\section{Massive Fermions and the Generalized Zeta Function}\label{Epstein}

\indent
The zeta function approach (ZFA) has been used to calculate the Casimir free energy for the massive fermion field between two plates and some solutions have been presented (see for example in\cite{r45Teo., r45Flachi.}). In this section, we compute explicitly the final results for the Casimir free energy and Casimir pressure for this problem using the ZFA, and also using the zero temperature subtraction approach (ZTSA), and show that, contrary to the massless case, they yield different results. Most importantly, we show that neither of these results are equivalent to the one obtained in Sec. \ref{massive} based on the fundamental approach. Moreover, we show that these discrepancies cannot be fixed completely by the renormalization program mentioned before (see e.g.~\cite{r17Geyer., r29Junji.}), since the extra unphysical terms are non-polynomial functions of $T$.

We use the first form of the free energy given by Eq.~(\ref{SZeta}), and present the sum over Matsubara frequencies as the difference between sum over all integers and even integers, as used in Eq.~(\ref{s21}). Then, we use the inhomogeneous zeta function on the Matsubara frequencies (see Appendix \ref{appendixA:The zeta function}). The result is
\begin{eqnarray}\label{s33}
&&F_{\mbox{\scriptsize Zeta}}(T,L) = \frac{A}{4 \sqrt{\pi ^3}} \sum\limits_{{n_1}>0} \lim\limits_{s \to 0}  \frac{\partial }{\partial s} \frac{1 }{\Gamma(s)} \left\lbrace \Gamma \left(s - \frac{3}{2}\right) \left(\omega_{n_1}^2 \right)^{\frac{3}{2} - s}	+ \right. \nonumber \\
&& \left.   8\sum\limits_{{n_0} = 1}^\infty     \left(\frac{\omega_{n_1}}{\beta {n_0}} \right)^{\frac{3}{2} - s}\left[ K_{\frac{3}{2} - s}\left( 2 \beta {n_0} \omega_{n_1} \right) -  2^{\frac{1}{2} - s}
K_{\frac{3}{2} - s}\left(\beta {n_0} \omega_{n_1}\right) \right] \right\rbrace  .
\end{eqnarray}
To explore the mechanism of removal of divergences from this point forward, it is useful to compare this expression with the analogous one that we have obtained for the massless case after using the inhomogeneous zeta function on the Matsubara frequencies, {\it i.e.}, Eq.~(\ref{s21c}). The first term in both expressions includes a divergent sum over the spatial modes and is a leftover from the use of inhomogeneous zeta function on the Matsubara frequencies. In the massless case, the spatial modes were regular and we could obtain the analytic continuation of its divergent term using a supplementary zeta function. In the present case, the modes are irregular and, as before, we compute the sum over the spatial modes using the Principle of the Argument theorem (see Appendix \ref{appendixB:The Argument Principle}), and obtain the following expression
\begin{eqnarray}\label{s34}
