  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.
