  We present the calculation of the Feynman path integral in real time for
tunneling in quantum mechanics and field theory, including the first quantum
corrections. For this purpose, we use the well-known fact that Euclidean saddle
points in terms of real fields can be analytically continued to complex saddles
of the action in Minkowski space. We also use Picard-Lefschetz theory in order
to determine the middle-dimensional steepest-descent surface in the complex
field space, constructed from Lefschetz thimbles, on which the path integral is
to be performed. As an alternative to extracting the decay rate from the
imaginary part of the ground-state energy of the false vacuum, we use the
optical theorem in order to derive it from the real-time amplitude for forward
scattering. While this amplitude may in principle be obtained by analytic
continuation of its Euclidean counterpart, we work out in detail how it can be
computed to one-loop order at the level of the path integral, i.e. evaluating
the Gau{\ss}ian integrals of fluctuations about the relevant complex saddle
points. To that effect, we show how the eigenvalues and eigenfunctions on a
thimble can be obtained by analytic continuation of the Euclidean eigensystem,
and we determine the path-integral measure on thimbles. This way, using
real-time methods, we recover the one-loop result by Callan and Coleman for the
decay rate. We finally demonstrate our real-time methods explicitly, including
the construction of the eigensystem of the complex saddle, on the archetypical
example of tunneling in a quasi-degenerate quartic potential.
