Several problems at the interface between the field-theoretical description of the Casimir effect and experiments on measuring the Casimir force are discussed. One of these problems is connected with the definition of the Casimir free energy in ideal metal rectangular boxes satisfying the general physical requirements. It is shown that the consideration of rectangular boxes with a partition (piston) does not negate the previously known results obtained for boxes without a piston. Both sets of results are found to be in mutual agreement. Another problem is related to the use of the proximity force approximation for the interpretation of the experimental data and to the search of analytical results beyond the PFA based on the first principles of quantum field theory. Next, we discuss concepts of experimental precision and of the measure of agreement between experiment and theory. The fundamental difference between these two concepts is clarified. Finally, recent approach to the thermal Casimir force taking screening effects into account is applied to real metals. It is shown that this approach is thermodynamically and experimentally inconsistent. The physical reasons of this inconsistency are connected with the violation of thermal equilibrium which is the basic applicability condition of the Lifshitz theory.

http://iopscience.iop.org/article/10.1088/1742-6596/161/1/012003/pdf

Experiment, theory and the Casimir effect

The Casimir effect for a massless scalar field on the helix boundary condition which is named as quantum spring is studied in our recent paper\cite{Feng}. In this paper, the Casimir effect of the quantum spring is investigated in $(D+1)$-dimensional spacetime for the massless and massive scalar fields by using the zeta function techniques. We obtain the exact results of the Casimir energy and Casimir force for any $D$, which indicate a $Z_2$ symmetry of the two space dimensions. The Casimir energy and Casimir force have different expressions for odd and even dimensional space in the massless case but in both cases the force is attractive. In the case of odd-dimensional space, the Casimir energy density can be expressed by the Bernoulli numbers, while in the even case it can be expressed by the $\zeta$-function. And we also show that the Casimir force has a maximum value which depends on the spacetime dimensions. In particular, for a massive scalar field, we found that the Casimir force varies as the mass of the field changes.

The Casimir force of Quantum Spring in the (D+1)-dimensional spacetime

We study the zero and finite temperature Casimir force acting on a perfectly conducting piston with arbitrary cross section moving inside a closed cylinder with infinitely permeable walls. We show that at any temperature, the Casimir force always tends to move the piston away from the walls and toward its equilibrium position. In the case of a rectangular piston, exact expressions for the Casimir force are derived. In the high-temperature regime, we show that the leading term of the Casimir force is linear in temperature and therefore the Casimir force has a classical limit. Due to duality, all these results also hold for an infinitely permeable piston moving inside a closed cylinder with perfectly conducting walls.

https://iopscience.iop.org/article/10.1088/1367-2630/11/1/013055/pdf

Repulsive Casimir force at zero and finite temperature

The Casimir effect for a massless scalar field on the helix boundary condition which is named as quantum spring is studied in our recent paper.27 Here, the Casimir effect of the quantum spring is investigated in (D+1)-dimensional spacetime for the massless and massive scalar fields by using the zeta function techniques. We obtain the exact results of the Casimir energy and Casimir force for any D, which indicate a Z2 symmetry of the two space dimensions. The Casimir energy and Casimir force have different expressions for odd and even dimensional space in the massless case but in both cases the force is attractive. In the case of odd-dimensional space, the Casimir energy density can be expressed by the Bernoulli numbers, while in the even case it can be expressed by the ζ-function. And we also show that the Casimir force has a maximum value which depends on the spacetime dimensions. In particular, for a massive scalar field, we found that the Casimir force varies as the mass of the field changes.

We derive rigorously explicit formulas of the Casimir free energy at finite temperature for massless scalar field and electromagnetic field confined in a closed rectangular cavity with different boundary conditions by zeta regularization method. We study both the low and high temperature expansions of the free energy. In each case, we write the free energy as a sum of a polynomial in temperature plus exponentially decay terms. We show that the free energy is always a decreasing function of temperature. In the cases of massless scalar field with Dirichlet boundary condition and electromagnetic field, the zero temperature Casimir free energy might be positive. In each of these cases, there is a unique transition temperature (as a function of the side lengths of the cavity) where the Casimir energy change from positive to negative. When the space dimension is equal to two and three, we show graphically the dependence of this transition temperature on the side lengths of the cavity. Finally we also show that we can obtain the results for a non-closed rectangular cavity by letting the size of some directions of a closed cavity going to infinity, and we find that these results agree with the usual integration prescription adopted by other authors.

Finite temperature Casimir energy in closed rectangular cavities: a rigorous derivation based on zeta function technique

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Finite temperature Casimir energy in closed rectangular cavities: a rigorous derivation based on zeta function technique