Collective translation waves

Summary: Full source feedback gives the normal propagation of spatial translation modes, with exact inertia and stiffness identities and a subluminal two-polarization branch at bound depth two.

Translate a complete bound field

A field bound in two spatial directions has two independent displacement directions. If those displacements vary along a perpendicular propagation coordinate, they provide a concrete candidate for a two-polarization wave. The propagation speed must follow from the complete field equation.

Use the scalar Dirac moment branch, with actual bound coordinates $x\in\mathbb R^d$ and an existing perpendicular coordinate $y$ in a containing sector. All coordinates are ordinary macroscopic directions in the one nested physical manifold. Set the underlying Dirac wave speed to one.

The normalized resting profile satisfies

$$H_b\phi=M\phi,\qquad H_b=-i\alpha_i\partial_i+\beta(m+a|x|^2),\qquad \int\phi^\dagger\phi\,dx=1. \tag{1}$$

Here $M$ is the profile's stationary frequency and $m$ its bare scalar coefficient. The full centered moment energy determines $a$:

$$Q_\sigma=\int\phi^\dagger\beta\phi\,dx,\qquad T_\sigma=\int |x|^2\phi^\dagger\beta\phi\,dx,\qquad a=\frac{g_0T_\sigma}{d}. \tag{2}$$

$Q_\sigma$ denotes scalar normalization, not electric charge. The source is local in $(t,y)$ and uses the centered scalar moments in the bound coordinates, as specified in Normal propagation of spatial Dirac modes.

A small displacement has spatial tangent

$$\eta_i=-\partial_i\phi,\qquad \Psi=e^{-iMt}\big[\phi+X_i(t,y)\eta_i+\cdots\big]. \tag{3}$$

The omitted terms include the velocity and normal-gradient response of the full spatial profile. Retaining only the displaced density would miss them, just as a general polarization texture can omit a nonzero Dirac complement.

Identify when the dipole source closes

Let $Y=\alpha_y$ anticommute with every bound $\alpha_i$ and with $\beta$. For even $d$, define the Hermitian involutions

$$\Gamma_b=i^{d/2}\alpha_1\cdots\alpha_d,\qquad \Xi=\beta\Gamma_b. \tag{4}$$

They obey

$$[\Xi,H_b]=0,\qquad [\Xi,\beta f(x)]=0,\qquad \{\Xi,Y\}=0. \tag{5}$$

Choose one complete spatial chirality, $\Xi\phi=\phi$. Every scalar source column $\beta f(x)\phi$ then has that same chirality, while its normal partner has the opposite chirality. This is a decomposition of spatial field components, with no additional physical dimensions.

Consequently the term linear in $qY$ between scalar source columns vanishes in the exact normal resolvent. The remaining terms are even in the normal wavenumber $q$. The parity $\beta\mathcal R$, where $\mathcal R\phi(x)=\phi(-x)$, further separates odd dipole sources from even second-moment sources. Thus all scalar shape sources can be retained while the dipole block closes.

This proof uses the selected even-dimensional chirality. At odd bound depth, normal propagation can mix dipole and shape sources through the field's mixed circulation. An isolated odd-dimensional dipole formula must not be substituted for that full coupled problem.

Keep the complete normal and conjugate response

The dipole source columns and their curvature are

$$b_i=\beta x_i\phi,\qquad \kappa=-\frac{2a}{Q_\sigma}. \tag{6}$$

The curvature comes from varying the scalar center, as derived in Translational Dirac response. For a circular eigenpolarization $\lambda$ of the radial circulation, let $w_{\lambda n}\ge0$ be the squared source projections on the complete bound eigenfields with energies $E_n$. Both angular Dirac families belong in this spectral sum.

The full bare dipole response is

$$\boxed{ \begin{aligned} \chi_\lambda(\omega,q) ={}&\sum_n w_{\lambda n} \frac{\omega+M+E_n}{(\omega+M)^2-E_n^2-q^2}\\ &+\sum_n w_{-\lambda n} \frac{-\omega+M+E_n}{(-\omega+M)^2-E_n^2-q^2}. \end{aligned}} \tag{7}$$

The second line retains the conjugate fluctuation. At $q=0$ this reduces to the usual two signed frequency resolvents. Equation (7) follows from the exact Clifford inverse, not a fixed-profile envelope dispersion.

The source-carrying modes solve

$$1-\kappa\chi_\lambda(\omega,q)=0. \tag{8}$$

Source-free modes and any canceled frozen poles remain part of the full field spectrum. For example, the normal partner of a degenerate stationary polarization can have zero dipole residue; it must not be counted as a dipole pole merely because it appears in the frozen denominator.

