Radiative corrections to the Casimir effect for the massive scalar field

In 1948 Casimir predicted an unexpected quantum field theory (QFT) effect: as a consequence of distorting the vacuum of the quantized electromagnetic field, two conducting and neutral parallel plates should attract each other with a force proportional to the inverse of the fourth power of their separation [1] (for general reviews on the Casimir effect, see Refs. [2]; for an introductory guide, see Ref. [3]). The first attempt to observe this phenomenon was made only in 1958 by Sparnaay [4]. However, the accuracy achieved by Sparnaay allowed him only to conclude that the experimental data was compatible with Casimir’s theoretical predictions. For different reasons, new experiments involving directly metal bodies were performed only very recently [5], but this time the high accuracy obtained and the excellent agreement between the experimental data and theory permit us to state safely that the Casimir effect is well established nowadays. The Casimir effect is not a peculiarity of the quantized electromagnetic field. In fact, the vacuum state (and its energy) of any relativistic quantum field, bosonic or fermionic, depends on the boundary conditions (BC) imposed on the fields or, more generally, on the classical background with which the fields interact. This makes the Casimir effect an important topic of research,