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 $\ensuremath{\square}$ to $\ensuremath{\square}f(\ensuremath{\square})$ 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.
Highlights
The idea of nonlocality is quite old in theoretical physics
In this paper we study quasiclassical solutions of the nonlocal massive scalar field equations
The kinetic part of the action for such a theory contains a function of the □-operator, which is chosen so that the theory does not contain new unphysical degrees of freedom
Summary
The idea of nonlocality is quite old in theoretical physics. Nonlocal modifications of the field theory were discussed already in the publications [1,2,3]. Initial data for the Hamilton-Jacobi equation specify a beam of trajectories in the phase space which forms a Lagrangian submanifold (for details see e.g., a remarkable book [45]) Knowledge of this Lagrangian submanifold allows one to construct the eikonal function SðxÞ and to find a solution of the transport equation for the amplitude uðxÞ by using the Liouville theorem. This WKB method is widely used in the standard quantum mechanics and field theory where the corresponding equations are second-order partial differential equations.
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