  Though theoretical treatments of quantum tunnelling within single-particle
quantum mechanics are well-established, at present, there is no quantum
field-theoretic description (QFT) of tunnelling. Due to the single-particle
nature of quantum mechanics, many-particle effects arising from quantum field
theory are not accounted for. Such many-particle effects, including
pair-production, have proved to be essential in resolving the Klein-paradox.
This work seeks to determine how quantum corrections affect the tunnelling
probability through an external field. We investigate a massive neutral scalar
field, which interacts with an external field in accordance with relativistic
quantum mechanics. To consider QFT corrections, we include another massive
quantised neutral scalar field coupling to the original via a cubic
interaction. This study formulates an all-order recursive expression for the
loop-corrected scalar propagator, which contains only the class of
vertex-corrected Feynman diagrams. This equation applies for general external
potentials. Though there is no closed-form analytic solution, we also
demonstrate how to approximate the QFT corrections if a perturbative coupling
to the quantised field is assumed.
