  The general formula for the probability of radiation of a twisted photon by a
classical current is derived. The general theory of generation of twisted
photons by undulators is developed. It is proved that the probability to record
a twisted photon produced by a classical current is equal to the average number
of twisted photons in a given state. The general formula for the projection of
the total angular momentum of twisted photons with given the energy, the
longitudinal projection of momentum, and the helicity is obtained. The symmetry
property of the average number of twisted photons produced by a charged
particle moving along a planar trajectory is found. The explicit formulas for
the average number of twisted photons generated by undulators both in the
dipole and wiggler regimes are obtained. It is established that, for the
forward radiation of an ideal right-handed helical undulator, the harmonic
number $n$ of the twisted photon coincides with its projection of the total
angular momentum $m$. As for the ideal left-handed helical undulator, we obtain
that $m=-n$. It is found that the forward radiation of twisted photons by a
planar undulator obeys the selection rule that $n+m$ is an even number. It
turns out that the average number of twisted photons produced by the undulator
and detected off the undulator axis is a periodic function of $m$ in a certain
spectral band of the quantum numbers $m$.
