Green function marginal

Physical scope

This static operator benchmark specifies the source normalization and distinguishes integration from restriction. Its physical use requires a specified electromagnetic action and observer response.

All coordinates here are ordinary macroscopic spatial coordinates in a flat model at finite depth within the one nested physical manifold. Writing $\mathbb R^d=\mathbb R^m\oplus\mathbb R^{d-m}$ chooses coordinates relative to the smaller submanifold; it gives the containing particle no inherent division into different kinds of dimensions.

Normalized identity

For integers $d>m>2$, set $p=d-m$, $r=|x|>0$, and

$$G_d(z)=\frac{\Gamma(d/2-1)}{4\pi^{d/2}}|z|^{2-d}, \qquad -\Delta_dG_d=\delta_d.$$

The coefficient follows from unit source flux. For $G_d(r)=c_dr^{2-d}$, integrating $-\Delta_dG_d$ over a ball gives $-\Omega_{d-1}r^{d-1}G_d'(r)=c_d(d-2)\Omega_{d-1}=1$. Using the unit-sphere area below and $\Gamma(d/2)=(d/2-1)\Gamma(d/2-1)$ gives the displayed $c_d$.

Then, with the full Euclidean fibre measure,

$$\boxed{\int_{\mathbb R^p}G_d(x,y)\,d^py=G_m(x).}$$

Indeed, polar coordinates and $u=s^2/r^2$ give

$$\begin{aligned} I_{d,m}(r) &=\Omega_{p-1}\int_0^\infty s^{p-1}(r^2+s^2)^{-(d-2)/2}\,ds\\ &=\pi^{p/2}\frac{\Gamma((m-2)/2)}{\Gamma((d-2)/2)}r^{2-m}, \qquad \Omega_{p-1}=\frac{2\pi^{p/2}}{\Gamma(p/2)}. \end{aligned}$$

The integral uses Euler's beta integral, DLMF §5.12. Multiplying by the coefficient in $G_d$ proves the boxed identity. For $d=m$ there is no fibre integration and the assertion is immediate. The equality also holds as locally integrable distributions in $x$; it does not assign a finite value to the point-source singularity at $x=0$.

At $m=3$, the unnormalized integral is $C_d/r$, with

$$C_d=\frac{\pi^{(d-2)/2}}{\Gamma((d-2)/2)}, \qquad (C_3,C_4,C_5,C_6,C_{10}) =\left(1,\pi,2\pi,\pi^2,\frac{\pi^4}{6}\right).$$

The normalized marginal is always $1/(4\pi r)$. Physical charges additionally depend on source couplings and kinetic normalization. The $1/r$ quantity is a potential; its radial derivative gives a $1/r^2$ force.

What changes the conclusion

For a normalized finite box weight, the weighted integral tends to zero as the box grows at fixed $r$; it is not the unweighted marginal. No probability normalization factor may be silently cancelled.

The radial integrand at large $s$ scales as $s^{1-m}$. For $m=2$ the potential integral diverges logarithmically, and for $m<2$ it diverges more strongly. A two-dimensional Green function requires its own additive-constant or reference prescription. The photon's bound depth two is not automatically the dimension of its static propagator.

Curved metrics, finite boundaries, fibre weights, and dynamical currents require a new calculation. In particular, observer inaccessibility alone does not specify which operation in the table is physically realized. The theorem permits the full marginal of the ordinary flat kernel to pass the static Coulomb test; it neither selects that kernel nor proves the complete field theory.

Verification and related

interaction_operator_checks.py checks independent radial quadratures, normalization, the force derivative, and finite-aperture counterexamples.

• Interaction operator tools — locates this projection benchmark among the interaction tools.
• Nested sector dynamics — inclusion alone supplies neither a global product nor a compatible projection of dynamics.