Transfer of the simplex operator to spatial spinors

Summary: A complete Killing-spinor frame transfers the scalar simplex operator to a local spinor operator, but its two Dirac branches and the real action distinguish this spectrum from an algebraic scalar mass.

An exact transfer, with the mass question kept explicit

The critical conformal scalar operator has normalized eigenvalues $S(\ell,d)=\binom{\ell+d-1}{d}$. It can be transferred to the full ordinary spinor bundle using Killing spinors. The transferred operator is local and self-adjoint, with coefficients fixed by the specified geometry.

Its direct insertion into the scalar mass position of the canonical Dirac Hamiltonian is not self-adjoint. Taking the real action changes the coefficient to the average of adjacent simplex values. Both facts follow from retaining the two Dirac signs and the full variation.

Its squared Dirac polynomial has the same simplex value on both signs and admits a Hermitian scalar insertion. The obstruction proved here concerns the stated round-sphere transfer, rather than all spatial spinor operators.

Throughout, $S_R^d$ is an assumed actual physical spatial geometry, $d=2h$, with undetermined radius $R$. All $d$ directions are ordinary macroscopic spatial directions. The calculation neither compactifies dimensions nor selects physical topology. Spinor components and a mathematical frame do not add spatial dimensions.

The complete Killing frame

Use skew-Hermitian Clifford multiplication and the self-adjoint spatial Dirac operator:

$$c(v)c(w)+c(w)c(v)=-2g(v,w)I,\qquad \mathscr D=c(e_a)\nabla_{e_a}. \tag{1}$$

Let $\beta$ be the parallel Hermitian chirality operator, with $\beta^2=I$ and $\{\beta,\mathscr D\}=0$. These are the canonical Hamiltonian conventions of the spatial mass reduction. Neither chirality nor a Dirac sign is identified with electric charge or an antiparticle.

A positive Killing spinor satisfies

$$\nabla_X\varepsilon=-\frac1{2R}c(X)\varepsilon,\qquad \mathscr D\varepsilon=\frac hR\varepsilon. \tag{2}$$

There are $r=2^h$ independent such spinors on the round sphere. Their pairwise inner products are constant, because differentiation gives two cancelling terms by skew-Hermiticity. Orthonormal values at one point therefore give a pointwise unitary frame $U$ of the entire spinor bundle.

The metric connection

$$\nabla^K_X=\nabla_X+\frac1{2R}c(X) \tag{3}$$

has these columns as parallel sections and has zero curvature on the round sphere. It is not the ordinary Clifford-compatible spin connection: the extra term in its Clifford compatibility equation is $[c(X),c(Y)]/(2R)$.

The scalar/Killing-spinor construction and its multiplicity correction are also given in Herzog and Shamir, Section 6.2, equations (6.12)–(6.14). Only their round-sphere spectral construction is used here.

Derive the full differential transfer

Let $\Delta$ be the nonnegative scalar Laplacian. For every smooth complex scalar function $f$, the Clifford product rule gives

$$\begin{aligned} \mathscr D(f\varepsilon) &=c(df)\varepsilon+\frac hR f\varepsilon,\\ \mathscr D(c(df)\varepsilon) &=(\Delta f)\varepsilon+\frac{1-h}{R}c(df)\varepsilon. \end{aligned} \tag{4}$$

The symmetric Hessian contracts to $\Delta f$ in the second line; $\sum_a c(e_a)c(df)c(e_a)=(d-2)c(df)$ supplies its connection term. Applying $\mathscr D$ again in the first line proves

$$\boxed{ L_K(f\varepsilon)=(\Delta f)\varepsilon,\qquad L_K=\mathscr D^2-\frac1R\mathscr D-\frac{h(h-1)}{R^2}. } \tag{5}$$

Equivalently, $L_K=(\nabla^K)^\dagger\nabla^K$. For a full coefficient vector $F=(f_1,\ldots,f_r)$ and $\psi=UF$, this gives $L_KU=U(\Delta\otimes I_r)$. All complex polarizations are retained.

