Local zeta regularization and the Casimir effect

Abstract

In this paper, whose aims are mainly pedagogical, we illustrate how to use the local zeta regularization to compute the stress-energy tensor of the Casimir effect. Our attention is devoted to the case of a neutral, massless scalar field in flat space-time, on a space domain with suitable (e.g., Dirichlet) boundary conditions. After a simple outline of the local zeta method, we exemplify it in the typical case of a field between two parallel plates, or outside them. The results are shown to agree with the ones obtained by more popular methods, such as point splitting regularization. In comparison with these alternative methods, local zeta regularization has the advantage to give directly finite results via analitic continuation, with no need to remove or subtract divergent quantities.

It differs from the previous arXiv version v1 only for misprint corrections. An enlarged version has been published in Progress of Theoretical Physics

References in corpus (2)

• Zeta Functions and the Casimir Energy
• Energy-momentum tensor of a Casimir apparatus in a weak gravitational field: scalar case

Cited by in corpus (7)

• Local Casimir Effect for a Scalar Field in Presence of a Point Impurity
• The semiclassical energy density of kinks and solitons
• Relative-Zeta and Casimir energy for a semitransparent hyperplane selecting transverse modes
• Local zeta regularization and the scalar Casimir effect III. The case with a background harmonic potential
• The Casimir energy anomaly for a point interaction
• Vacuum polarization with zero-range potentials on a hyperplane
• A note on "Algebraic approach to Casimir force between two $\delta$-like potentials" (K. Ziemian, Ann. Henri Poincaré, Online First, 2021)