&&\hspace{-8mm}F_{\mbox{\scriptsize Zeta}}(T,L) =  \frac{A }{2 \sqrt{\pi ^5}} \lim\limits_{s \to 0}  \frac{\partial }{\partial s} \frac{1}{\Gamma(s)}    \left\lbrace \int_{ 0}^{ \infty} dp \left[  L \omega(p)  +\Ln\left( 1 + \frac{\omega(p) - m}{\omega(p)  + m}  e^{ - 2 L \omega(p)}\right)\right] \right. \nonumber\\ 
&& \hspace{-8mm} \left. \left[  \Gamma \left(s - \frac{1}{2}\right) p^{2 - 2s} -   \sum\limits_{{n_0} = 1}^\infty \frac{4 \pi p^{\frac{3}{2} - s}}{\left(n_0 \beta \right)^{\frac{1}{2} - s}}  \left[ 2 J_{\frac{1}{2} - s}\left(2 \beta {n_0} p\right) - 2^{\frac{1}{2} - s} J_{\frac{1}{2} - s}\left( \beta {n_0} p\right) \right]  \right] \right\rbrace  ,
\end{eqnarray}
where $\omega(p)=\sqrt{p^2 + m^2}$. The terms which include the logarithm function, are finite in the domain of integration. So, for these terms, we take the derivative with respect to $s$, take the limit $s \to 0$, and obtain a result which is identical to $F_{\mbox{\scriptsize Casimir}}$ as given by Eq.~(\ref{s27}), 
\begin{eqnarray}\label{s34b}
&&\hspace{-8mm} F_{\mbox{\scriptsize Zeta}}(T,L) = F_{\mbox{\scriptsize Casimir}}(T,L) +  \frac{AL }{2 \sqrt{\pi ^5}} \lim\limits_{s \to 0}  \frac{\partial }{\partial s} \frac{\Gamma \left(s - \frac{1}{2}\right)}{\Gamma(s)}    \int_{0}^{  \infty} dp \omega(p) \left\lbrace  p^{2 - 2s}  -\right.\nonumber \\
&&\hspace{-8mm}  \left. 4 \pi \sum\limits_{{n_0} = 1}^\infty \frac{ p^{\frac{3}{2} - s}}{\left(n_0 \beta \right)^{\frac{1}{2} - s}}   \left[ 2 J_{\frac{1}{2} - s}\left(2 \beta {n_0} p\right)- \sqrt{2^{\left(1- 2 s\right)}} J_{\frac{1}{2} - s}\left( \beta {n_0} p\right) \right]   \right\rbrace. 
 \end{eqnarray}
 The remaining terms are extra unphysical terms which are leftovers from the use of the inhomogeneous zeta function. The first of these terms includes a divergent integral and is identical to the first term of the $F_{\mbox{\scriptsize bounded}}(T,L)$, given by Eq.~(\ref{s1003bv}). As shown in Sec.~\ref{massive}, these terms are both equal to the analytic continuation of $F_{\mbox{\scriptsize free}}(0,L)$ which is equal to $[A L m^4/(32 \pi^2)]  \left[ 3 - 4 \ln(mL)\right]$. 
This is the analytic continuation embedded in the first form of free energy Eq.~(\ref{s4}), and has rendered this term finite.
Finally, taking the derivative with respect to $s$, and taking the limit  $s \to 0$, we can express the final result as follows 
\begin{equation}\label{s35d}
F_{\mbox{\scriptsize Zeta}}(T,L) = F_{\mbox{\scriptsize Casimir}}(T,L) + \frac{A L m^4}{32 \pi^2}  \left[ 3 - 4 \ln(mL) \right]+ \Delta F_{\mbox{\scriptsize free}}(T,L).
\end{equation}
The order of the terms presented above is the same as in Eq.~(\ref{s34b}), {\it i.e.}, the second term is the finite term mentioned above, and the last term is the thermal correction to the free energy of the free massive case, {\it i.e.}, $\Delta F_{\mbox{\scriptsize free}}(T,L)$ given in Eq.~(\ref{s25}). 

