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

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 $\lambda {\varphi }^4$ theory submitted to Robin boundary condition on a plate.

16 pages, 2 figures. Version 2: contains a new section on the renormalization of the two-point Green function in the presence of a flat boundary. Accepted for publication in J. Phys. A

References in corpus (14)

• New Developments in the Casimir Effect
• Precision Measurement of the Casimir Force from 0.1 to 0.9 microns
• Measurement of the Casimir force between parallel metallic surfaces
• The `Friction' of Vacuum, and other Fluctuation-Induced Forces
• Nonlinear Micromechanical Casimir Oscillator
• Stiction, Adhesion Energy and the Casimir Effect in Micromechanical Systems
• Improved Precision Measurement of the Casimir Force Using Gold Surfaces
• Improved Precision Measurement of the Casimir Force
• Casimir effect for scalar fields under Robin boundary conditions on plates
• Fluctuation - induced forces in critical fluids
• On the energy-momentum tensor for a scalar field on manifolds with boundaries
• Boundary dynamics and multiple reflection expansion for Robin boundary conditions
• Alternative Dimensional Reduction via the Density Matrix
• When the Casimir energy is not a sum of zero-point energies

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