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Regular Article - Theoretical Physics

Thermal Casimir effect with general boundary conditions J. M. Muñoz-Castañedaa , L. Santamaría-Sanzb , M. Donairec , M. Tello-Frailed Departamento de Física Teórica, Atómica y Óptica, Valladolid University, Valladolid, Spain

Received: 9 April 2020 / Accepted: 13 August 2020 / Published online: 29 August 2020 © The Author(s) 2020

Abstract In this paper we study the system of a scalar quantum field confined between two plane, isotropic, and homogeneous parallel plates at thermal equilibrium. We represent the plates by the most general lossless and frequencyindependent boundary conditions that satisfy the conditions of isotropy and homogeneity and are compatible with the unitarity of the quantum field theory. Under these conditions we compute the thermal correction to the quantum vacuum energy as a function of the temperature and the parameters encoding the boundary condition. The latter enables us to obtain similar results for the pressure between plates and the quantum thermal correction to the entropy. We find out that our system is thermodynamically stable for any boundary conditions, and we identify a critical temperature below which certain boundary conditions yield attractive, repulsive, and null Casimir forces.

1 Introduction Since its theoretical prediction in 1948 [1,2] the Casimir effect has been extensively studied, both theoretically [3–6] and experimentally [7–10]. In its original formulation the Casimir force is a consequence of the interaction energy due to the coupling between the quantum vacuum fluctuations of the electromagnetic field with the charged current fluctuations of the plates [11,12]. For separation distances between plates much larger than any other length scale which determines the electric response of the plates, only the longwavelength transverse modes of the electromagnetic field are relevant to the interaction, and they can be mimicked by the normal modes of a scalar field [4,13,14]. a e-mail:

[email protected] (corresponding author)

b e-mail:

[email protected]

c e-mail:

[email protected]

d e-mail:

[email protected]

Recently there has been renewed interest in the thermal Casimir effect motivated by its applications to the design of nano-electronic devices [15–18], the appearance of negative self-entropies in Casimir-like systems [19–25], technological applications, and cosmological problems [26]. In most of the cases the focus has been on the dependence of the Casimir effect at finite temperature with the geometry, and not much attention has been paid to the dependence on the physical properties of the boundaries appearing in the system. The quantum vacuum energy at zero temperature of a massless scalar field confined between two parallel plates with general boundary conditions was studied in [27], using the theory of selfadjoint extensions for the Laplace–Beltrami operator developed in [28]. The most remarkable result of Ref. [27] is the computation of the quantum vacuum energy for a scalar field confined betwee