Next, we calculate the Casimir free energy using the ZTSA~\cite{r22Rav.}, as defined in Eq.~(\ref{s16}). We use the results presented in Sec.\ \ref{massive} for the free energy of the bounded and free cases, given by Eqs.~(\ref{s1003}, \ref{s25}), to obtain 
\begin{eqnarray}\label{s31}
F_{\mbox{\scriptsize ZTSA}}(T,L)&=&F_{\mbox{\scriptsize Casimir}}(T,L) + \Delta F_{\mbox{\scriptsize free}}(T,L).
\end{eqnarray}
It is worth mentioning that the massless limit of $F_{\mbox{\scriptsize Zeta}}(T,L)$ and $F_{\mbox{\scriptsize ZTSA}}(T,L)$ for the massive cases, given by Eqs.~(\ref{s35d}, \ref{s31}), coincide exactly with their massless cases, given by Eq.~(\ref{s21bb}). 

Now we can compare the three different results obtained for the Casimir free energy using the fundamental definition, {\it i.e.,} $F_{\mbox{\scriptsize Casimir}}$ given by Eq.~(\ref{s27}), the zeta function approach, {\it i.e.,} $F_{\mbox{\scriptsize Zeta}}$ given by Eq.~(\ref{s35d}), and the zero temperature subtraction approach, {\it i.e.,} $F_{\mbox{\scriptsize ZTSA}}$ given in Eq.~(\ref{s31}). Comparing $F_{\mbox{\scriptsize Zeta}}$ with $F_{\mbox{\scriptsize ZTSA}}$, we observe that, contrary to the massless case, the results are not equivalent: there is an extra term in $F_{\mbox{\scriptsize Zeta}}$, which is its second term in Eq.~(\ref{s35d}) and is temperature-independent. As mentioned above, this extra term is an analytic continuation of the divergent term which appears after using the inhomogeneous Epstein zeta function. Next, we compare these two results with $F_{\mbox{\scriptsize Casimir}}$. First, as can be seen from the Eqs.~(\ref{s27}, \ref{s35d}, \ref{s31}), $F_{\mbox{\scriptsize Casimir}}$ does not include the extra temperature independent term in $F_{\mbox{\scriptsize Zeta}}$, mentioned above. Second,  $F_{\mbox{\scriptsize Casimir}}$ does not include the thermal correction term of the free case, {\it i.e.}, $\Delta F_{\mbox{\scriptsize free}}(T,L)$ given by Eq.~(\ref{s25}), which appears in both $F_{\mbox{\scriptsize Zeta}}$ and $F_{\mbox{\scriptsize ZTSA}}$. Note that this extra term is a non-polynomial function of $T$, the high temperatures limit of which is     
\begin{equation}\label{s34a} 
\Delta F_{\mbox{\scriptsize free}}(T,L) \xrightarrow[T\gg m]{TL\gg 1}  - \left( \frac{7 A L \pi^2}{180 } \right) T^4    +  \left( \frac{A L m^2}{12 } \right) T^2 - \frac{A L m^4}{8 {\pi}^2}\left[\ln \left( \frac{\pi T}{m}\right) -\gamma \right], 
\end{equation}
where $\gamma \simeq 0.5772$ is the Euler-Maschernoi constant. This expansion can also be obtained by the heat kernel coefficients.

As mentioned above, it has long been recognized that the use of the ZFA yields extra unphysical terms. To remedy this, Geyer et al.~\cite{r17Geyer.} defined a renormalization program in which the polynomial terms obtained using the heat kernel coefficients with powers greater or equal to two are subtracted. In their work on bosonic cases, they emphasized that all of the mentioned terms are of quantum character and do not include the classical term which is proportional to temperature. We now explore the results of this renormalization program. Below, we state the renormalization program as presented in reference~\cite{r17Geyer.},
\begin{equation}\label{s35a}
F^{\mbox{\scriptsize ren}}= E_{0}^{\mbox{\scriptsize ren}} + \Delta_{\mbox{\scriptsize T}} F_{0} 
- \alpha_{0} \left( k_{\mbox{\scriptsize B}} T\right)^4 -\alpha_{1} \left( k_{\mbox{\scriptsize B}} T\right)^3  - \alpha_{2} \left(  k_{\mbox{\scriptsize B}} T\right)^2 . 
\end{equation}
The coefficients of these terms depend on geometrical characteristics of the configuration and can be expressed in terms of heat kernel coefficients. Therefore, based on this renormalization program, the physical Casimir free energy for a massive fermion field confined between two parallel plates obtained using zeta function is as follows 
\begin{equation}\label{s36}
F_{\mbox{\scriptsize Zeta}}^{\mbox{\scriptsize ren}}(T,L)= F_{\mbox{\scriptsize Zeta}} +
      \left( \frac{7 A L \pi^2}{180 } \right) T^4    -  \left( \frac{A L m^2}{12 } \right) T^2 ,
\end{equation}
where $ F_{\mbox{\scriptsize Zeta}}$ is given by Eq.~(\ref{s35d}). One can analogously define a renormalized ZTSA free energy as follows  
\begin{equation}\label{s36b}
F_{\mbox{\scriptsize ZTSA}}^{\mbox{\scriptsize ren}}(T,L) = F_{\mbox{\scriptsize ZTSA}} +
      \left( \frac{7 A L \pi^2}{180 } \right) T^4    -  \left( \frac{A L m^2}{12 } \right) T^2 ,
\end{equation}
where $ F_{\mbox{\scriptsize ZTSA}}$ is given by Eq.~(\ref{s31}).

