Self-consistent nested Dirac pair

Summary: A common scalar moment interaction determines both nested Dirac profiles and their relative deformation, with reciprocal source variation and no independent coupling coefficient for the pair.

Let both spatial profiles respond to one interaction

The single-profile calculation derives a scalar confining source from the full spatial Dirac field. Here the same interaction determines two profiles on nested spatial sectors. The higher field changes the lower source and the lower field changes the higher source; neither profile is held fixed during self-consistency.

Take $Q_p\subset Q_{p+q}$ and local flat coordinates $x\in\mathbb R^p$, $y\in\mathbb R^q$ in the containing sector. The distinction between $x$ and $y$ is relative to this particular lower partner. All $p+q$ directions are ordinary, equivalent kinds of macroscopic spatial direction. For a six-dimensional field with a five-dimensional partner, this is a description of their nesting, not an inherent division of the six-dimensional particle's space.

Adopt orthogonal projected channels of the same field $\Psi_\infty$, with the appropriate tangential Dirac kinetic terms, specified spatial measures and separately fixed unit norms. These are explicit assumptions of a projected scalar model. Nested reciprocity describes the required measures and inclusion maps; a probability-normalized extension over an infinite fiber cannot silently be replaced by a constant of unit amplitude. No additional interaction particle is introduced.

Specify the interaction below as a function of the projected channel densities. A coherent unprojected superposition also requires the transition-density terms of its full source. Integrated kinetic orthogonality alone does not remove those pointwise scalar cross terms. Connecting the projected model to that coherent field requires the corresponding source projection.

The calculation establishes stationary solutions of this projected model. It does not identify them as the actual electron orbit, a neutrino profile, or a stable two-particle bound state.

The common energy fixes both source matrices

Let $M_p$ and $M_{p+q}$ be separately centered scalar moments $\int \xi\xi^T\Phi^\dagger\beta\Phi$ on their respective sectors. Use the isometric spatial inclusion $I:\mathbb R^p\to\mathbb R^{p+q}$ and define the containing-space source matrix

$$S=IM_pI^T+M_{p+q}. \tag{1}$$

The common scalar moment energy is

$$\boxed{\mathcal E =T_p+T_{p+q}+m_pQ_p+m_{p+q}Q_{p+q} +\frac{g_0}{2}\operatorname{tr}S^2,\qquad g_0>0.} \tag{2}$$

Here $T_d=\langle\Phi_d,-i\alpha\cdot\nabla\Phi_d\rangle$ and $Q_d=\int\Phi_d^\dagger\beta\Phi_d$ is a scalar integral, not an electric charge. Expanding (2) gives both self terms and $g_0\operatorname{tr}(M_pI^TM_{p+q}I)$ as the cross term. There is one coefficient $g_0$ for all three terms.

The projected bilocal variation then gives scalar source matrices

$$A_p=g_0I^TSI,\qquad A_{p+q}=g_0S. \tag{3}$$

The higher-to-lower source is the restriction of the common matrix. The lower-to-higher source is its reciprocal inclusion. The spatial matrices and both field profiles must be varied together.

A closed stationary symmetry class

Consider real profiles whose scalar moments have the form

$$M_p=t_pI_p,\qquad M_{p+q}=\begin{pmatrix}t_xI_p&0\\0&t_yI_q\end{pmatrix}. \tag{4}$$

Equations (1)–(3) imply

$$\boxed{a=g_0(t_p+t_x),\qquad b=g_0t_y.} \tag{5}$$

The lower Dirac operator has scalar mass $m_p+a|x|^2$; the higher one has scalar mass $m_{p+q}+a|x|^2+b|y|^2$. In particular, the coefficient along every shared spatial direction is the same for both fields. The higher field may acquire a different spatial width relative to its lower partner without assigning its directions different physical kinds.

This is a stationary symmetry class. Arbitrary complex perturbations can generate additional mixed spatial moments, so a stability calculation must include those responses rather than assuming that (4) closes all time-dependent motions.

Two-frequency fields in a readjusted nested pair extends the real stationary profiles to exact opposite-frequency components in both channels. Their currents can oscillate with independent clocks while both scalar sources retain the same self-consistency equations.

