Scalar field theory under Robin boundary conditions: two-point function and energy-momentum tensor

TL;DR

This paper analyzes a four-dimensional scalar field with Robin boundary conditions on parallel plates, deriving boundary corrections to the energy-momentum tensor and calculating the Casimir energy, confirming consistency with existing results.

Contribution

It introduces a path integral approach with auxiliary fields for Robin boundary conditions and computes the Casimir energy in this setup, including boundary corrections.

Findings

Boundary corrections to the energy-momentum tensor are derived.

Casimir energy is computed as a function of plate separation and Robin parameters.

Boundary contribution to vacuum energy vanishes, aligning with previous results.

Abstract

We reconsider four-dimensional scalar field theory in presence of Robin boundary conditions on two parallel plates. These boundary conditions are directly imposed in the path integral definition of the theory via auxiliary fields living on the plates. We discuss how this leads to boundary corrections to the standard energy momentum tensor operator. Via a dimensional reduction to an effective three-dimensional boundary theory, we compute the Casimir energy in terms of the plate separation and the two Robin parameters, as well as the scalar field propagator in the presence of the plates. Coincidentally, the boundary contribution vanishes in the expectation value for the vacuum energy, thereby giving results in full accordance with other energy expressions in the literature for the same setup. We also discuss for which values of the Robin parameters this energy is real-valued.