Casimir effect for the scalar field under Robin boundary conditions: A functional integral approach

TL;DR

This paper develops a functional integral approach to define and analyze the Casimir effect for a scalar field with Robin boundary conditions, revealing conditions for energy minima and discussing renormalization in related quantum field theories.

Contribution

It introduces a method to implement Robin boundary conditions dynamically via the action and derives the Casimir energy for this setup, including special cases and energy minima.

Findings

Casimir energy depends on boundary parameters and plate separation.

For certain parameters, the energy exhibits a minimum at specific distances.

The formalism allows for one-loop renormalization of Green functions under Robin conditions.

Abstract

In this work we show how to define the action of a scalar field in a such a way that Robin boundary condition is implemented dynamically, i.e., as a consequence of the stationary action principle. We discuss the quantization of that system via functional integration. Using this formalism, we derive an expression for the Casimir energy of a massless scalar field under Robin boundary conditions on a pair of parallel plates, characterized by constants $c_1$ and $c_2$. Some special cases are discussed; in particular, we show that for some values of $c_1$ and $c_2$ the Casimir energy as a function of the distance between the plates presents a minimum. We also discuss the renormalization at one-loop order of the two-point Green function in the $\lambda\phi^4$ theory submitted to Robin boundary condition on a plate.