Nonperturbative Quantum Propagators in Bounded Spaces
Abstract
We outline a new approach to calculating the quantum mechanical propagator in the presence of geometrically nontrivial Dirichlet boundary conditions based upon a generalisation of an integral transform of the propagator studied in previous work (the socalled ``hit function''), and a convergent sequence of Padé approximants. In this paper the generalised hit function is defined as a manypoint propagator and we describe its relation to the sum over trajectories in the Feynman path integral. We then show how it can be used to calculate the Feynman propagator. We calculate analytically all such hit functions in $D=1$ and $D=3$ dimensions, giving recursion relations between them in the same or different dimensions and apply the results to the simple cases of propagation in the presence of perfectly conducting planar and spherical plates. We use these results to conjecture a general analytical formula for the propagator when Dirichlet boundary conditions are present in a given geometry, also explaining how it can be extended for application for more general, nonlocalised potentials. Our work has resonance with previous results obtained by Grosche in the study of path integrals in the presence of delta potentials. We indicate the eventual application in a relativistic context to determining Casimir energies using this technique.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.04969
 Bibcode:
 2021arXiv211004969E
 Keywords:

 Quantum Physics;
 High Energy Physics  Theory;
 Mathematical Physics
 EPrint:
 26 pages,6 figures, 5 appendices