Tunnelling amplitudes through localised external potentials from Feynman diagram summation

TL;DR

This paper develops a quantum field theory approach to calculate tunnelling amplitudes for a scalar particle through potential barriers, bridging a gap between quantum mechanics and QFT methods.

Contribution

It demonstrates that summing Feynman diagrams in QFT can reproduce tunnelling amplitudes consistent with relativistic quantum mechanics for simple potentials.

Findings

Summation of Feynman diagrams recovers known tunnelling amplitudes.

Provides a QFT-based framework for tunnelling calculations.

Highlights the need to include many-particle effects for full QFT treatment.

Abstract

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.