Keep the full anisotropic Dirac field

The higher profile requires more than the two radial functions of a central field. Let $r=|x|$, $s=|y|$, choose a unit rest polarization $w$ with $\beta w=w$, and put $C_x=\alpha(\widehat x)$, $C_y=\alpha(\widehat y)$. They obey $C_x^2=C_y^2=I$ and $C_xC_y=-C_yC_x$.

An angularly normalized full spatial field is

$$\boxed{\Phi_{p+q}(x,y)= \frac{F_{00}w+iF_{10}C_xw+iF_{01}C_yw+F_{11}C_xC_yw} {\sqrt{\Omega_{p-1}\Omega_{q-1}}},} \tag{6}$$

with real functions $F_{ab}(r,s)$. For $q=1$, the second angular integration is the two signs of the one existing spatial coordinate. It does not create a further space.

The $00$ and $11$ components have positive $\beta$ sign; the $10$ and $01$ components have negative sign. The scalar density is therefore

$$\sigma=\frac{F_{00}^2+F_{11}^2-F_{10}^2-F_{01}^2} {\Omega_{p-1}\Omega_{q-1}}. \tag{7}$$

This identity holds pointwise for every normalized $w$. Cross terms with opposite $\beta$ signs vanish, and the remaining cross Clifford product $C_xC_y$ is anti-Hermitian, so its expectation contributes no real scalar term for these real profiles. The norm has the same expression with all four squares positive.

Consequently

$$pt_x=\int r^2\sigma\,dx\,dy,\qquad qt_y=\int s^2\sigma\,dx\,dy. \tag{8}$$

This stationary scalar source remains degenerate in the rest polarization $w$. Both objects can be polarized, but this term by itself does not prefer a particular neutral rotation weight. Their response to a non-isotropic spatial source still retains polarization, as in Dirac shape response.

A reproducible product radial operator

For each radial coordinate use the normalized even and odd-angular Laguerre functions in the single-profile note. Let $K_p,K_q$ represent the corresponding upper-to-lower radial derivatives, and let $R_p^{(0)},R_p^{(1)},R_q^{(0)},R_q^{(1)}$ represent multiplication by the squared radii in those bases. The product basis is ordered $00,10,01,11$. Define

$$X=K_p\otimes I,\qquad Y=I\otimes K_q,$$

$$D_{ab}=(-1)^{a+b} \left[mI+a_{\rm s}R_p^{(a)}\otimes I +b_{\rm s}I\otimes R_q^{(b)}\right], \tag{9}$$

where $a_{\rm s},b_{\rm s}$ denote the physical coefficients called $a,b$ in (5), to distinguish them from the component indices. Direct spatial differentiation of (6) gives

$$\boxed{H= \begin{pmatrix} D_{00}&X^T&Y^T&0\\ X&D_{10}&0&Y^T\\ Y&0&D_{01}&-X^T\\ 0&Y&-X&D_{11} \end{pmatrix}.} \tag{10}$$

The minus signs follow from anticommuting $C_x$ and $C_y$. Removing $F_{11}$ changes the anisotropic Dirac equation. In the isotropic limit $a=b$, the central positive branch has $F_{11}=0$ and reproduces the ordinary radial solution in $p+q$ dimensions. Away from that limit the mixed upper component is generally nonzero.

A dimensionless self-consistency equation

For $m_p=m_{p+q}=0$, write $\rho=b/a$. At shared scale $a=1$, compute the lower per-axis scalar moment $t_p$ and the higher moments $t_x(\rho),t_y(\rho)$ from their full Dirac profiles. Equations (5) reduce to

$$\boxed{\rho=\frac{t_y(\rho)}{t_p+t_x(\rho)},\qquad a=\left[g_0\big(t_p+t_x(\rho)\big)\right]^{3/5}.} \tag{11}$$

The first equation determines the relative deformation without a pair coefficient. The second determines the common spatial scale for any specified $g_0$. Both frequencies scale as $a^{1/3}$ and all scalar second moments as $a^{-2/3}$.

For $p=5$, $q=1$, the converged branch at $a=1$ gives

For example $g_0(0.814814980+0.777880017)/5=1$ and $g_0(0.200160151)=0.628369372$, checking both source equations with the same coefficient. The higher mixed upper component has norm fraction about $7.25694\times10^{-5}$ and is retained throughout the calculation.

