Operator product expansion and convergence

For two scalar primaries, an exchanged scalar contributes a distance power determined by its dimension minus the two external dimensions. Derivatives of that exchanged primary supply further powers and angular dependence. Their relative coefficients follow from conformal symmetry once the primary coefficient is given. Spinning exchanges require tensor structures as well. Keeping only primary terms generally fails to reproduce the full correlator, because each primary represents an entire family of descendants.

In a normalized identical-scalar OPE the identity coefficient is fixed by the two-point function. Other primary coefficients are related to normalized three-point data. Signs of individual real OPE coefficients can depend on operator sign conventions; positivity becomes a statement about suitable squared coefficients in reflection-positive identical-operator four-point functions. An OPE coefficient is not a classical probability of a microscopic particle reaction, even when its square enters a positive spectral expansion.

Center radial quantization at one of the two fused insertions. If their separation is smaller than the distance from that center to every remaining insertion, a sphere can enclose the pair while leaving all others outside. The inside path integral prepares a state, and expanding it in a complete dilatation eigenbasis is precisely the OPE. The outside path integral supplies a bra. Hilbert-space convergence then controls the overlap with that external configuration.

This radius test is sufficient, not a universal maximal domain. A different conformal frame can sometimes supply a better separating sphere. Approaching the boundary of a chosen expansion can make convergence slow, and a collision with another operator creates a physical singularity. Interchanging limits, derivatives and infinite sums requires appropriate control. The theorem concerns full sums under its assumptions, whereas a practical finite list of primaries or descendant levels introduces a separate approximation.

Radial propagation suppresses large scaling dimensions by exponential factors at fixed separated geometry. Degeneracy and coefficient growth also contribute, so a bare geometric power cannot automatically be called the full remainder. Rigorous convergence results combine positivity and spectral estimates to control tails under stated conditions. Near the edge of a channel's domain, reorganizing into another channel or a better radial coordinate can be more effective than simply adding a few arbitrary low operators.

A geometric series is a useful transparent analogy because its tail is exactly known. It demonstrates how a convergent sum can still be poorly approximated near its boundary, but its coefficient assumptions must not be transferred silently to a CFT. The spectral lab similarly illustrates changing the number of retained terms in a fixed sine sum. It has no primary dimensions, OPE coefficients, separating spheres or analytic-continuation prescription and does not certify a bootstrap truncation.