is also an exact finite-cutoff identity useful for verification. Keep field frequencies $\omega+(2n+1)M$, $-N\le n<N$, and moment frequencies $\omega+2jM$, $-N\le j\le N$. Let $D$ be the block diagonal frozen field resolvent denominator, and construct $S_F,T_F$ by matching the harmonics in (3)–(5). Then

$$\boxed{\mathbb P_N=T_FD^{-1}S_F,\qquad \det(D-S_F\mathbb K_NT_F) =\det D\,\det(I-\mathbb P_N\mathbb K_N).} \tag{10}$$

The frozen determinant factor retains source-free field modes and pole cancellations. A truncated moment equation and a differently truncated field equation need not agree at their frequency boundaries. Matching both cutoffs gives (10) exactly.

A periodic field can have a simpler scalar readout at rest

Periodic field coefficients do not guarantee a nonzero sideband in every observable. At $q=0$, let $P_E$ be the complete spectral projector of $H_b$ at eigenvalue $E$ in a finite spatial regulator. Since $\alpha_yP_E\alpha_y=P_{-E}$, the positive sideband in (6) vanishes identically away from the poles if and only if every residue satisfies

$$\boxed{C_-^\dagger P_EC_+ =\left(C_-^\dagger P_{-E}C_+\right)^T.} \tag{11}$$

The negative sideband has the corresponding conjugate relation. This criterion follows by expanding both resolvents in spectral projectors. It checks an identity at every pole rather than inferring cancellation from a few sampled frequencies. A continuum statement would require the corresponding equality of spectral measures and valid operator domains.

For the centered Cartesian radial examples at depths three, five and six, all complete projectors satisfy (11) for the scalar second moments and dipoles. The direct source response has no $\omega\pm2M$ scalar output at rest, even though its field perturbation contains both carriers and its full field operator is periodic. This is a verified property of the declared regulators; it is not an assumption about an arbitrary source, an arbitrary nonlocal probe or a physical photon current.

For identical rest polarizations, write $C$ for the unit-profile source columns and $\chi_1$ for their bare single-frequency response. Since $C_+=aC$ and $C_-=-b\alpha_yC$, the diagonal harmonic obeys the exact normal-partner identity

$$\boxed{\Pi_0(\omega,q) =|a|^2\chi_1(\omega,q)-|b|^2\chi_1(\omega,q)^T.} \tag{12}$$

This identity follows directly from (7), including the opposite normal momentum. Different occupied rest polarizations require their respective single-frequency responses. Equation (12) concerns the diagonal harmonic; at nonzero $q$ it does not remove the other blocks of (9).

In particular, if a shape channel has $\chi_1=A(\omega)I+B(\omega)T$, with $T^T=-T$, then

$$\boxed{\Pi_0=sA(\omega)I+nB(\omega)T, \qquad n=|a|^2+|b|^2.} \tag{13}$$

The signed scalar balance and canonical norm multiply different parts of the response. Replacing both by a single small scalar source would lose its polarization-dependent dynamics.

Check growth in the full periodic field

When (11) removes the rest scalar sidebands, there is an autonomous source realization with four full spatial blocks:

$$\begin{gathered} L_4=\operatorname{diag}(H_b-M,H_b+M,-H_b^*+M,-H_b^*-M),\\ T_4=(C_+^\dagger,C_-^\dagger,C_+^T,C_-^T),\qquad S_4=\Sigma_4T_4^\dagger,\qquad \Sigma_4=\operatorname{diag}(I,I,-I,-I),\\ \boxed{\mathcal A_4=L_4+S_4\kappa T_4.} \end{gathered} \tag{14}$$

Its resolvent gives exactly $\Pi_0$. For a mode $(u_+,u_-,v_+,v_-)$ of frequency $\omega$, reconstruct the physical ordinary and conjugate fields as

$$\boxed{Z(t)=e^{-i\omega t} \begin{pmatrix} e^{-iMt}u_++e^{iMt}u_-\\ e^{iMt}v_++e^{-iMt}v_- \end{pmatrix}.} \tag{15}$$

For a nonreal source-carrying pole, (11) makes its unwanted scalar sidebands vanish, so (15) solves (4) at $q=0$. Source-free modes of $L_4$ have real frequencies and cannot create exponential growth. Conversely, the closed rest moment equation separates the harmonics of a growing Floquet solution into poles of this same source resolvent. This is a finite-dimensional rest-response reduction under (11), not a claim of a global symmetry for arbitrary fields or propagating modes.

The period is $P=\pi/M$. The vector multiplying $e^{-i\omega t}$ in (15) changes sign after $P$, so its full evolution multiplier is $-e^{-i\omega P}$. Independent integration of (4) over a full period checks this multiplier and the growth rate $\operatorname{Im}\omega$. The evolution preserves the indefinite canonical form even when a mode grows; conservation of that form is not positive-energy stability.

At unit number norm, scalar balance $s=0.3$, one common $g_0=6.5$ and zero bare mass, the total-degree-one Cartesian regulators give:

These values verify the reduction in the declared spatial regulators. They are not converged particle frequencies. In these examples the periodic field is unstable, so a small residue of one slow response pole cannot be interpreted as a stable weakly coupled particle. Slow shape poles and their signed response derives the slow frequency and residue together and verifies their canonical sign.

Propagation and the second object's response remain essential

In the same spatial calculations, normal momentum makes the scalar shape sidebands nonzero. Their first small-$q$ term is linear in $q$; the positive and negative normal momenta give opposite sideband amplitudes for the tested preparation. Thus the rest cancellation does not authorize using a stationary scalar susceptibility for the whole propagating periodic field.

A reciprocal nested interaction uses the curvature of the common energy, including its off-diagonal shape restriction and scalar-center terms. The readjusted nested pair supplies the stationary source requirement. If two periodic backgrounds have different bound frequencies, their common response generally needs both frequency indices; a single period cannot be assumed for an irrational frequency ratio. Both objects' spatial polarizations and their feedback must be retained.

The source-frequency analysis explains why an actual photon output map is still necessary. A scalar moment cancellation is a statement about that readout, while a photon could sample a different spatial current with different surviving sidebands. Spatial current response derives those output columns from the same field and verifies the normal continuity identity before and after source feedback.

Verification

periodic_dirac_response_checks.py constructs complete centered Cartesian field sources with one common interaction coefficient. Complex finite variations verify (4), including all center, breathing and shape terms. An independent temporal Fourier transform reconstructs the full periodic field operator.

Complete spectral projectors check (11). Dense normal Dirac resolvents independently check (6)–(7). Explicit time-dependent forced fields retain both spatial polarizations; their temporal Fourier transforms recover the predicted moment sidebands. Matched full-field and moment harmonic cutoffs check (9)–(10), retaining ordinary and conjugate fluctuations. Full-period evolution independently verifies the reconstructed growing modes, their Floquet multipliers and preservation of the canonical form. Reports use timestamped filenames and checks remain active under Python optimization. These calculations establish the stated algebra and regulated response; they do not identify physical particle masses or interaction rates.

Related

• Two-frequency spatial Dirac currents — supplies the exact nonlinear background and its changing current.
• Source frequencies in spatial coupling — distinguishes the carrier from the spatial clock sampled in an interaction.
• Normal propagation of spatial Dirac modes — supplies full normal resolvents and the centered source conventions.
• Translational Dirac response — derives the scalar-center curvature retained throughout this calculation.