  Quantum tunneling in a many-body system is much more non-trivial than that in
a one-body system. The most characteristic phenomenon is the mixed tunneling,
which has been studied in many fields for decades. For instance, let us
consider a system where there are two coupled particles and only one of them
feels a potential barrier. Quantum tunneling of such a system is not described
by either Euclidean or Lorentzian time evolution and the exponent of the WKB
wave function becomes complex. Recently, a similar phenomenon, polychronic
tunneling, has been proposed in quantum gravity, which enhances the decay rate
of a meta-stable vacuum by many orders of magnitude. In this paper, we present
path integral formalism that is applicable to such systems. The formalism can
be directly extended to quantum gravity and has some implications on the
problem of time in quantum gravity. We also discuss a possible relation to the
conventional path integral.
