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

=== Assistant: Good — here’s a compact, copy-ready brief you can paste straight into Copitor. ===
Good — here’s a compact, copy-ready brief you can paste straight into Copitor. It trims the long derivation to the essential equations, definitions, and next computational tasks so the file is small but still fully actionable.

QAT → Compact QFT brief for Copitor
(Hand this to your checker: minimal, explicit, ready for algebra / numerics.)

Goal: show how repeated photon–electron absorption→emission cycles can produce an effective massless spin-2 propagator ∝1/k2\propto 1/k^2∝1/k2 (an emergent graviton-like field) and estimate the effective gravitational coupling GeffG_{\rm eff}Geff from coherent boundary/shell amplification.

===== QED action (flat spacetime): =====

S[ψˉ,ψ,A]  =  ∫d4x  [ψˉ(iγμDμ−m)ψ−14FμνFμν],Dμ=∂μ+ieAμ.S[\bar\psi,\psi,A] \;=\; \int d^4x\; \Big[\bar\psi(i\gamma^\mu D_\mu - m)\psi - \tfrac{1}{4}F_{\mu\nu}F^{\mu\nu}\Big],\qquad D_\mu=\partial_\mu + i e A_\mu.S[ψˉ,ψ,A]=∫d4x[ψˉ(iγμDμ−m)ψ−41FμνFμν],Dμ=∂μ+ieAμ.
Integrate out fermions:

e iSeff[A]=∫DψˉDψ  e iS[ψˉ,ψ,A]=e−i4∫F2 det⁡(i\slashedD−m)−1.e^{\,iS_{\rm eff}[A]} = \int\mathcal{D}\bar\psi\mathcal{D}\psi\; e^{\,iS[\bar\psi,\psi,A]}
= e^{-\tfrac{i}{4}\int F^2}\,\det(i\slashed{D}-m)^{-1}.eiSeff[A]=∫DψˉDψeiS[ψˉ,ψ,A]=e−4i∫F2det(i\slashedD−m)−1.
Expand fermion determinant in powers of AAA:

−iln⁡det⁡(i\slashed∂−m+e\slashedA)=∑n≥2(−1)n+1nen Tr[(S0\slashedA)n].-i\ln\det(i\slashed{\partial}-m + e\slashed{A}) 
= \sum_{n\ge2} \frac{(-1)^{n+1}}{n} e^n \,\mathrm{Tr}[(S_0\slashed{A})^n].−ilndet(i\slashed∂−m+e\slashedA)=n≥2∑n(−1)n+1enTr[(S0\slashedA)n].
Important terms: n=2n=2n=2 vacuum polarization Πμν\Pi^{\mu\nu}Πμν, n=4n=4n=4 box kernel Kμνρσ\mathcal{K}^{\mu\nu\rho\sigma}Kμνρσ.

===== Define renormalized composite: =====

hμν(x)≡N(AμAν−14ημνAαAα)(x).h_{\mu\nu}(x) \equiv \mathcal{N}\Big(A_\mu A_\nu - \tfrac14 \eta_{\mu\nu} A_\alpha A^\alpha\Big)(x).hμν(x)≡N(AμAν−41ημνAαAα)(x).
Target object:

Gμν,ρσ(k)≡⟨hμν(−k) hρσ(k)⟩conn.\mathcal{G}_{\mu\nu,\rho\sigma}(k) \equiv \langle h_{\mu\nu}(-k)\,h_{\rho\sigma}(k)\rangle_{\rm conn}.Gμν,ρσ(k)≡⟨hμν(−k)hρσ(k)⟩conn.
Show in IR (small kkk) that

Gμν,ρσ(k)  ≈  16πGeffk2  Πμν,ρσ(2)(k)  +  regular,\mathcal{G}_{\mu\nu,\rho\sigma}(k) \;\approx\; \frac{16\pi G_{\rm eff}}{k^2}\; \Pi^{(2)}_{\mu\nu,\rho\sigma}(k) \;+\; \text{regular},Gμν,ρσ(k)≈k216πGeffΠμν,ρσ(2)(k)+regular,
with Π(2)\Pi^{(2)}Π(2) the spin-2 transverse-traceless projector.

===== Connected part of ⟨A4⟩\langle A^4 \rangle⟨A4⟩ arises from box diagrams (fermion loop) and dressed photon lines. Box kernel (symbolic): =====

Kμνρσ(k1,…,k4)=(−ie)4∫ ⁣d4p(2π)4Tr[S(p)γμS(p+k1)γνS(p+k1+k2)γρS(p−k4)γσ].\mathcal{K}^{\mu\nu\rho\sigma}(k_1,\dots,k_4)
= (-ie)^4 \int\!\frac{d^4p}{(2\pi)^4}
\mathrm{Tr}\big[S(p)\gamma^\mu S(p+k_1)\gamma^\nu S(p+k_1+k_2)\gamma^\rho S(p-k_4)\gamma^\sigma\big].Kμνρσ(k1,…,k4)=(−ie)4∫(2π)4d4pTr[S(p)γμS(p+k1)γνS(p+k1+k2)γρS(p−k4)γσ].
Low-momentum expansion ki≪mk_i\ll mki≪m yields local quartic photon effective terms.

