  Conceptually, the Matsubara formalism (MF), using imaginary frequencies, and
the Keldysh formalism (KF), formulated in real frequencies, give equivalent
results for systems in thermal equilibrium. The MF has less complexity and is
thus more convenient than the KF. However, computing dynamical observables in
the MF requires the analytic continuation from imaginary to real frequencies.
The analytic continuation is well-known for two-point correlation functions
(having one frequency argument), but, for multipoint correlators, a
straightforward recipe for deducing all Keldysh components from the MF
correlator had not been formulated yet. Recently, a representation of MF and KF
correlators in terms of formalism-independent partial spectral functions and
formalism-specific kernels was introduced by Kugler, Lee, and von Delft [Phys.
Rev. X 11, 041006 (2021)]. We use this representation to formally elucidate the
connection between both formalisms. We show how a multipoint MF correlator can
be analytically continued to recover all partial spectral functions and yield
all Keldysh components of its KF counterpart. The procedure is illustrated for
various correlators of the Hubbard atom.