Translation invariance and commuting momentum generators give

$$\kappa\chi_\lambda(0,0)=1,\qquad \partial_\omega\chi_\lambda(0,0)=0. \tag{9}$$

Expand the dipole response at the origin as

$$\chi_\lambda=\frac1\kappa+ \frac12\chi_{\omega\omega}\omega^2+ \chi_{q^2}q^2+ O(\omega^3,\omega q^2,q^4). \tag{10}$$

The inverse force response then has leading coefficients

$$\boxed{ \mathcal I=-2a^2\chi_{\omega\omega},\qquad \mathcal K=4a^2\chi_{q^2},\qquad c_{\rm tr}^2=\frac{\mathcal K}{\mathcal I}.} \tag{11}$$

Indeed, a scalar force coefficient $f_i$ adds energy $f_i\delta P_i=Q_\sigma f_iX_i$. The full dipole response becomes $\delta P_i=Q_\sigma^2f_i/(\mathcal I\omega^2-\mathcal Kq^2)$ to this order. This fixes the normalization of both coefficients.

Derive the coefficients directly in physical space

Put $L=H_b-M$. The spatial Dirac identities

$$[H_b,x_i]=-i\alpha_i,\qquad \{H_b,\alpha_i\}=-2i\partial_i \tag{12}$$

give explicit inhomogeneous solutions:

$$\boxed{ L^{-1}\eta_i=Mx_i\phi-\frac i2\alpha_i\phi,\qquad (H_b+M)^{-1}\eta_i=-\frac i2\alpha_i\phi.} \tag{13}$$

These inverses are taken on the relevant complementary subspaces. Equivalently, apply the displayed operators to the right sides to verify the equations directly. Kernel components have zero dipole residue and do not contribute to the coefficients.

The velocity correction is $iL^{-1}\eta_i$. Its first scalar-density variation is zero pointwise: the $iMx_i\phi$ term is a phase variation, and $\beta\alpha_i$ is anti-Hermitian. The full source therefore permits this generalized translation partner. The normal correction lies in the opposite $\Xi$ sector and has no first-order scalar-source feedback by (5).

The complete ordinary/conjugate response gives

$$\mathcal I=2\operatorname{Re}\langle\eta_i,L^{-1}\eta_i\rangle, \qquad \mathcal K=2\operatorname{Re} \langle\eta_i,(H_b+M)^{-1}\eta_i\rangle. \tag{14}$$

Define the actual bound kinetic contribution in direction $i$:

$$T_i=\langle\phi,\alpha_i(-i\partial_i)\phi\rangle. \tag{15}$$

Localization and unit normalization imply $2\operatorname{Re}\langle-\partial_i\phi,x_i\phi\rangle=1$. Integration by parts in the other term gives $-i\langle-\partial_i\phi,\alpha_i\phi\rangle=T_i$. Substitution into (13)–(14) proves

$$\boxed{\mathcal I=M+T_i,\qquad \mathcal K=T_i,\qquad c_{\rm tr}^2=\frac{T_i}{M+T_i}.} \tag{16}$$

For a radial field, $T_i=T/d$ with $T=\langle\phi,-i\alpha_i\partial_i\phi\rangle$. The quadratic scalar virial identity is $T=2aT_\sigma$, while $M=T+mQ_\sigma+aT_\sigma$. Thus

$$\boxed{ T_i=\frac{2(M-mQ_\sigma)}{3d},\qquad m=0\ \Longrightarrow\ c_{\rm tr}^2=\frac{2}{3d+2}.} \tag{17}$$

This also explains the previously computed translation inertias without fitting their numerical values.

Two transverse polarizations do not fix the speed

For the zero-bare-mass branch, the small-wavenumber speeds are:

The depth-two modes are genuine source-carrying, gapless spatial displacements with two transverse polarizations. Within this specified scalar moment model they propagate at half the underlying wave speed. Changing the common scalar coupling only rescales the zero-bare-mass profile; it does not change (17).

This is distinct from the lightlike propagation required by the photon construction. The action's covariance in $(t,y)$ does not make every collective excitation lightlike about a finite-frequency, finite-density background. Likewise, covariance under rotations within the bound space does not complete the kernel's dynamics under rotations mixing bound and free directions.

At higher orders the circular branches can differ. Positivity of $\mathcal I$ and $\mathcal K$ describes this low-frequency displacement sector; it does not remove separate radial instabilities or establish a stable physical particle.

The same full velocity partner determines the low-frequency current and magnetic moment in Soft spatial current response.

Null spatial pulses supplies a constructive stationary-background comparison with equal time and propagation norms, while keeping its component action distinct from this scalar Dirac branch.

Verification

collective_translation_wave_checks.py evaluates both full angular radial families and their normal partners. It compares the exact coefficients with independent spatial derivatives and radial/angular quadrature, and verifies the source equation against the full normal and conjugate matrices.

A separate Cartesian Hermite construction retains every scalar second moment and both scalar-center sources at bound depth two. Independent nonlinear source variations verify its curvature. Increasing its total degree from 24 through 30 to 36 brings the displacement frequency toward the independently calculated radial result. Reports are timestamped and all checks remain active under Python optimization.