  Currently there is no general theory of quantum tunnelling of a particle
through a potential barrier which is compatible with QFT. We present a complete
calculation of tunnelling amplitudes for a scalar field for some simple
potentials using quantum field-theoretic methods. Using the perturbative
S-matrix formalism, starting with the Klein-Gordon Lagrangian, we show that an
infinite summation of Feynman diagrams can recover tunnelling amplitudes
consistent with relativistic quantum mechanics. While this work does not
include many-particle effects arising from a fully quantised QFT, it is
necessary to investigate QFT corrections to tunnelling amplitudes.
