Normal propagation of spatial Dirac modes

Summary: Retaining the normal Dirac partner gives exact free-direction propagation, source-preserving boosts and a full moment-response equation for collective fluctuations.

A bound sector can propagate through a containing spatial direction

Choose bound coordinates $x\in\mathbb R^d$ and one perpendicular coordinate $y$ in a containing sector of the same nested physical manifold. The coordinate $y$ is another ordinary macroscopic spatial direction. For the lower field it is a free direction relative to its bound spatial profile; it is not an internal or auxiliary dimension.

Use a Clifford representation appropriate to that containing space, with Hermitian $\alpha_1,\ldots,\alpha_d,\alpha_y,\beta$. The containing representation may include several copies of a lower representation. Its field components and the mode decompositions below do not add further physical coordinates.

Let a normalized full spatial bound mode obey

$$H_b\phi=M\phi,\qquad H_b=-i\alpha_i\partial_i+\beta\mathfrak m(x), \qquad M>0,\qquad\int\phi^\dagger\phi\,dx=1. \tag{1}$$

$M$ is this model's positive bound frequency. Its use as a rest parameter in the calculation does not identify it with a measured particle mass or replace the actual electron orbit. The full bound field, including its lower radial components, is the one derived in Rotation-neutral Dirac fields.

The omitted partner carries the normal propagation

Because $\alpha_y$ anticommutes with every $\alpha_i$ and with $\beta$,

$$\{H_b,\alpha_y\}=0,\qquad H_b(\alpha_y\phi)=-M\alpha_y\phi. \tag{2}$$

The two normalized fields $\phi,\alpha_y\phi$ are orthogonal. For a free normal momentum $k$, the full operator $H_b+k\alpha_y$ acts on their span as

$$\begin{pmatrix}M&k\\k&-M\end{pmatrix}. \tag{3}$$

Consequently its positive frequency and normalized spatial mode are

$$\boxed{E=\sqrt{M^2+k^2},\qquad \phi_k=\sqrt{\frac{E+M}{2E}}\,\phi +\frac{k}{\sqrt{2E(E+M)}}\alpha_y\phi.} \tag{4}$$

The normal velocity is $\langle\phi_k,\alpha_y\phi_k\rangle=k/E$. A projection onto $\phi$ alone would instead give the constant operator $M$, losing this propagation. Exact elimination gives

$$H_{\rm eff}(E,k)=M+\frac{k^2}{E+M},\qquad Z(E,k)=1+\frac{k^2}{(E+M)^2}. \tag{5}$$

The physical velocity reconstructed with this norm is $[2k/(E+M)]/Z=k/E$. This is a direct spatial example of the dressed-source construction. The negative-frequency partner here is a component of the same field's Dirac dynamics, not an additional interaction particle.

The scalar source also gives an exact zero-mode condition. Since $\{H_b,\beta\}=2\mathfrak m(x)I$, a normalized bound eigenfield obeys

$$\boxed{M\langle\beta\rangle =\int\mathfrak m(x)\,\phi^\dagger\phi\,dx.} \tag{5a}$$

For the present nonnegative quadratic scalar profiles, the right side is strictly positive for every nonzero square-integrable field. There is therefore no localized zero-frequency bound mode in this source class. The positive branch has $\langle\beta\rangle>0$ and its normal dispersion in (4) remains massive. The massless zero mode required by the photon construction needs its own source dynamics. Transverse localization and preservation of normal propagation alone do not establish that zero mode. Equation (5a) concerns the bound eigenproblem (1). The full real-linear fluctuation operator also varies the source and can have collective zero modes, including the translation mode. Their normal dispersion and any identification with a photon require that complete fluctuation calculation.

A normal boost preserves the scalar binding source

To retain self-consistency, specify the source's spatial dependence. Use the scalar moment interaction locally in the free coordinate:

$$\mathcal E_{\rm int} =\frac{g_0}{2}\int dy\,\operatorname{tr}M_\sigma(y)^2, \qquad M_\sigma(y)=\int (x-X(y))(x-X(y))^T \Psi^\dagger\beta\Psi\,dx. \tag{6}$$

The interaction is nonlocal in the stated bound coordinates and local in $(t,y)$. Its action is invariant under boosts in the $(t,y)$ plane: those boosts leave $x$ unchanged, while $\overline\Psi\Psi$ is a scalar. This is an explicit assumption about the kernel in the free directions, not a claim of a completed Lorentz-covariant action in every spatial plane.

For rapidity $\eta$, put

$$S_\eta=e^{\eta\alpha_y/2},\qquad E=M\cosh\eta,\qquad k=M\sinh\eta.$$

The exact uniformly propagating field is

$$\boxed{\Psi_\eta(t,x,y) =e^{i(ky-Et)}S_\eta\phi(x).} \tag{7}$$

Since $S_\eta^\dagger\beta S_\eta=\beta$,

$$\Psi_\eta^\dagger\beta\Psi_\eta=\phi^\dagger\beta\phi \tag{8}$$

pointwise. Its scalar center and every bound scalar moment remain unchanged. Thus if the rest profile solves the scalar source equation, the same source binds (7) with the same coefficient $g_0$. No confining potential has to be retuned for this boost.

