Anonymous
11/9/2025, 2:05:19 PM
No.16841405
>>16841353
If you actually couple [math]\delta A[/math] to a "gravitational potential" [math]\Phi_g[/math] via a Lagrangian term [math]\lambda,\delta A\cdot \Phi_g[/math], you either (i) break [math]U(1)[/math] gauge symmetry (Proca‑like, ruining AB holonomy and contradicting precision bounds on photon mass), or (ii) introduce a Stueckelberg field [math]\chi[/math] with [math]A\mapsto A+d\chi[/math], [math]\chi\mapsto \chi-\sigma[/math] to keep gauge invariance, in which case the observable is a massive scalar mixed with [math]\delta A[/math], not gravity, and the AB sector is unaffected (monodromy remains a [math]\pi_1[/math]-character). None of this produces a coordinate‑independent scalar equal to [math]\partial_\mu A^\mu[/math]. In the weak‑field Newtonian limit, gravity is encoded by the metric perturbation [math]h_{00}=-2\Phi_g/c^2[/math], i.e. a component of a symmetric 2‑tensor (a section of [math]S^2 T^*X[/math]), living in a completely different functorial world than [math]A\in \Omega^1(X)[/math]; there is no natural transformation of sheaves sending a gauge‑equivalence class of [math]A[/math] to a diffeomorphism‑invariant scalar [math]\Phi_g[/math].
If you actually couple [math]\delta A[/math] to a "gravitational potential" [math]\Phi_g[/math] via a Lagrangian term [math]\lambda,\delta A\cdot \Phi_g[/math], you either (i) break [math]U(1)[/math] gauge symmetry (Proca‑like, ruining AB holonomy and contradicting precision bounds on photon mass), or (ii) introduce a Stueckelberg field [math]\chi[/math] with [math]A\mapsto A+d\chi[/math], [math]\chi\mapsto \chi-\sigma[/math] to keep gauge invariance, in which case the observable is a massive scalar mixed with [math]\delta A[/math], not gravity, and the AB sector is unaffected (monodromy remains a [math]\pi_1[/math]-character). None of this produces a coordinate‑independent scalar equal to [math]\partial_\mu A^\mu[/math]. In the weak‑field Newtonian limit, gravity is encoded by the metric perturbation [math]h_{00}=-2\Phi_g/c^2[/math], i.e. a component of a symmetric 2‑tensor (a section of [math]S^2 T^*X[/math]), living in a completely different functorial world than [math]A\in \Omega^1(X)[/math]; there is no natural transformation of sheaves sending a gauge‑equivalence class of [math]A[/math] to a diffeomorphism‑invariant scalar [math]\Phi_g[/math].