  We argue that every CFT contains light-ray operators labeled by a continuous
spin J. When J is a positive integer, light-ray operators become integrals of
local operators over a null line. However for non-integer J, light-ray
operators are genuinely nonlocal and give the analytic continuation of CFT data
in spin described by Caron-Huot. A key role in our construction is played by a
novel set of intrinsically Lorentzian integral transforms that generalize the
shadow transform. Matrix elements of light-ray operators can be computed via
the integral of a double-commutator against a conformal block. This gives a
simple derivation of Caron-Huot's Lorentzian OPE inversion formula and lets us
generalize it to arbitrary four-point functions. Furthermore, we show that
light-ray operators enter the Regge limit of CFT correlators, and generalize
conformal Regge theory to arbitrary four-point functions. The average null
energy operator is an important example of a light-ray operator. Using our
construction, we find a new proof of the average null energy condition (ANEC),
and furthermore generalize the ANEC to continuous spin.
