Polarized Dirac shape response

Summary: Full spatial Dirac quadrupole sources split into two calculable angular families whose polarization weights, radial spectra and reciprocal source feedback determine the allowed shape response.

Calculate the shapes the field can actually change into

The transition-density interaction compares both objects' changing spatial patterns. A complete calculation also needs the motions those patterns can undergo. Here those motions are calculated from the full first-order Dirac operator and the same scalar moment source which determines the stationary profile.

This is a conditional model on ordinary spatial coordinates $x\in\mathbb R^d$ within the one nested physical manifold. Field components, angular basis functions and shape coefficients are descriptions of this field, not further physical dimensions. The radial profile is the stated scalar model's solution; it does not replace the actual electron orbit.

Use Hermitian Clifford matrices $\alpha_i,\beta$ and a unit rest polarization $w$ with $\beta w=w$. Write the normalized stationary field as

$$\Phi(x)=\frac{f(r)w+i g(r)\alpha(\widehat x)w} {\sqrt{\Omega_{d-1}}},\qquad \int_0^\infty(f^2+g^2)r^{d-1}dr=1. \tag{1}$$

Its self-consistent central operator is

$$H_0=-i\alpha\cdot\nabla+\beta(m+ar^2),\qquad H_0\Phi=E\Phi. \tag{2}$$

Scalar Dirac moment modes derives $a=g_0T/d$, where $T=\int r^2(f^2-g^2)r^{d-1}dr$. The common coefficient $g_0$ also fixes the feedback below.

A quadrupole has two different Dirac angular responses

Let $A$ be a real symmetric traceless spatial matrix. Its source vertex is

$$V_A(x)=\beta(x^TAx),\qquad v_A=V_A\Phi. \tag{3}$$

Both signs in $\beta$ matter. In particular the source acts on the lower radial component as well as the upper one.

For any $x$, define spinor-valued homogeneous polynomials

$$\begin{aligned} p_A(x)&=(x^TAx)w,\\ N_A(x)&=\frac{2}{d+2}\alpha(Ax)w,\\ M_A(x)&=p_A(x)-\alpha(x)N_A(x). \end{aligned} \tag{4}$$

They obey

$$\boxed{\alpha\cdot\partial\,M_A=0,\qquad \alpha\cdot\partial\,N_A=0,\qquad p_A=M_A+\alpha(x)N_A.} \tag{5}$$

A polynomial annihilated by $\alpha\cdot\partial$ is called monogenic. Here (5) follows directly: $\alpha\cdot\partial p_A=2\alpha(Ax)w$, $\alpha\cdot\partial N_A=2\operatorname{tr}A\,w/(d+2)=0$, and $\alpha\cdot\partial(\alpha(x)N_A)=(d+2)N_A$. Thus the two functions are derived from the actual source.

On the unit sphere, equation (3) becomes

$$\sqrt{\Omega_{d-1}}\,v_A =r^2\left[fM_A-i g\alpha(\widehat x)M_A\right] +r^2\left[f\alpha(\widehat x)N_A-i gN_A\right]. \tag{6}$$

The first family has upper angular degree two and lower degree three. The second has upper degree two and lower degree one. Both retain the original $d$ spatial coordinates. Their different radial Dirac operators give different spectra even though they arise from the same quadrupole.

The target's polarization fixes the angular weights

Choose a real orthonormal basis $A_r$ of traceless symmetric matrices, $\operatorname{tr}(A_rA_s)=\delta_{rs}$, and define

$$J_{ij}=w^\dagger(-i\alpha_i\alpha_j)w\quad(i<j), \qquad J_{ji}=-J_{ij}. \tag{7}$$

$J$ is the ordinary spatial circulation tensor of the prepared field. Its action on the shape basis is the real skew matrix

$$\mathcal K_{rs} =\sum_{i<j}J_{ij}[A_r,A_s]_{ij} =\frac12\operatorname{tr}\left(A_r[J,A_s]\right). \tag{8}$$

Use the normalized spherical average $\langle1\rangle_S=1$ and set $c_0=2/[d(d+2)]$, $c_1=4/[d(d+2)^2]$. Direct second and fourth spherical moments give

$$\boxed{G_N=\big[\langle N_r^\dagger N_s\rangle_S\big] =c_1(I+i\mathcal K),\qquad G_M=\big[\langle M_r^\dagger M_s\rangle_S\big] =c_0I-G_N.} \tag{9}$$

The cross average $\langle M_r^\dagger\alpha(\widehat x)N_s\rangle_S$ vanishes. For example, the first identity follows by averaging $\alpha(A_r\widehat x)\alpha(A_s\widehat x)$ and retaining both its scalar part $\operatorname{tr}(A_rA_s)$ and its bivector part in (8). Both Gram matrices are positive semidefinite because they are actual spatial inner products.

For a complex incident shape $c_rA_r$, the two source strengths are $c^\dagger G_Mc$ and $c^\dagger G_Nc$, with their radial factors restored below. Thus the target polarization $J$ and the incident shape and phase $c$ enter together. A common physical rotation of $J$ and every $A_r$ leaves the prediction unchanged.

For pure rest polarizations at depths five and six, diagonalizing $i\mathcal K$ gives:

Here helicity labels a complex rotating spatial quadrupole relative to $J$. It is not an independently assigned particle charge. The five-dimensional null direction supplies the half-helicity channels. At $\lambda=-1$, the $N$ family has zero source weight, but the $M$ family remains available. A zero in one family is therefore not a zero of the whole interaction.

