  We investigate the nature of resonant tunneling in Quantum Field Theory.
Following the pioneering work of Banks, Bender and Wu, we describe quantum
field theory in terms of infinite dimensional quantum mechanics and utilize the
``Most probable escape path'' (MPEP) as the class of paths which dominate the
path integral in the classically forbidden region. Considering a 1+1
dimensional field theory example we show that there are five conditions that
any associated bound state in the classically allowed region must satisfy if
resonant tunnelling is to occur, and we then proceed to show that it is
impossible to satisfy all five conditions simultaneously.
