Non-perturbative Quantum Propagators in Bounded Spaces

Abstract

We outline a new approach to calculating the quantum mechanical propagator in the presence of geometrically non-trivial Dirichlet boundary conditions based upon a generalisation of an integral transform of the propagator studied in previous work (the so-called "hit function"), and a convergent sequence of Padé approximants. In this paper the generalised hit function is defined as a many-point 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 [1][2][3]. We indicate the eventual application in a relativistic context to determining Casimir energies using this technique.

Worldline approach for spinor fields in manifolds with boundaries

“…These approaches are based on delta-like potentials instead of the method of images. On the other hand, worldline numerical computations have been employed on the same areas [10,12,[69][70][71], mostly considering rigid boundaries and scalar fields. The approach provided in the present article could be used to study these phenomena analytically for spinor fields, as well to test numerical computations in manifolds with rigid boundaries.…”

“…This approach was introduced in the worldline context in [10] (for a similar mechanism for Neumann boundary conditions, see [11]). However, this procedure is generally not suitable for usual perturbative calculations: indeed, such procedures would lead to an expansion in positive powers of λ and the limit λ → ∞ usually appears to be ill-defined (for a strong coupling approach involving Padé approximants, see [12]). An exception occurs for the free scalar field in flat space, where a resummation that leads to the correct heat-trace expansion for the Dirichlet propagator in the limit of infinite coupling is possible [13] (for a similar resummation involving the Neumann propagator, see [14]).…”

Effective action for delta potentials: Spacetime-dependent inhomogeneities and Casimir self-energy

“…[23] (and later proved in [24]) has been used to establish several properties of fermionic condensates in curved space [25][26][27][28]. Similar resummation techniques have also been employed in the investigation of Casimir self-interactions under spacetime-dependent boundary conditions [29][30][31] and, more in general, for background fields that are simple enough to obtain a closed expression either for the effective action [32][33][34] or for the corresponding heat kernel [35,36].…”