These numbers describe the declared stationary scalar model. They are not a particle mass fit or an empirical neutrino coupling prediction.

Energy and reciprocal variation provide independent checks

For each massless partner, a norm-preserving spatial dilation gives its kinetic expectation as twice its scalar source expectation. Summing the stationary equations yields

$$E_p+E_{p+q}=6\mathcal E_{\rm int},\qquad \boxed{\mathcal E=\frac56(E_p+E_{p+q}).} \tag{12}$$

The total energy at the displayed scale is approximately $4.296173765$. The full Dirac energy is not globally bounded below, so this identity does not establish a global ground state or nonlinear stability.

For independent variations of both partner fields, their cross Hessian is

$$\delta_p\delta_{p+q}\mathcal E =g_0\operatorname{tr} \left(I\delta M_pI^T\delta M_{p+q}\right). \tag{13}$$

The same bilinear is obtained by differentiating either source in (3). It is the explicit reciprocal response of the two solved profiles. The general tensor form remains valid when their orientations and shape perturbations are restored; the stationary table uses the symmetry (4).

Normal propagation retains the deformed scalar source

The higher field's four real profiles also supply the normal frame needed by Reciprocal scalar envelopes. Choose an ordinary free coordinate in a sector containing the whole pair, with Clifford matrix $\alpha_z$ anticommuting with the bound matrices and $\beta$. It is an existing direction of the same physical manifold.

Write the matrix in the numerator of (6) as

$$P=F_{00}I+iF_{10}C_x+iF_{01}C_y+F_{11}C_xC_y.$$

The two anticommuting spatial matrices give the exact quaternion identity

$$\boxed{P^\dagger P= (F_{00}^2+F_{10}^2+F_{01}^2+F_{11}^2)I.} \tag{14}$$

Since $\beta\alpha_z$ commutes with both $C_x$ and $C_y$, the mixed scalar normal-partner block vanishes. For an orthonormal rest-polarization matrix $V$ with $\beta V=V$, put $F=PV/\sqrt{\Omega_{p-1}\Omega_{q-1}}$ and $W=(F,\alpha_zF)$. Then

$$\boxed{W(x,y)^\dagger\beta W(x,y) =\sigma(x,y)(\sigma_3\otimes I),} \tag{15}$$

where $\sigma$ is (7). This identity holds for arbitrary complex normal amplitudes and every selected rest polarization. Thus their scalar density multiplies the same deformed spatial source, rather than introducing unaccounted interference terms.

Angular integration removes the mixed normal current. The two partner families have the canonical spatial norm. When any real profile parameter $s$ varies, the angular average of the projected derivative contains only the radial norm derivative:

$$\int W^\dagger\partial_sW\,dx\,dy=0 \tag{16}$$

for a normalized profile family. This justifies the first-order kinetic normalization used in the two-envelope reduction.

An exact uniform normal boost preserves the scalar density pointwise, as in Normal Dirac propagation. Boosting the deformed partner therefore leaves both reciprocal scalar source matrices unchanged for a uniform pair. Varying envelope densities can still drive omitted spatial profiles through the residual described in the scalar envelope calculation; (16) does not establish full dynamical closure.

Verification

nested_dirac_pair_checks.py compares (10) directly with finite spatial derivatives of the full Clifford field for several nested depths, including arbitrary complex test coefficients. Product radial quadrature checks its moments and norm. The isotropic limit is independently compared with the central Dirac solver. Increasing radial basis size and changing both numerical widths test the anisotropic branch and the coupled self-consistency equation.

Finite differences of the frequency verify both scalar moment derivatives. Finite differences of the pair energy and of each partner's source verify (13). Several specified values of the one common coefficient test the scaling in (11). Reports have timestamped filenames and all conditions remain active under Python optimization.

The complete anisotropic normal frame is also checked pointwise for complex amplitudes, with angular normalization and real profile derivatives. Independent full spatial derivatives verify boosted propagation of the actual self-consistent deformed partner.

Related

• Nested moment coupling — derives the common moment interaction across the sector hierarchy.
• Nested sector reciprocity — supplies the spatial restriction and inclusion with consistent measures.
• Self-consistent scalar Dirac moment modes — supplies the central member of the pair and its radial representation.