Nonlocal scalar field in an external potential: WKB approximation

TL;DR

This paper develops WKB solutions for a nonlocal scalar field theory with an external potential, analyzing how nonlocality influences energy levels and barrier penetration probabilities compared to local theories.

Contribution

It introduces a method to construct WKB solutions for a nonlocal scalar field with a modified kinetic operator, ensuring physical consistency and exploring its effects on quantum phenomena.

Findings

Nonlocality alters energy levels of trapped scalar fields.

Barrier penetration probabilities are affected by the nonlocal form factor.

Conditions are identified under which the nonlocal theory remains physically consistent.

Abstract

We consider a nonlocal theory of a scalar massive field in a flat spacetime background in the presence of an external potential and construct WKB solutions for this theory. We use a model in which the kinetic part of the scalar field action is modified by changing $\Box$ to $\Box f(\Box)$ operator. We discuss conditions when the corresponding form factor $f$ is chosen so that the theory does not contain new unphysical degrees of freedom. We applied the obtained WKB solutions for study energy levels of the field trapped by a one-dimensional potential and the probability of the barrier penetration. This allows us to illustrate how the effects of the nonlocality change the known results obtained for the local field theory.