  Computing dynamical properties of strongly interacting quantum many-body
systems poses a major challenge to theoretical approaches. Usually, one has to
resort to numerical analytic continuation of results on imaginary frequencies,
which is a mathematically ill-defined procedure. Here, we present an efficient
method to compute the spectral functions of the two-component Fermi gas near
the strongly interacting unitary limit directly in real frequencies. To this
end, we combine the Keldysh path integral that is defined in real time with the
self-consistent T-matrix approximation. The latter is known to predict
thermodynamic and transport properties in good agreement with experimental
observations in ultracold atoms. We validate our method by comparison with
thermodynamic quantities obtained from imaginary time calculations and by
transforming our real-time propagators to imaginary time. By comparison with
state-of-the-art numerical analytic continuation of the imaginary time results,
we show that our real-time results give qualitative improvements for dynamical
quantities. Moreover, we show that no significant pseudogap regime exists in
the self-consistent T-matrix approximation above the critical temperature
$T_c$, an issue that has been under significant debate. We close by pointing
out the versatile nature of our method as it can be extended to other systems,
like the spin- or mass-imbalanced Fermi gas, other Bose-Fermi models, 2D
systems as well as systems out of equilibrium.
