Angular kernel parity

The valid integral-operator statement

On the ordinary spatial unit sphere $S^{d-1}$, $d\geq2$, use probability-normalized measure $d\sigma$ and define

$$(Tf)(x)=\int_{S^{d-1}}(x\cdot y)^2 f(y)\,d\sigma(y).$$

If $f(-y)=-f(y)$, changing $y$ to $-y$ proves $Tf=0$. In fact the complete nonzero spectrum consists of

$$T1=\frac1d,\qquad T(y^TAy)(x)=\frac{2}{d(d+2)}x^TAx \quad(A=A^T,\ \operatorname{tr}A=0).$$

This follows from rotational invariance and

$$\int y_iy_jy_ky_l\,d\sigma =\frac{\delta_{ij}\delta_{kl}+\delta_{ik}\delta_{jl} +\delta_{il}\delta_{jk}}{d(d+2)}.$$

Rotational invariance and index symmetry force the numerator's three delta pairings to have one common coefficient. Contracting $i=j$ and $k=l$ gives $\int|y|^4d\sigma=1$, fixing that coefficient to $1/[d(d+2)]$. Contracting with the traceless matrix $A$ gives the quadratic eigenvalue.

The range contains only constants and traceless quadratics. The corresponding eigenvalues have multiplicities $1$ and $(d-1)(d+2)/2$; all other spherical harmonics, including even degrees four and higher, lie in the kernel. The trace is $1/d+[(d-1)(d+2)/2],2/[d(d+2)]=1$.

Why a density interaction is different

In a quartic density interaction, the mean-field operation is

$$U_\rho(x)=g(T\rho)(x),\qquad \psi(x)\longmapsto U_\rho(x)\psi(x),$$

rather than $\psi\mapsto gT\psi$. Squaring an odd wave function produces an even density, which can source $U_\rho$.

For an exact example take $\psi(x)=\sqrt d\,x_1$, so $\int|\psi|^2d\sigma=1$. Then

$$\rho=d x_1^2,\qquad U_\rho(x)=\frac{g}{d+2}(1+2x_1^2),\qquad \langle\psi|U_\rho|\psi\rangle =\frac{g(d+8)}{(d+2)^2}>0 \quad(g>0).$$

Thus an even kernel gives a nonzero mean-field shift to an odd state in this explicit scalar model. An even multiplication operator preserves parity; it need not annihilate the odd subspace.

Application to projected fields

Using this angular nullspace for a projected field requires an explicit identification of that field with the scalar angular functions. Physical spin-one polarization has its own representation. A mass-protection argument using the nullspace must establish that the full projected action and its corrections preserve the same structure.

The photon module instead keeps $P_\gamma\mathcal M P_\gamma=0$ as the condition on the full mass operator. The $n=0$ counting label alone does not prove it.

Gauge invariance supplies a different constraint. For a consistently gauged, anomaly-free quantum theory with a symmetry-preserving prescription, $q_\mu\Pi^{\mu\nu}=0$ permits nonzero transverse vacuum polarization. It does not require $\Pi^{\mu\nu}=0$ or a constant measured coupling. A regular transverse coefficient leaves the usual massless pole; singular terms or symmetry-breaking backgrounds require additional analysis. See Wingate's QED lectures, §6.2 for an explicit nonzero vacuum-polarization calculation.

In a simultaneous matter-and-connection gauge transformation the covariant kinetic term is invariant: the matter phase variation and connection variation cancel. On-shell current conservation and the quantum Ward identities require their respective field and measure assumptions.

Verification and related

interaction_operator_checks.py checks the sphere spectrum and the odd-state mean-field example by independent circle and sphere quadratures, plus the nonzero transverse tensor.

• Interaction operator tools — summarizes the angular result and its application.
• Projected bilocal variation — derives the actual mean-field operation from a density action.