Equations (4) and (7) use different density conventions. The canonical mode in (4) has unit norm per chosen normal normalization interval. The boosted field in (7) has the Lorentz-transformed density. Dividing (7) by $\sqrt{\cosh\eta}$ restores unit laboratory density and also divides its scalar source by $\cosh\eta$. A nonlinear source then has to be solved at that changed scalar density. The distinction is developed in Scalar envelope of a bound Dirac field.

Retain the local current when transforming the density

Write $\rho=\phi^\dagger\phi$ and $j_y=\phi^\dagger\alpha_y\phi$. The exact pointwise transformations are

$$\boxed{\rho_\eta=\cosh\eta\,\rho+\sinh\eta\,j_y, \qquad (j_y)_\eta=\sinh\eta\,\rho+\cosh\eta\,j_y.} \tag{9}$$

The unboosted mode can have a nonzero local normal current even when its integrated normal current vanishes. For a central radial profile

$$\phi=\frac{f(r)w+i g(r)\alpha(\widehat x)w}{\sqrt{\Omega_{d-1}}}, \qquad\beta w=w,$$

define $J_{iy}=w^\dagger(-i\alpha_i\alpha_y)w$. Direct Clifford multiplication gives

$$j_y(x)=\frac{2f(r)g(r)}{\Omega_{d-1}} \sum_iJ_{iy}\widehat x_i,\qquad \int j_y\,dx=0. \tag{10}$$

This is an ordinary spatial current of the field. Calling it an electric current would require the physical charge generator from the action.

Its first spatial moment is

$$\int x_i j_y\,dx=F_0J_{iy},\qquad F_0=\frac2d\int_0^\infty r^d f(r)g(r)\,dr. \tag{11}$$

The full Dirac equation supplies an independent identity. Since $\{H_b,x_i\alpha_y\}=-i\alpha_i\alpha_y$,

$$\boxed{F_0=\frac{1-2G/d}{2M},\qquad G=\int_0^\infty g(r)^2r^{d-1}dr.} \tag{12}$$

One can also derive it by integrating the two radial Dirac equations: $\int r^d(M+\mathfrak m)fg=d(1-G)/2$ and $\int r^d(M-\mathfrak m)fg=(d-2)G/2$.

For $v=\tanh\eta$, the centroid of the boosted number density is therefore

$$\boxed{(X_\rho)_i=vF_0J_{iy}.} \tag{13}$$

The scalar source center stays fixed by (8). The number-density centroid can shift because the boost mixes density with the spatially varying current. The effect depends on the actual field polarization and containing Clifford representation; it cannot be inferred from the bound depth alone. For example a selected even-dimensional rest chirality makes the mixed $J_{iy}$ vanish, while the tested pure odd-dimensional rest preparations have nonzero mixed circulation.

Collective propagation requires variation of the full source

The normal partner solves the frozen bound eigenproblem exactly. A collective fluctuation also changes the binding source. For a finite moment interaction this additional response can be retained without restricting the field to two normal amplitudes.

Let $b_A(x)$ be the full spatial source vectors at a stationary profile, and let $\kappa$ be the real symmetric curvature matrix of its moments. Write the extra, real-linear source derivative as

$$\delta N_{\rm extra}[\eta] =\sum_{A,B}b_A\kappa_{AB}h_B, \qquad h_A=2\operatorname{Re}\langle b_A,\eta\rangle. \tag{14}$$

The frozen operator $H_b\eta$ is retained in addition to (14). For the centered isotropic scalar moment model, take a Frobenius-orthonormal basis $A$ of all symmetric spatial tensors and include

$$b_A=\beta(x^TAx)\phi,\qquad b_i=\beta x_i\phi. \tag{15}$$

The shape block of $\kappa$ is $g_0I$; its dipole block is $-2aI/Q_\sigma$, where $\mathfrak m=m+a|x|^2$ and $Q_\sigma=\langle\phi,\beta\phi\rangle\ne0$. The second block is fixed by varying the scalar center. It must be included along with the breathing and traceless shapes. This is the full centered source variation derived in Translation response. The finite list of moments describes spatial patterns in the existing sector, not additional physical coordinates.

Let $C$ denote the column operator with columns $b_A$, and set $L_q=H_b+q\alpha_y-M$. With the ordinary and conjugate perturbation convention

$$\delta\Psi=e^{-iMt} \left(u(x)e^{i(qy-\omega t)} +v(x)^*e^{-i(qy-\omega^*t)}\right),$$

the full Fourier equation is

$$\boxed{ \omega\begin{pmatrix}u\\v\end{pmatrix} =\left[\mathcal B_0(q)+\mathsf S\kappa\mathsf T\right] \begin{pmatrix}u\\v\end{pmatrix}, \quad \mathcal B_0(q)=\begin{pmatrix}L_q&0\\0&-L_{-q}^*\end{pmatrix}, \quad \mathsf S=\begin{pmatrix}C\\-C^*\end{pmatrix}, \quad\mathsf T=(C^\dagger,C^T).} \tag{16}$$

Here the star on a matrix denotes entrywise conjugation. In particular, the conjugate block keeps the opposite normal momentum. The moment amplitude is $h=\mathsf T(u,v)^T$.