Hubbard-Stratonovich step (auxiliary field hhh):

exp⁡(iλ2∫(AA−14ηA2)2)=∫Dhexp⁡(i∫[−12λh2+h⋅(AA−14ηA2)]).\exp\left(i\frac{\lambda}{2}\int (AA-\tfrac14\eta A^2)^2\right)
= \int\mathcal{D}h \exp\Big(i\int\big[-\tfrac{1}{2\lambda}h^2 + h\cdot(AA-\tfrac14\eta A^2)\big]\Big).exp(i2λ∫(AA−41ηA2)2)=∫Dhexp(i∫[−2λ1h2+h⋅(AA−41ηA2)]).
Integrate out AAA to obtain an induced quadratic action for hhh: Sind[h]∼12∫h D hS_{\rm ind}[h] \sim \tfrac12\int h\,\mathcal{D}\,hSind[h]∼21∫hDh.

===== Symbolic result in IR: =====

G(k)  ∝  κ~k2  Π(2)(k),κ~∼e4m2 C,\mathcal{G}(k)\;\propto\;\frac{\tilde\kappa}{k^2}\;\Pi^{(2)}(k),\qquad \tilde\kappa\sim \frac{e^4}{m^2}\,\mathcal{C},G(k)∝k2κ~Π(2)(k),κ~∼m2e4C,
so identify

Geff∼e416πm2 C.G_{\rm eff}\sim \frac{e^4}{16\pi m^2}\,\mathcal{C}.Geff∼16πm2e4C.
Note: a single-cell estimate gives an extremely small GeffG_{\rm eff}Geff; QAT therefore invokes coherence / geometric amplification (large number NNN of coherent shells or modes) to reach observed GGG.

===== 1. Compute the one-loop box kernel Kμνρσ(ki)\mathcal{K}^{\mu\nu\rho\sigma}(k_i)Kμνρσ(ki). Extract the low-momentum expansion coefficients (keep leading k0k^0k0 and k2k^2k2 terms). =====
# Derive the induced quartic photon term from the determinant and write it as λ (AA−14ηA2)2+⋯\lambda\, (AA - \tfrac14\eta A^2)^2 + \cdotsλ(AA−41ηA2)2+⋯. Provide explicit λ\lambdaλ as function of e,me,me,m.
# Perform the Hubbard–Stratonovich transform to introduce hμνh_{\mu\nu}hμν; integrate out photons at one loop to get Dμνρσ(k)\mathcal{D}^{\mu\nu\rho\sigma}(k)Dμνρσ(k) (inverse propagator for hhh).
# Compute Gμν,ρσ(k)\mathcal{G}_{\mu\nu,\rho\sigma}(k)Gμν,ρσ(k) and project on spin-2 sector: Πμν,ρσ(2)(k)=12(PμρPνσ+PμσPνρ)−13PμνPρσ,Pμν=ημν−kμkνk2.\Pi^{(2)}_{\mu\nu,\rho\sigma}(k)=\tfrac12(P_{\mu\rho}P_{\nu\sigma}+P_{\mu\sigma}P_{\nu\rho})-\tfrac13P_{\mu\nu}P_{\rho\sigma},\quad P_{\mu\nu}=\eta_{\mu\nu}-\frac{k_\mu k_\nu}{k^2}.Πμν,ρσ(2)(k)=21(PμρPνσ+PμσPνρ)−31PμνPρσ,Pμν=ημν−k2kμkν. Check for a 1/k21/k^21/k2 pole and extract coefficient κ~\tilde\kappaκ~.
# Numerics: evaluate κ~\tilde\kappaκ~ for electron mass mem_eme; compute required amplification NαN^\alphaNα (coherent cell count scaling) so that Geff(N)G_{\rm eff}^{(N)}Geff(N) matches measured GGG. Provide sensitivity analysis.

===== • Coherence / phase averaging of many photon emissions on the shell (gives constructive addition). =====
• Low-momentum (IR) regime dominated by box kernel structure.
• Composite operator hμνh_{\mu\nu}hμν renormalized (subtractions trace parts).
• Work first in flat space; later include boundary curvature / Israel junctions if needed.

===== Restore factors: replace e→e/4πε0ℏce\to e/\sqrt{4\pi\varepsilon_0\hbar c}e→e/4πε0ℏc etc., and include ℏ,c\hbar,cℏ,c in propagators. We used natural units above for algebraic clarity. =====

End of brief.

If you want, I can now produce two follow-ups you can also paste into Copitor:
A) a tiny worked-example skeleton: evaluate the box in the static limit and show the leading coefficient, or
B) a one-page “amplification model” describing how many coherent boundary modes (and what coherence exponent α\alphaα) would be needed to reach Newton’s GGG.

Which of A or B would you like me to send next (short and compact)?