[Paper Review] Vacuum Stress Tensor of a Scalar Field in a Rectangular Waveguide

Robson B. Rodrigues, Svaiter, N. F.+1 more|ArXiv.org|Oct 31, 2001

Gyrotron and Vacuum Electronics Research3 citations

TL;DR

This paper computes the renormalized canonical and improved vacuum stress-energy tensors for a massless scalar field in an infinitely long rectangular waveguide using the zeta function regularization and heat kernel method. It reveals strong position dependence in local Casimir forces, with repulsive and attractive behaviors near edges and walls, and shows that edge divergences persist even with an external configuration, necessitating physical boundary models for resolution.

ABSTRACT

Using the heat kernel method and the analytic continuation of the zeta function, we calculate the canonical and improved vacuum stress tensors, ${T_{μν}(\vec{x})}$ and ${Θ_{μν}(\vec{x})}$, associated with a massless scalar field confined in the interior of an infinitely long rectangular waveguide. The local depence of the renormalized energy for two special configurations when the total energy is positive and negative are presented using ${T_{00}(\vec{x})}$ and ${Θ_{00}(\vec{x})}$. From the stress tensors we obtain the local Casimir forces in all walls by introducing a particular external configuration. It is shown that this external configuration cannot give account of the edge divergences of the local forces. The local form of the forces is obtained for three special configurations.

Motivation & Objective

• To compute the renormalized vacuum stress-energy tensors for a massless scalar field in an infinitely long rectangular waveguide.
• To investigate the local structure of Casimir forces in geometries with edges and corners, which are known to introduce complications not present in simple parallel plates.
• To examine the role of edge and surface divergences in the local force calculation and assess whether external configurations can eliminate them.
• To establish a connection between the local stress tensor and the global Casimir energy via volume integration.
• To analyze the dependence of local forces on waveguide dimensions and position, particularly near edges and walls.

Proposed method

• Applies the zeta function regularization method to compute the vacuum expectation values of the canonical and improved stress-energy tensors.
• Uses the heat kernel expansion to systematically handle divergences and perform analytic continuation of the zeta function.
• Implements boundary conditions (Dirichlet) on the waveguide walls to define the mode spectrum and eigenmodes.
• Derives the local force density from the discontinuity of the stress-energy tensor across boundaries using the relation $ F_{ ext{local}} = \langle T_{\mu\nu} \rangle_{\text{out}} - \langle T_{\mu\nu} \rangle_{\text{in}} $.
• Evaluates the resulting infinite sums over mode quantum numbers using analytic techniques, including symmetry reduction and summation identities.
• Compares the finite part of the local energy density $ \langle T_{00} \rangle $ with the known global Casimir energy, showing agreement up to a size-dependent constant.

Experimental results

Research questions

• RQ1 How does the vacuum stress-energy tensor of a massless scalar field behave locally in a rectangular waveguide with edges and corners?
• RQ2 Can an external configuration eliminate the edge and surface divergences in the local Casimir force calculation?
• RQ3 What is the dependence of the local Casimir force on position along the waveguide walls, particularly near edges?
• RQ4 How do the local forces relate to the global Casimir energy, and what is the role of the finite part of the energy density?
• RQ5 Under what conditions does the local force become uniform or exhibit non-trivial spatial structure?

Key findings

• The vacuum stress-energy tensor is strongly position-dependent, reflecting the influence of edges and corners in the rectangular geometry.
• The local Casimir force on the wall at $ x_1 = 0 $ is repulsive (negative) and exhibits a minimum at the center and maxima near the edges, consistent with non-uniform stress distribution.
• The force on the wall at $ x_2 = 0 $ is attractive (positive) with a single maximum at the center, showing asymmetric behavior compared to the $ x_1 $-wall.
• When $ b = 2a $, the force on the $ x_1 = 0 $ wall remains attractive and non-uniform, while the force on the $ x_2 = 0 $ wall is small and vanishes as $ b \to \infty $, approaching the uniform parallel plate result.
• The integral of the finite part of $ \langle T_{00}(\vec{x}) \rangle $ over the waveguide volume yields the global Casimir energy plus a constant $ C(a,b) $, confirming consistency between local and global descriptions.
• Edge divergences in the local force do not cancel with those in the stress tensor, and the chosen external configuration fails to regularize them, indicating the need for physical models of boundary material response or quantum boundary conditions.