Quantum field-theoretic approaches to quantum tunnelling through external potentials

Abstract

Although single-particle quantum tunnelling is well-understood in the context of nonrelativistic quantum mechanics, there is no comprehensive description of such tunnelling mechanism in a quantum field theory (QFT) framework. In contrast to quantum mechanics which can include non-local interactions, QFT necessarily describes all interactions locally via a coupling between quantised fields, allowing for many-particle effects like pair-production which are not accounted for in quantum mechanics. This thesis formulates a description of quantum tunnelling using QFT, by first investigating a massive neutral scalar field interacting with an external localised potential that is not quantised. In this scenario, it is shown that the scattering matrix formalism in QFT can recover tunnelling probabilities consistent with relativistic quantum mechanics for simple potentials. Crucially, the scattering matrix must be computed non-perturbatively in order to reproduce tunnelling faithfully. To incorporate QFT corrections, a coupling to an intermediate quantised neutral scalar field is added to the Lagrangian. An integral equation for the dressed loop-corrected scalar propagator is derived, though it is limited to a certain class of loop-corrected Feynman diagrams. This calculation is applicable to arbitrary external potentials. An approximate form of this propagator is found by assuming that the coupling to the quantised field can be treated perturbatively, while retaining a non-perturbative approach to the interaction with the external field. It is demonstrated that its magnitude can be bounded in the regime where the mass of the intermediate particle is asymptotically large compared to other energy scales. This thesis clarifies the question of tunnelling within QFT, by answering key questions about how to extract scattering matrix elements corresponding to tunnelling amplitudes, and why a non-perturbative treatment is warranted. Additionally, this work provides a foundation for numerically modelling quantum tunnelling using QFT, and extending the concept beyond simple neutral scalar fields to more realistic Lagrangians.