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 Incoming links: Scalar-primary two-point function

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 354 / 1 / b / i / Solution 2026-09-28

At spacelike separation

x_{12}^{2} > 0

, conformal symmetry fixes the scalar-primary two-point function to

⟨ O (x_{1}) O (x_{2})⟩ = \frac{C _{O}}{( x _{12}^{2} ) ^{Δ}} .

Applying the d'Alembertian at

x_{2}

gives

F_{0} = \frac{2 C _{O} Δ ( 2Δ + 2 - d )}{( x _{12}^{2} ) ^{Δ + 1}}

away from contact terms. Thus, up to a real multiplicative constant,

F_{0} \propto (x_{12}^{2})^{- Δ - 1}

. The scalar conformal unitarity bound makes the coefficient nonnegative and makes it vanish when the bound is saturated, as expected for the null free-field equation of motion.

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