Quantum mechanics

Pure mathematics

Mathematical physics

Biology

Scattering amplitude

Particle physics

Crossing

Paleontology

Structural basin

Physics

Amplitude

Inversion (geology)

Mathematics

Point reflection

Mathematical analysis

Geometry

Series (stratigraphy)

Conformal map

Meromorphic function

Complex plane

Journal of High Energy Physics

We derive a Lorentzian OPE inversion formula for the principal series of $sl(2,\mathbb{R})$. Unlike the standard Lorentzian inversion formula in higher dimensions, the formula described here only applies to fully crossing-symmetric four-point functions and makes crossing symmetry manifest. In particular, inverting a single conformal block in the crossed channel returns the coefficient function of the crossing-symmetric sum of Witten exchange diagrams in AdS, including the direct-channel exchange. The inversion kernel exhibits poles at the double-trace scaling dimensions, whose contributions must cancel out in a generic solution to crossing. In this way the inversion formula leads to a derivation of the Polyakov bootstrap for $sl(2,\mathbb{R})$. The residues of the inversion kernel at the double-trace dimensions give rise to analytic bootstrap functionals discussed in recent literature, thus providing an alternative explanation for their existence. We also use the formula to give a general proof that the coefficient function of the principal series is meromorphic in the entire complex plane with poles only at the expected locations.

A crossing-symmetric OPE inversion formula