Self-consistent scalar Dirac moment modes

Summary: The scalar moment kernel admits self-consistent radial Dirac modes with a calculable source and virial law, while the complete conjugate response detects an instability in several massless-background branches.

A spatial profile derived from the existing source term

A coupling vertex needs actual spatial field profiles. This note applies the nonlocal scalar moment interaction to a full first-order Dirac field, retaining both radial components and the change in its own source.

The model is a conditional flat spatial calculation on $\mathbb R^d$ within the one nested macroscopic manifold. It does not replace a particle's proposed global spatial geometry or the actual Coulomb-plus-precession electron orbit. The computed modes are stationary solutions of the stated model; their existence alone does not establish stable physical particles.

Use Hermitian Dirac matrices $\alpha_i,\beta$, $\hbar=c=1$, and a bare scalar coefficient $m\geq0$. Define

$$\sigma(x)=\Psi(x)^\dagger\beta\Psi(x),\qquad Q_\sigma=\int\sigma(x)dx,\qquad X=\frac{\int x\sigma(x)dx}{Q_\sigma},$$

on a domain where $Q_\sigma\ne0$, and let

$$M_\sigma=\int(x-X)(x-X)^T\sigma(x)dx. \tag{1}$$

The scalar density need not be positive. The field norm is separately $\int\Psi^\dagger\Psi=1$. Adopt the energy

$$\boxed{\mathcal E[\Psi] =\langle\Psi,(-i\alpha\cdot\nabla+m\beta)\Psi\rangle +\frac{g_0}{2}\operatorname{tr}M_\sigma^2, \qquad g_0>0.} \tag{2}$$

This is the scalar channel of the transition-density interaction with the centered dot-square kernel. Centering has the variational role derived in centroid-relative self-binding. Because the first central scalar moment is zero, variations of $X$ cancel in the first variation of (1). The exact field equation is

$$i\partial_t\Psi= \left[-i\alpha\cdot\nabla+ \beta\left(m+g_0(x-X)^TM_\sigma(x-X)\right)\right]\Psi. \tag{3}$$

All coefficients in the response come from the same field. Equation (3) has no additional interaction particle or externally supplied confining potential. Its full energy is not bounded below over arbitrary Dirac fields, so a lowest positive-frequency branch is not a global energy minimum.

The radial self-consistency equation

Use the angularly normalized radial field in Rotation-neutral Dirac fields, with upper radial function $f$ and lower function $ig\alpha(\widehat x)$. The scalar moment is isotropic:

$$M_\sigma=\frac{T}{d}I, \qquad T=\int_0^\infty r^2(f^2-g^2)r^{d-1}dr. \tag{4}$$

Writing

$$a=\frac{g_0T}{d}, \tag{5}$$

the stationary radial equations become

$$\boxed{(E-m-ar^2)f=g'+\frac{d-1}{r}g, \qquad(E+m+ar^2)g=-f'.} \tag{6}$$

Thus a pure scalar quadratic mass profile is generated by the same moment whose spatial mode it confines. Given a normalized positive-frequency solution of (6) with $T>0$, equation (5) determines the coefficient required for self-consistency. Alternatively one can fix $g_0$ and solve for $a$. The latter form is needed to compare sectors under one common interaction.

These radial modes retain all $d$ position coordinates. Their rest spinor polarization can carry different rotation weights while sharing the same radial source. The scalar interaction therefore does not select the neutral weights over the other polarizations.

Scaling and the total energy

Under the norm-preserving dilation $\Psi_\lambda(x)=\lambda^{d/2}\Psi(\lambda x)$, the Dirac kinetic expectation scales as $\lambda$ and the moment energy as $\lambda^{-4}$. The bare mass expectation is unchanged. Stationarity consequently requires

$$\boxed{T_D=4\mathcal E_{\rm int}=2aT,} \qquad T_D=\langle\Psi,-i\alpha\cdot\nabla\Psi\rangle. \tag{7}$$

Multiplying the stationary field equation by $\Psi^\dagger$ and integrating then gives

$$E=3aT+m\langle\beta\rangle, \qquad \mathcal E=E-\frac{aT}{2}. \tag{8}$$

The spectral frequency and the total nonlinear energy are different. For $m=0$, rescaling the radial coordinate gives

$$E(a)=\epsilon_d a^{1/3},\qquad T=\frac{E}{3a},\qquad \boxed{a=\left(\frac{g_0\epsilon_d}{3d}\right)^{3/5}, \quad E=\epsilon_d\left(\frac{g_0\epsilon_d}{3d}\right)^{1/5}, \quad\mathcal E=\frac56E.} \tag{9}$$

Here $\epsilon_d$ is the lowest positive radial frequency at $a=1$, $m=0$. The powers in (9) follow from the first-order kinetic term and the quadratic spatial moment; they are not the scaling of a scalar Schrödinger oscillator. No particle mass or occupied mass-formula index has been assigned to this radial frequency.