Away from the frozen resolvent poles, eliminate the full spatial field, retaining all its components. Its moment response is

$$\boxed{\Pi(\omega,q) =C^\dagger(\omega-L_q)^{-1}C -C^T(\omega+L_{-q}^*)^{-1}C^*.} \tag{17}$$

An imposed source coefficient $f$ then gives

$$\boxed{h=[I-\Pi(\omega,q)\kappa]^{-1}\Pi(\omega,q)f.} \tag{18}$$

This small matrix contains the full bound-field response through its resolvents. It is not the response obtained by freezing the radial profiles inside an envelope ansatz. Both ordinary and conjugate fluctuations, scalar-center motion and shape response remain present.

The normal anticommutation identity gives a useful exact evaluation:

$$\begin{aligned} (\omega-L_q)^{-1} &=[\omega+M+H_b+q\alpha_y] [(\omega+M)^2-H_b^2-q^2]^{-1},\\ (\omega+L_{-q}^*)^{-1} &=[\omega-M-H_b^*+q\alpha_y^*] [(\omega-M)^2-(H_b^*)^2-q^2]^{-1}. \end{aligned} \tag{19}$$

All scalar terms multiply the identity. This expresses normal momentum dependence in terms of the actual squared bound operator; it introduces no independent interaction field.

A homogeneous source-carrying mode away from frozen poles satisfies $\det(I-\Pi\kappa)=0$. Modes in the source kernel must also be kept. For any finite spatial regulator, the full characteristic identity is

$$\det[\omega-\mathcal B(q)] =\det[\omega-\mathcal B_0(q)]\det[I-\Pi(\omega,q)\kappa]. \tag{20}$$

At a frozen pole, the factors can cancel singularities; the complete operator or their continued product determines the spectrum. Dropping the frozen factor would lose source-free modes. For example, phase symmetry gives the collective zero direction $(i\phi,-i\phi^*)$ at $q=0$ even when $M>0$.

Equations (16)–(20) supply a full-source route to the normal dispersion of collective modes. Identifying a transverse massless photon still requires solving that spectrum for the appropriate spatial solution. A zero frequency at $q=0$ alone does not establish a linear lightlike branch or its physical polarization.

The same construction applies to the nested pair under its specified projected-channel assumptions. Use both full spatial perturbation spaces and the curvature of their common energy. For the second-moment shapes, its block is

$$\kappa_{\rm shape}=g_0 \begin{pmatrix}I&\mathcal R\\\mathcal R^\dagger&I\end{pmatrix}, \qquad \mathcal R(C)=I_{ab}^TCI_{ab}, \tag{21}$$

where $I_{ab}$ is the actual lower-to-higher spatial inclusion and $\mathcal R^\dagger$ uses the Frobenius moment norm. Center variations supply the additional dipole blocks. The source column operator is block diagonal over the two projected channels; the off-diagonal curvature in (21) carries their mutual response. Equations (17)–(18) consequently retain both objects' full spatial variations and repeated source feedback. For finite pair coupling the background must be the jointly readjusted pair, so these equations cannot use two unchanged isolated profiles.

Two-frequency spatial Dirac currents combines both normal partners into an exact nonlinear field with a changing spatial current and a constant scalar source. It retains the complete boost phase and the response of its periodic background.

Periodic spatial source response extends these full resolvents to a two-frequency background, retaining the frequency sidebands of its source and both ordinary and conjugate field variations.

Collective translation waves solves the source-carrying dipole branch for selected even-dimensional chiral fields. The complete source and normal partners give exact inertia and stiffness; at bound depth two the two displacement polarizations have low-frequency speed one-half in Dirac units. A collective zero alone therefore does not identify a photon in this scalar model.

Verification

normal_dirac_transport_checks.py constructs the containing Clifford representation and directly differentiates the boosted spatial field at depths three, five and six. It checks the complete bound-plus-normal Dirac equation, the pointwise scalar source, the same self-consistent moment coefficient, density/current transformation, and the polarization-dependent centroid. Independent radial integration checks (12), and reconstruction checks (4)–(5).

Independent Cartesian Hermite regulators at depths three and five verify the full centered nonlinear source derivative, both normal resolvents, the driven response (18) and the complete characteristic identity (20). The projected squared-coordinate operators retain their full-basis matrix elements. These are finite-regulator algebra checks, not converged collective particle spectra; the full phase zero mode is retained.

The exact uniform boost does not cure a bound-profile instability. The radial stability limits of the scalar branch remain those in the profile note, and spatially varying envelopes require the additional response terms described in the envelope reduction.

Related

• Self-consistent scalar Dirac moment modes — supplies the full bound profiles and their source equation.
• Dressed spatial source vertices — explains the reconstruction and kinetic norm in (5).
• Scalar envelope of a bound Dirac field — derives the response when the normal amplitude and scalar density change.