Editing Openai/6897769e-4ee4-800f-aba5-69cca34f701c (section)

=== 1. Compute the one-loop four-photon kernel (box diagram) for a single fermion species, extract the low-momentum expansion Kμνρσ(k)\mathcal{K}^{\mu\nu\rho\sigma}(k)Kμνρσ(k). This gives the precise coefficient (no guesswork) in front of quartic photon terms. ===
# Perform Hubbard–Stratonovich transform on the induced quartic term, introduce hμνh_{\mu\nu}hμν, integrate out photons to second order and obtain Dμνρσ(k)\mathcal{D}^{\mu\nu\rho\sigma}(k)Dμνρσ(k) explicitly.
# Extract Gμν,ρσ(k)\mathcal{G}_{\mu\nu,\rho\sigma}(k)Gμν,ρσ(k) and project on spin-2 part; verify whether it has a simple pole ∝1/k2\propto 1/k^2∝1/k2 in the IR.
# Estimate numerically the effective GeffG_{\rm eff}Geff produced by a single boundary cell (one atom shell). Then model amplification by NNN coherent cells: Geff(N)∼NαGeff(1)G_{\rm eff}^{(N)} \sim N^\alpha G_{\rm eff}^{(1)}Geff(N)∼NαGeff(1) and explore what α\alphaα is required to reach observed GGG.
# Nonrelativistic limit: from amplitude between static currents, compute the effective potential V(r)V(r)V(r) and check whether it gives 1/r1/r1/r (Newtonian) or another power law; study the role of coherence in modifying the power law.