A reproducible radial representation

Put $x=wr^2$ and $\alpha=d/2-1$. Use normalized upper and lower bases

$$f_n(r)=N_n^{(f)}e^{-x/2}L_n^\alpha(x),\qquad g_n(r)=N_n^{(g)}r e^{-x/2}L_n^{\alpha+1}(x), \tag{10}$$

with

$$(N_n^{(f)})^2=\frac{2w^{\alpha+1}n!}{\Gamma(n+\alpha+1)}, \qquad (N_n^{(g)})^2=\frac{2w^{\alpha+2}n!}{\Gamma(n+\alpha+2)}. \tag{11}$$

The derivative identity is

$$-f_n'=\sqrt{w(n+d/2)}\,g_n+\sqrt{wn}\,g_{n-1}. \tag{12}$$

Let $K$ be this upper-to-lower matrix. Multiplication by $r^2$ gives tridiagonal matrices $R_f,R_g$ with diagonal entries $(2n+d/2)/w$ and $(2n+d/2+1)/w$. Their upper off-diagonals are respectively $-\sqrt{(n+1)(n+d/2)}/w$ and $-\sqrt{(n+1)(n+d/2+1)}/w$. The finite radial Hamiltonian is

$$H_a=\begin{pmatrix}mI+aR_f&K^T\\K&-mI-aR_g\end{pmatrix}. \tag{13}$$

The basis width $w$ is a numerical parameter, not a physical source coefficient. Changing it and increasing the number of functions provides independent convergence controls.

At $a=1$, $m=0$, the converged positive branch gives:

The table fixes the source scale $a$ to display the dimensionless profiles. It does not use a common $g_0$ across its rows. Equation (9) performs that common-coefficient comparison without independently fitting each depth.

The complete radial fluctuation problem

The source changes when the spatial field changes. In the finite radial representation set $B=\operatorname{diag}(R_f,-R_g)$ and let $\phi$ be a real normalized stationary coefficient vector. Then $T=\phi^TB\phi$, $a=g_0T/d$, and $v=B\phi$. Writing a perturbation as $\eta=x+iy$ in the rotating frame gives

$$\boxed{\frac{d}{dt}\begin{pmatrix}x\\y\end{pmatrix} =\begin{pmatrix}0&L_-\\-L_+&0\end{pmatrix} \begin{pmatrix}x\\y\end{pmatrix},} \tag{14}$$

where

$$L_-=H_a-EI,\qquad L_+=L_-+\frac{2g_0}{d}vv^T. \tag{15}$$

This retains the ordinary and conjugate source variations in Dirac source linearization. The phase mode has $L_-\phi=0$. On the fixed-norm tangent space, $\phi^Tx=0$; quotienting the phase direction removes the associated zero-mode numerical ambiguity. It does not remove growing physical radial perturbations.

For the $m=0$, $a=1$ branch, converged radial growth rates include $1.028404447$ at depth three, $1.027065172$ at four, $0.954281303$ at five, and $0.816171505$ at six, in the model's time units. These are finite nonzero rates, not small numerical real parts of a phase zero mode. Direct nonlinear evolution of the depth-five field reproduces the growing linearized perturbation, with the remainder decreasing when the initial perturbation is reduced. Norm and the full nonlinear energy remain conserved during that growth.

Consequently these massless-background branches cannot be identified as stable particles on the strength of their stationary profiles. They remain well-defined source and response calculations for the stated model. A radial calculation alone also does not establish stability against every spatial angular perturbation, even where it detects no radial growth.

Polarized Dirac shape response calculates the quadrupole motions of these same profiles. Translational Dirac response retains their moving center and derives its zero mode and inertial response.

Self-consistent nested Dirac pair lets both a lower and a containing spatial field determine their profiles through one common scalar coefficient.

Radial dynamics of two-frequency fields embeds this entire radial equation into the full opposite-frequency spatial family, retaining its source, kinetic weight and any radial instability.

Verification

spatial_dirac_mode_checks.py checks the basis and kinetic matrices by independent Gauss–Laguerre quadrature, compares positive frequencies at several basis sizes and widths, and substitutes the reconstructed functions directly into both equations (6). It checks self-consistency, the dilation identity, the distinction between total energy and frequency, and nonlinear perturbation evolution.

As an independent spectral control, the same radial method is tested with equal scalar and vector quadratic backgrounds. Eliminating the lower component there gives the analytic equation $(E-m)^2(E+m)=2a(4n+d)^2$ for radial level $n$. This control belongs to the solvable class described by Ginocchio; it tests the discretization and is not substituted for the scalar source in (3).

Related

• Rotation-neutral Dirac fields in physical space — gives the full angular field and its conserved total rotation.
• Transition-density coupling — supplies the nonlocal scalar source and full variations.
• Centroid-relative self-binding — explains the translation-preserving center variation.