Spatial response of a conformal scalar-current interaction

Summary: An exact zero-current Killing field has a closed full-variation response to a conformal scalar-current interaction, whose simplex coefficient enters collective frequencies and makes this branch unstable at high harmonic degree.

Test the scalar-current route in the field equation

The conformal scalar operator has a simplex spectrum, but an interaction in which it acts on the scalar current must be varied and solved as a nonlinear field equation. There is an exact spatial background on which this response can be calculated. The simplex value enters the collective response; it is not directly the background frequency or an observed particle mass.

Specify the actual physical geometry as a round six-dimensional sphere with undetermined radius $R$. Its six directions are ordinary macroscopic spatial directions, with no compactification. This is a conditional geometry, not a determination of the electron's physical topology.

Use the ordinary complex rank-eight spinor bundle, the self-adjoint spatial Dirac operator $\mathscr D$ and Hermitian chirality $\beta$. Initially set $R=1$ and normalize spatial measure to total volume one. The proposed energy is

$$E(\psi)=\langle\psi,\mathscr D\psi\rangle +\frac12\langle J,(g_0I+g_1\widehat P_6)J\rangle, \qquad J=\psi^\dagger\beta\psi. \tag{1}$$

Here $\widehat P_6=\Delta(\Delta+4)(\Delta+6)/6!$ and

$$\widehat P_6Y_{\ell,a}=p_\ell Y_{\ell,a},\qquad p_\ell=S(\ell,6),\qquad p_0=0. \tag{2}$$

The $g_0$ term is the local scalar interaction. The derivative-current term proportional to $g_1$ is an additional candidate, not a reduction already supplied by the master action. Its full variation gives

$$i\partial_t\psi=\mathscr D\psi+ \bigl[(g_0I+g_1\widehat P_6)J\bigr]\beta\psi. \tag{3}$$

Both ordinary and conjugate variations of the source are retained, as required by the complete scalar response.

A full spatial field with zero scalar current

Six-dimensional Clifford multiplication can be represented by real skew-symmetric $8$ by $8$ matrices $c_i$. A concrete construction uses

$$X=\begin{pmatrix}0&1\\1&0\end{pmatrix},\quad J_2=\begin{pmatrix}0&1\\-1&0\end{pmatrix},\quad Z=\begin{pmatrix}1&0\\0&-1\end{pmatrix}. \tag{4}$$

The six tensor words $IIJ_2,\ IJ_2X,\ XJ_2Z,\ J_2IZ,\ J_2XX,\ J_2ZX$ give the $c_i$. They obey $c_ic_j+c_jc_i=-2\delta_{ij}I$. Take $\beta=i c_1c_2\cdots c_6$. It is Hermitian, purely imaginary and antisymmetric, and anticommutes with each $c_i$.

In the complete Killing frame, choose a constant real unit vector $v$ and put $\varepsilon=Uv$. The explicit stereographic frame $U=(I-x_ic_i)/\sqrt{1+|x|^2}$ is real. Thus, globally,

$$\begin{aligned} \mathscr D\varepsilon&=3\varepsilon,& \varepsilon^\dagger\varepsilon&=1,& \varepsilon^\dagger\beta\varepsilon&=0,\\ \varepsilon^\dagger c(X)\varepsilon&=0,& \varepsilon^\dagger c(X)\beta\varepsilon&=0. \end{aligned} \tag{5}$$

The bilinear identities follow from antisymmetry; the mixed matrix $c(X)\beta$ is also antisymmetric. They are properties of this particular spatial field, not imposed zeros for an observed particle coupling. Choosing this real family does not remove complex perturbations.

Let $\xi=\beta\varepsilon$, so $\mathscr D\xi=-3\xi$. Equation (3) has the exact solution

$$\boxed{\psi_0(t,q)=e^{-3it}\varepsilon(q)} \tag{6}$$

for every $g_0,g_1$. Its scalar current is zero and its ordinary norm is nonzero. Neither its Dirac sign nor its zero scalar current identifies an antiparticle, electric neutrality or an observed particle.

The complete response on an invariant spatial subspace

Put $\psi=e^{-3it}(\varepsilon+\eta)$. Since the background current vanishes, the complete linearization is

$$i\partial_t\eta=(\mathscr D-3)\eta +2\xi(g_0I+g_1\widehat P_6) \operatorname{Re}(\xi^\dagger\eta). \tag{7}$$

Take any real degree-$\ell$ scalar harmonic $u$, normalized by $\int u^2=1$, with $\ell\ge1$ and $\Lambda=\ell(\ell+5)$. Define

$$\chi_0=u\xi,\qquad \chi_1=c(du)\xi. \tag{8}$$

All physical derivatives of the spinor are included. The Killing equation and the scalar harmonic equation give

$$\begin{aligned} (\mathscr D-3)\chi_0&=-6\chi_0+\chi_1,\\ (\mathscr D-3)\chi_1&=\Lambda\chi_0-\chi_1. \end{aligned} \tag{9}$$

The bilinear identities (5) also give $\xi^\dagger\chi_1=0$ pointwise. For arbitrary complex $z_0,z_1$,

$$\operatorname{Re}\bigl[\xi^\dagger(z_0\chi_0+z_1\chi_1)\bigr] =u\,\operatorname{Re}z_0. \tag{10}$$

Thus the complex span of (8) is invariant under the full real-linear equation (7). It includes the spatial derivative partner and both phase quadratures. It is an invariant subspace of the full field problem, not a claim that all perturbations have this form.

