Optical Absorption Measures Interaction-Dressed Quantum Metric and Berry Curvature — E8 Intelligence Research

FINDING: Optical absorption directly measures the interaction-dressed quantum metric and Berry curvature in solids, unifying band geometry with many-body physics. | MATH: Berry curvature Ω_n(k) = i⟨∇_k u_n| × |∇_k u_n⟩; quantum metric g_μν(k) = Re⟨∂_μ u_n|(1−|u_n⟩⟨u_n|)|∂_ν u_n⟩; optical conductivity σ_μν(ω) ∝ ∫ d²k Σ_n g_μν^n(k) δ(ℏω − E_n(k)) + Ω-term (circular dichroism); dressed quantities: Ω̃ = Ω + Σ_interaction corrections via vertex functions Γ. | CONNECTION: g_μν is the real part of the quantum geometric tensor Q_μν = g_μν + (i/2)Ω_μν — a Hermitian metric on the Bloch bundle. This is a Kähler-type structure: the imaginary antisymmetric part (Berry curvature) plays the role of a symplectic form, the real symmetric part (quantum metric) the Riemannian metric. Ratio |Ω|/√det(g) defines a local "geometric phase density" — in ideal two-band systems this saturates at values tied to the Chern number C = (1/2π)∫Ω d²k. For Dirac cones (graphene, topological insulators), g and Ω are isot Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com