  We analyze vacuum tunneling in quantum field theory in a general formalism by
using the Wigner representation. In the standard instanton formalism, one
usually approximates the initial false vacuum state by an eigenstate of the
field operator, imposes Dirichlet boundary conditions on the initial field
value, and evolves in imaginary time. This approach does not have an obvious
physical interpretation. However, an alternative approach does have a physical
interpretation: in quantum field theory, tunneling can happen via classical
dynamics, seeded by initial quantum fluctuations in both the field and its
momentum conjugate, which was recently implemented in Ref. [1]. We show that
the Wigner representation is a useful framework to calculate and understand the
relationship between these two approaches. We find there are two, related,
saddle point approximations for the path integral of the tunneling process: one
corresponds to the instanton solution in imaginary time and the other one
corresponds to classical dynamics from initial quantum fluctuations in real
time. The classical approximation for the dynamics of the latter process is
justified only in a system with many degrees of freedom, as can appear in field
theory due to high occupancy of nucleated bubbles, while it is not justified in
single particle quantum mechanics, as we explain. We mention possible
applications of the real time formalism, including tunneling when the instanton
vanishes, or when the imaginary time contour deformation is not possible, which
may occur in cosmological settings.