Derive the radial operators without removing angular components

For a monogenic polynomial of degree $\ell$, the ordinary angular Dirac pair has radial equations

$$\begin{aligned} (E-M)F&=G'+\frac{d+\ell-1}{r}G,\\ (E+M)G&=-F'+\frac\ell rF. \end{aligned} \tag{10}$$

Writing $F=r^\ell f_\ell$, $G=r^\ell g_\ell$ converts the radial integration weight to $r^{d+2\ell-1}dr$. Its matrix is the previously derived radial matrix evaluated at the numerical parameter $d+2\ell$. This parameter accounts for powers of the same radius; it introduces no additional spatial directions.

Let $H_D$ denote that radial matrix with the unchanged scalar coefficients $m,a$. For the two families in (6),

$$H_M=H_{d+4},\qquad H_N=-XH_{d+2}X,\qquad X=\begin{pmatrix}0&I\\I&0\end{pmatrix}. \tag{11}$$

The second formula follows by exchanging the upper and lower radial angular roles. Its signs can also be checked directly with $\alpha\cdot\partial(F\alpha(\widehat x)N) =(F'+(d+\ell-1)F/r)N$.

Project the two radial source pairs $(r^2f,-r^2g)$ onto their respective orthonormal physical radial bases, obtaining real vectors $u_M,u_N$. Their complete-basis norms satisfy

$$\|u_M\|^2=\|u_N\|^2 =\int_0^\infty r^4(f^2+g^2)r^{d-1}dr. \tag{12}$$

In particular the total source norm has the isotropic value $c_0\int r^4(f^2+g^2)r^{d-1}dr$ for every unit complex shape. The polarization dependence enters how that strength is divided between families with different dynamics, not a depth-specific zero inserted in the total norm.

Keep both ordinary and conjugate feedback

Put $L=\operatorname{diag}(H_M-E,H_N-E)$ and define

$$u_\lambda= \begin{pmatrix} \sqrt{c_0-c_1(1+\lambda)}\,u_M\\ \sqrt{c_1(1+\lambda)}\,u_N \end{pmatrix}. \tag{13}$$

A real spatial source pairs helicities $\lambda$ and $-\lambda$. The full ordinary/conjugate fluctuation block is

$$\boxed{\mathcal B_\lambda= \begin{pmatrix} L+g_0u_\lambda u_\lambda^T&g_0u_\lambda u_{-\lambda}^T\\ -g_0u_{-\lambda}u_\lambda^T&-L-g_0u_{-\lambda}u_{-\lambda}^T \end{pmatrix}.} \tag{14}$$

Its coefficient is the same $g_0$ as in the stationary source. Multiplication by $\operatorname{diag}(I,-I)$ gives a Hermitian energy Hessian. The matrix itself need not be Hermitian or have a positive energy, because the full Dirac fluctuation problem retains both signs of the spectrum.

For $z$ away from the indicated poles define the bare shape response

$$\chi_\lambda(z)= u_\lambda^T(z-L)^{-1}u_\lambda +u_{-\lambda}^T(-z-L)^{-1}u_{-\lambda}. \tag{15}$$

Eliminating the responding field from (14) gives

$$\boxed{\chi^{\rm full}_\lambda(z) =\frac{\chi_\lambda(z)}{1-g_0\chi_\lambda(z)}.} \tag{16}$$

This is the response to an additional specified scalar shape source. It is also a building block for a reciprocal pair calculation: each object's induced shape becomes part of the other's source. A scattering probability still requires the physical incoming/outgoing mode map, appropriate boundary conditions and flux normalization.

A spectral example from the self-consistent profiles

At $a=1$, $m=0.5$, the first positive frequencies of the finite quadrupole block converge as follows:

These are model fluctuation frequencies, not fitted particle masses or weak-interaction rates. The table fixes $a$, so the self-consistent $g_0$ differs between depths. Both geometries retain nonzero shape responses. Their differing helicity spectra provide a calculable distinction without establishing anomalously weak five-dimensional coupling.

No growing quadrupole modes were found in these finite calculations or in the corresponding $m=0$ examples. The latter backgrounds nevertheless have the radial instability documented in the profile note. Stability of one angular response does not remove an instability elsewhere.

Dynamical opening of spatial shape channels uses this complete response to calculate how a higher shape with zero static restriction can drive a lower polarized field at nonzero frequency.

Incoming spatial shape modes compares these active modes at depths five and six using one common interaction coefficient, checks their moment restrictions and distinguishes their incident amplitudes from forces which first create a higher-sector response.

Verification

angular_dirac_response_checks.py constructs full Clifford fields, differentiates their spatial polynomials, and independently integrates their angular Gram matrices with a positive spherical rule. It tests both radial operators against differentiated physical radial functions, source completeness, low-frequency convergence, and the full matrix resolvent against (16). The centered full-field energy is separately varied by finite differences. Conditions remain active with Python optimization enabled.

Related

• Rotation-neutral Dirac fields — supplies the full spatial field and its circulation tensor.
• Self-consistent scalar Dirac moment modes — supplies the stationary profiles and their radial stability test.
• Translational Dirac response — applies the same method to center motion and its exact zero mode.