Generalized Schlömilch formulas and thermal Casimir effect of a fermionic rectangular box

Abstract

Schlömilch's formula is generalized and applied to the thermal Casimir effect of a fermionic field confined a three-dimensional rectangular box. The analytic expressions of the Casimir energy and Casimir force are derived for arbitrary temperature and edge sizes. The low and high temperature limits and finite temperature cases are considered for the entire parameter space spanned by edge sizes and/or temperature. In the low temperature limit, it is found that for typical rectangular box, the effective 2-dimensional parameter space spanned by the two edge size ratios can be split into four regions. In one region, all three forces between three pairs of faces are attractive, and in another two regions, the force along the longest edge becomes repulsive and in the last region the force along both the longest and medium sized edge becomes repulsive. Three forces cannot be made simultaneously repulsive. For the waveguide under low temperature, the Casimir force along the longer side of the waveguide cross-section transforms from attractive to repulsive when the aspect ratio of the cross-section exceed a critical value. For the parallel plate scenario under low temperature, our results agrees with previous works. For high tempera limit, it is shown that both the Casimir energy and force approach zero due to the high temperature suppression of the quantum fluctuation responsible for the Casimir energy. For the finite temperature case, we separate the parameter space into four subcases (C1 to C4) and various edge size and temperature effects are analyzed. In general, we found that in all cases the Casimir energy is always negative, while the Casimir force at any finite or low temperature can be either repulsive or attractive depending on the sizes of the edges. For the case (C1) that is similar to parallel plates with relatively high temperature, it is found that the Casimir force is always attractive, regardless the change of the plate separation. At the given temperature, The Casimir energy/force densities approach the infinite parallel plate limit even when the plate edge size is 2 times the plate separation. For the case (C2) that is similar to a waveguide with relatively high temperature, the Casimir force along the longer side of the waveguide cross-section transform from attractive to repulsive when this side exceed a critical value. This critical point forms a boundary in the parameter space when the shorter edge of the waveguide cross-section changes and the boundary values decreases with respect to temperature increase. Case (C3) covers the low temperature parallel plate, typical rectangular box and waveguide geometries. For the waveguide case, the force along the waveguide longitude also transform from attractive to repulsive when the waveguide length exceed certain critical value. These critical values changes with respect to temperature in a nontrivial way. For the typical waveguide case (C4) at low temperature, the Casimir energy density along the longitudinal direction is a constant while force density dec...

Casimir free energy for massive fermions: a comparative study of various approaches

“…Therefore the differences between the results of ZFA, ZTSA and the fundamental approach for the massive case are not resolved by the renormalization programs of the former two. On the other hand, as we shall show, the extra terms in the massless case are simple polynomials and can be removed by the renormalization programs introduced [39,40], or cancel out in the piston method for the Casimir pressure [39,[41][42][43]. Hence, for the massless case, the results of the fundamental approach, the ZFA (supplemented with the piston approach or its renormalized version), and the renormalized version of ZTSA are all equivalent.…”

“…To fully explore this approach, we consider four different ways of using the zeta function and show that they yield equivalent results. Moreover, we shall compute the Casimir free energy using the Schlömilch formulas approach (SFA) [40], as the second representative of the analytic continuation approach, and show that its results are equivalent to those of the ZFA. Finally we use the zero temperature subtraction approach (ZTSA) [67], and show that its results are not equivalent to those of the ZFA and SFA.…”

“…As stated above, we also obtain the Casimir free energy using the Schlömilch formula approach (SFA) [40], as a second representative of the analytic continuation approach. In fact, this approach is used for obtaining only the thermal corrections of the Casimir effect, and the zero temperature part should be calculated separately using other methods, e.g., the ZFA.…”

“…Most of the work mentioned above have used ZFA. In 2018, Mo and Jia [40] used SFA to calculate the thermal correction of the Casimir free energy for an electromagnetic field in a conducting rectangular box. To eliminate the high temperature divergences, they defined a renormalized free energy by subtracting the free black-body term along with any possible terms proportional to T 2 and T 3 , with reference to Geyer's work [39].…”

“…As we shall show, all even powers of T , including ln(T ), are due the thermal free energy of the free case which has not been subtracted, and the odd powers of T , except for the classical term, are due to a nonextensive term generated by the analytic continuation. We then subtract terms with powers of T greater or equal to two, in accordance with the renormalization programs introduced [39,40].The remaining linear term is the correct classical term, while the remaining T ln(T ) and ln(T ) terms are unphysical. Therefore the differences between the results of ZFA, ZTSA and the fundamental approach for the massive case are not resolved by the renormalization programs of the former two.…”