Conformal Field Theory — Free Boson Correlator
Two-point functions, operator product expansion, and conformal Ward identities
Conformal field theories (CFTs) are quantum field theories invariant under conformal transformations (angle-preserving maps). In 2D, the conformal group is infinite-dimensional — all holomorphic functions. Primary operators O with conformal dimension h transform as O → (∂z'/∂z)^h O. The free boson has h=0 but its derivative ∂φ has h=1. The two-point function is completely fixed by conformal symmetry: ⟨O(z₁)O(z₂)⟩ = C / |z₁−z₂|^(4h). The visualization shows the magnitude of the 2-point correlator as a heat map: it diverges as the two operators approach each other (OPE singularity) and falls off as a power law. The three-point function adds a third operator; its coefficient C₁₂₃ encodes the operator product expansion (OPE) data that defines the CFT.