Its physical Gram matrix is $\operatorname{diag}(1,\Lambda)$. In the orthonormal basis $(\chi_0,\chi_1/\sqrt{\Lambda})$, define

$$L_\ell=\begin{pmatrix}-6&\sqrt{\Lambda}\\ \sqrt{\Lambda}&-1\end{pmatrix},\qquad a_\ell=g_0+g_1p_\ell,\qquad E_{00}=\begin{pmatrix}1&0\\0&0\end{pmatrix}. \tag{11}$$

Writing the coefficient vector as $z=x+iy$, the equations are

$$\dot x=L_\ell y,\qquad \dot y=-(L_\ell+2a_\ell E_{00})x. \tag{12}$$

The quadratic canonical energy is $z^\dagger L_\ell z+2a_\ell x_0^2$. The response frequencies therefore satisfy

$$\boxed{ \Omega^4-T_\ell\Omega^2+D_\ell=0, } \tag{13}$$

where

$$\begin{aligned} T_\ell&=37+2\Lambda-12a_\ell,\\ D_\ell&=(\Lambda-6)(\Lambda-6+2a_\ell). \end{aligned} \tag{14}$$

These are eigenvalues of $L_\ell(L_\ell+2a_\ell E_{00})$. They are not $\Omega=p_\ell$. At $\ell=1$, one squared frequency is zero and the other is $49-12a_1$. At $\ell=0$, the derivative partner vanishes; that separate block has squared frequency $36-12g_0$ and a phase-space description of one complex amplitude.

High harmonics give an exact instability result for this branch

For $\ell\ge2$, exponential growth occurs if $D_\ell<0$, or if $T_\ell<0$, or if $T_\ell^2-4D_\ell<0$. A negative squared frequency gives real growth; nonreal squared frequencies give a complex quartet.

At large degree, with fixed finite $g_0$,

$$p_\ell\sim\frac{\ell^6}{720},\qquad T_\ell\sim-\frac{g_1}{60}\ell^6,\qquad D_\ell\sim\frac{g_1}{360}\ell^8. \tag{15}$$

If $g_1>0$, the trace eventually becomes negative. If $g_1<0$, the determinant eventually becomes negative. Therefore every nonzero $g_1$ gives an unstable harmonic of this exact background in the unrestricted continuum problem. The invariant subspace (8) is enough to establish instability of the full linearized field.

For $g_0=0$, exact rational arithmetic gives these first unstable degrees within the family:

These are dimensionless model benchmarks, not fitted particle indices. A finite cutoff can omit the unstable degrees, but then its physical origin and the discarded response must be justified. Other backgrounds, different kinetic operators or a derived nonlocal response require their own analysis; (15) is not a no-go theorem for all IDWT dynamics.

An exact nonzero local scalar branch has complete spatial spectral stability before the derivative-current term is added. Its full response has also been tested: the added term leaves that constant-current background unchanged but again destabilizes its high harmonics. Thus the source response matters even when a local self-consistent scalar coefficient is already present.

Restore the physical scale

Let the actual sphere have volume $\mathcal V$ and fix canonical norm $Q$. The background becomes $\sqrt{Q/\mathcal V}\,e^{-3it/R}\varepsilon$. If the dimensional interaction coefficients multiplying $J^2/2$ and $J\widehat P_6J/2$ are $\kappa$ and $\eta$, respectively, then

$$g_0=\frac{R\kappa Q}{\mathcal V},\qquad g_1=\frac{R\eta Q}{\mathcal V}. \tag{16}$$

Equations (11)–(15) use time $t/R$; physical response frequencies are $\Omega/R$. No radius, canonical norm, master coefficient or occupied particle mode is selected by this calculation.

Verification

conformal_current_response_checks.py checks the full real Clifford representation, Killing fields and all bilinear cancellations. Independent stereographic differentiation verifies (9) for zonal and angular harmonics, with arbitrary complex perturbation coefficients. The nonlinear current on this perturbation plane and its complete first variation are checked directly.

Exact matrix algebra derives (13)–(15). Independent variations of the canonical quadratic energy and the full four-real-component generator verify the frequencies and growth conditions. Exact rational scans give the table. The normalized scalar spectrum enters a specified nonlinear response; it has not been derived as the master mass operator.