AdS/CFT — Holographic Dictionary

Maldacena 1997  |  Zbulk[φ₀] = ⟨e∫φ₀OCFT
bulk field φ(z,x) CFT operator O(x)
AdS radial z RG scale μ
bulk mass m²L² scaling dim Δ=d/2+√(d²/4+m²L²)
black hole horizon thermal state TH
RT surface area entanglement entropy S
AdS/CFT correspondence (Maldacena 1997) equates a gravitational theory in (d+1)-dimensional anti-de Sitter space to a conformal field theory on the d-dimensional boundary. The bulk-to-boundary propagator K(z,x;x') = (z/(z²+|x-x'|²))^Δ encodes how boundary sources excite bulk fields. The Ryu-Takayanagi formula (2006) identifies the entanglement entropy of a boundary region A with the area of the minimal bulk geodesic (in 2+1D: length) homologous to A: S(A) = Area(γ_A)/(4G_N). This visualizes the Poincaré patch of AdS₂ with the boundary at z→0, showing bulk-to-boundary propagators and RT geodesics as semicircles.