Put $A=R\mathscr D$ and define

$$Q_d(A)=\prod_{j=1}^h(A-j)(A+j-1),\qquad \widehat Q_d=\frac{Q_d(A)}{d!}. \tag{6}$$

Every transferred scalar factor becomes

$$R^2L_K+(h-j)(h+j-1) =A^2-A-j(j-1)=(A-j)(A+j-1). \tag{7}$$

Thus the normalized critical scalar operator $\widehat P_d=R^dP_{d,R}/d!$ obeys

$$\boxed{\widehat Q_d\,U=U(\widehat P_d\otimes I_r).} \tag{8}$$

This is an exact unitary transfer, not a projection onto one spinor line. In particular,

$$\begin{aligned} Q_6(A)&=(A-3)A(A^2-1)(A^2-4),\\ Q_{10}(A)&=(A-5)A(A^2-1)(A^2-4)(A^2-9)(A^2-16). \end{aligned} \tag{9}$$

The factor $C_{d-1}(A)=A\prod_{j=1}^{h-1}(A^2-j^2)$ is the standard odd conformal Dirac power of Clerc and Ørsted, Theorem 7.1. The extra factor $A-h$ in $Q_d=(A-h)C_{d-1}$ is essential. The transferred operator uses the Killing connection; it is not the standard critical spinorial conformal intertwiner with its half-integer shift.

Both spectral branches and their multiplicities

For a degree-$\ell$ scalar harmonic, let $\Lambda=\ell(\ell+d-1)$. On the formal pair $(f\varepsilon,Rc(df)\varepsilon)$, (4) reads

$$A=\begin{pmatrix}h&\Lambda\\1&1-h\end{pmatrix}. \tag{10}$$

Its eigenvalues are $\ell+h$ and $-(\ell+h-1)$. At $\ell=0$, the derivative partner vanishes and only the positive Killing eigenvalue remains. For higher degrees, both branches have the same value $S(\ell,d)$ under $\widehat Q_d$.

Using instead the standard Dirac level $k=0,1,\ldots$, the two signs give

$$\boxed{ \begin{array}{ll} A=+(k+h):&\widehat Q_d=S(k,d),\\ A=-(k+h):&\widehat Q_d=S(k+1,d), \end{array}} \tag{11}$$

where $S(0,d)=0$. The polynomial is symmetric under $A\mapsto1-A$, not under $A\mapsto-A$. At fixed $\ell\ge1$, the equal-value spaces are positive level $\ell$ and negative level $\ell-1$. Their dimension is

$$r\binom{\ell+d-1}{d-1}+r\binom{\ell+d-2}{d-1} =r\left[\binom{\ell+d}{d}-\binom{\ell+d-2}{d}\right]. \tag{12}$$

The bracket is the scalar harmonic multiplicity, in agreement with the complete frame (8). The zero eigenspace is the rank-$r$ positive Killing space. No modes have been discarded or counted twice. Identifying a Dirac level with an occupied particle mode remains an additional physical requirement.

What the real mass action actually varies into

For self-adjoint $M$, the scalar Hamiltonian contribution $\beta M$ is self-adjoint only if $[M,\beta]=0$. Here

$$\beta Q_d(A)\beta=Q_d(-A)\ne Q_d(A). \tag{13}$$

Hence $\mathscr D+\mu\beta\widehat Q_d$, for real nonzero $\mu$, does not preserve the canonical norm. This is a failure of that mass insertion, not of the self-adjoint operator $\widehat Q_d$ itself.

