[Paper Review] The generalized Abel-Plana formula with applications to Bessel functions and Casimir effect

A. A. Saharian|ArXiv.org|Aug 8, 2007

Quantum Electrodynamics and Casimir EffectPhysics and Astronomy156 references115 citations

TL;DR

This paper introduces a generalized Abel-Plana formula (GAPF) that extends the applicability of the classical Abel-Plana summation to complex series involving zeros of Bessel functions and modified Bessel functions with imaginary order. The method enables cutoff-independent evaluation of vacuum expectation values (VEVs) in the Casimir effect for spherical, cylindrical, and accelerated boundaries, separating boundary-induced contributions into strongly convergent integrals and reducing renormalization to bulk-only procedures.

ABSTRACT

One of the most efficient methods for the evaluation of the vacuum expectation values for physical observables in the Casimir effect is based on using the Abel-Plana summation formula. This enables to derive the renormalized quantities in a manifestly cutoff independent way and to present them in the form of strongly convergent integrals. However, applications of the Abel-Plana formula, in its usual form, are restricted by simple geometries when the eigenmodes have a simple dependence on quantum numbers. The author generalized the Abel-Plana formula which essentially enlarges its application range. Based on this generalization, formulae have been obtained for various types of series over the zeros of combinations of Bessel functions and for integrals involving these functions. It has been shown that these results generalize the special cases existing in literature. Further, the derived summation formulae have been used to summarize series arising in the direct mode summation approach to the Casimir effect for spherically and cylindrically symmetric boundaries, for boundaries moving with uniform proper acceleration, and in various braneworld scenarios. This allows to extract from the vacuum expectation values of local physical observables the parts corresponding to the geometry without boundaries and to present the boundary-induced parts in terms of integrals strongly convergent for the points away from the boundaries. As a result, the renormalization procedure for these observables is reduced to the corresponding procedure for bulks without boundaries. The present paper reviews these results. We also aim to collect the results on vacuum expectation values for local physical observables such as the field square and the energy-momentum tensor in manifolds with boundaries for various bulk and boundary geometries.

Motivation & Objective

• To extend the applicability of the Abel-Plana formula beyond simple geometries with linear quantum number dependence.
• To derive summation formulae for series over zeros of combinations of Bessel functions and modified Bessel functions with imaginary order.
• To enable manifestly cutoff-independent computation of vacuum expectation values (VEVs) for field square and energy-momentum tensor in bounded spacetimes.
• To decompose VEVs into bulk parts and boundary-induced parts, simplifying renormalization by isolating geometry-dependent contributions.
• To apply the generalized formalism to diverse physical systems: spherical and cylindrical boundaries, uniformly accelerated plates, and braneworld scenarios.

Proposed method

• Derive a generalized Abel-Plana formula (GAPF) applicable to functions with complex analytic structure and zeros on the imaginary axis.
• Apply the GAPF to series over zeros of $ Z_{i ilde{\omega}}(u,v) $, $ \bar{K}_{iz}(\eta) $, and $ Z_{iz}(u,v) $, which arise in eigenfrequency spectra for boundary problems.
• Use the GAPF to decompose Wightman functions into a bulk part and a boundary-induced part, both expressed as absolutely convergent integrals.
• Apply the resulting summation formulae to mode-summed Casimir energy and stress calculations in various geometries, including $ R^D \times S^1 $, parallel plates, spherical shells, and cylindrical boundaries.
• Utilize the formalism to compute VEVs of the field square and energy-momentum tensor (EMT) in regions with boundaries, separating the divergent bulk contribution from the finite boundary-induced part.
• Apply the GAPF to radiation problems, such as helical motion in a dielectric cylinder, by summing over eigenmodes via the derived series formulae.

Experimental results

Research questions

• RQ1 How can the Abel-Plana formula be generalized to handle series over zeros of Bessel functions with complex or imaginary orders?
• RQ2 Can the generalized formula be used to extract finite, cutoff-independent expressions for vacuum expectation values in the Casimir effect with non-trivial symmetries?
• RQ3 To what extent can the boundary-induced contributions in VEVs be isolated and expressed as rapidly convergent integrals?
• RQ4 How does the GAPF facilitate renormalization in systems with spherical, cylindrical, or accelerated boundaries by reducing it to the bulk-only case?
• RQ5 What are the implications of the GAPF for braneworld models and vacuum polarization in curved or non-trivial spacetimes like cosmic strings or Rindler wedges?

Key findings

• The generalized Abel-Plana formula (GAPF) successfully extends the classical Abel-Plana method to series over zeros of Bessel functions and modified Bessel functions with imaginary order, such as $ \bar{K}_{iz}(\eta) $ and $ Z_{iz}(u,v) $.
• The GAPF allows the Wightman function to be decomposed into a bulk part and a boundary-induced part, with the latter expressed as a strongly convergent integral, enabling cutoff-independent evaluation of VEVs.
• For scalar fields in a global monopole background with spherical boundaries, the vacuum energy-momentum tensor (EMT) and field square VEVs are computed via the GAPF, with boundary-induced parts given by convergent integrals.
• In the case of two uniformly accelerated plates in the Rindler vacuum, the VEVs of the field square and EMT are separated into single-plate and interaction terms, with the latter derived using the GAPF and expressed in terms of convergent integrals.
• For braneworld scenarios in AdS bulk with two flat branes, the eigenfrequencies are zeros of a combination of Bessel functions, and the GAPF enables the extraction of single-brane and interaction contributions to the VEVs.
• The formalism is applied to electromagnetic Casimir densities in cylindrical and spherical geometries, yielding finite, renormalized expressions for the EMT and field square in regions between boundaries, with explicit formulae provided for both scalar and electromagnetic fields.