Taking the real quadratic energy and varying every complex component gives instead

$$\frac{\delta\,\operatorname{Re}\langle\psi,\mu\beta\widehat Q_d\psi\rangle} {\delta\psi^\dagger} =\frac\mu2(\beta\widehat Q_d+\widehat Q_d\beta)\psi =\mu\beta\widehat Q_d^{\,\mathrm{even}}\psi, \tag{14}$$

where

$$\boxed{ \widehat Q_d^{\,\mathrm{even}} =\frac{A^2}{d!}\prod_{j=1}^{h-1}(A^2-j^2). } \tag{15}$$

Its coefficient on either sign of Dirac level $k$ is $[S(k,d)+S(k+1,d)]/2$.

The change depends on the level, so a common scale does not undo it. Adding $\mu\widehat Q_d$ directly to the Hamiltonian is self-adjoint, but gives frequencies $(k+h)/R+\mu S(k,d)$ and $-(k+h)/R+\mu S(k+1,d)$. That is a different equation from a scalar Dirac mass insertion.

No finite polynomial in $A^2$ agrees with $Q_d(A)$ at every positive Dirac level: infinitely many equal values would force a polynomial identity, whereas $Q_d$ has a nonzero odd part. This limitation does not classify all geometric operators or nonlinear effective actions.

A nonlocal spectral candidate and the kinetic term

The spectral operator

$$\widehat M_d=\frac1{d!}Q_d(|A|) \tag{16}$$

is positive, self-adjoint and commutes with $\mathscr D$ and $\beta$. It has value $S(k,d)$ on both signs of level $k$. The factor $|A|=\sqrt{A^2}$ makes it spatially nonlocal. No elimination of master-field modes has yet been shown to produce the complete operator.

An explicit normal Dirac extension does obtain the absolute Dirac operator from ordinary spatial continuation of a full parent spinor. The same reduction induces a nontrivial kinetic norm: a constant bulk scalar coefficient remains constant after canonical normalization. The assumed half-cylinder boundary and nonlinear normal profiles remain to be derived from the master field.

Even this specified candidate retains the spatial kinetic contribution. The canonical Hamiltonian $\mathscr D+\mu\beta\widehat M_d$ gives

$$\boxed{ \omega_{k,\pm} =\pm\sqrt{\frac{(k+h)^2}{R^2}+\mu^2S(k,d)^2}. } \tag{17}$$

Reading the second term alone as an observed rest mass requires a physical separation and state identification that have not been derived.

The local scalar interaction survives the change of frame

A unitary change of field variables does not turn the written local quartic into a high-order differential interaction. Under $\psi=UF$,

$$J=\psi^\dagger\beta\psi=F^\dagger B(q)F,\qquad B(q)=U(q)^\dagger\beta U(q). \tag{18}$$

At unit radius, in stereographic coordinates, put $C=x_ic_i$, $s=|x|^2$, with constant Clifford matrices $c_i$. An explicit frame is

$$U(x)=\frac{I-C}{\sqrt{1+s}},\qquad B(x)=\frac{(1-s)I+2C}{1+s}\,\beta. \tag{19}$$

It has $U^\dagger U=I$, $B^2=I$, and its columns satisfy (2). The current retains a spatially varying polarization matrix even when $F$ is constant. Replacing it by $F^\dagger F$ would change the physical interaction.

The local energy $\kappa\int J^2/2$ therefore still varies into $\kappa JBF$ in this frame. It does not become $\widehat P_dF$. The physical derivative calculation likewise shows that a change of representation does not create an interaction eigenvalue.

Equation (8) closes a mathematical scalar-to-spinor construction. An admissible mass term inherited from the master action, occupied states and common sector normalization remain separate requirements.

Verification

killing_spinor_simplex_checks.py uses (19) and independently differentiates the full stereographic Dirac operator twice. Arbitrary complex spinor values, gradients and symmetric Hessians verify (5) in dimensions two, four, six and ten. That spatial check does not use a scalar eigenvalue formula.

Exact symbolic matrix products check (6)–(12). Separate complex variations verify the real mass action, norm change of the direct insertion, adjacent-count average and kinetic identity (17). These checks establish the specified mathematical constructions, not a physical mass mechanism.