To illustrate the differences between the five expressions for the Casimir free energy, we plot them in Fig.~(\ref{fp21}). As can be seen in this figure, the free energies obtained via the ZFA, and the ZTSA decrease without bound as temperature increases, while the Casimir free energy goes to zero at high temperatures. Moreover, the free energies obtained by applying the renormalization program, {\it i.e.}, $F_{\mbox{\scriptsize Zeta}}^{\mbox{\scriptsize ren}}$, $F_{\mbox{\scriptsize ZTSA}}^{\mbox{\scriptsize ren}}$, do not go to zero as temperature increases, and in fact diverge, due to the subtraction of only the first two terms of Eq.~(\ref{s34a}), which are proportional to $T^2$ and $T^4$. The divergence is due to the remaining $\ln(T/m)$ term. The zero temperature limit of both ZTSA results are compatible with that of the  $F_{\mbox{\scriptsize Casimir}}$, while those of the ZFA are not. This is due to the extra constant term in the $F_{\mbox{\scriptsize Zeta}}$, and $F_{\mbox{\scriptsize Zeta}}^{\mbox{\scriptsize ren}}$, given by the second term in Eq.~(\ref{s35d}).

So far, we have illustrated that the conventional renormalization programs for the ZFA or ZTSA, based on subtracting powers of $T^2$ and higher in the high temperature expansions, do not in general yield the correct results. However, we can still utilize the facility of the zeta function method within the fundamental approach, and devise a new renormalization program. To do this, we need to calculate the free energy of the free massive case by applying the zeta function. We use the first form given by Eq.~(\ref{s4}), integrate over $t$, replace the sum over odd Matsubara frequencies with a sum over all integers minus even integers, which yields
\begin{eqnarray}
F_{\mbox{\scriptsize free}}(T,L)= 4TAL\int\frac{d^3 k}{\left(2 \pi\right)^3}  \lim\limits_{s \to 0} \frac{\partial }{\partial s} \sum\limits_{{n_0} =  1 }^\infty \Bigg\lbrace &\left[ \left(\frac{n_0 \pi}{\beta} \right)^2+M^2\right]^{-s}  - \nonumber\\
&\left[ \left(\frac{2n_0 \pi}{\beta} \right)^2+M^2\right]^{-s}\Bigg\rbrace ,\nonumber 
\end{eqnarray}
where $M^2=k^2+m^2$ and $k^2=K_T^2+k_L^2$. Finally we use the inhomogeneous zeta function to obtain
\begin{eqnarray}\label{s36eew}
&&F_{\mbox{\scriptsize Zeta}}^{\mbox{\scriptsize free}} (T,L) =  \frac{A L}{2\sqrt{\pi^5}} \lim\limits_{s \to 0} \frac{\partial}{\partial s} \int_{0}^{\infty} \frac{dk k^2}{\Gamma (s)} \left[ \omega(k)^{(1-2s)} + \right. \nonumber \\
&&\left. \sum\limits_{j = 1}^{\infty} \left(\frac{\omega(k)}{j \beta}\right)^{\frac{1}{2}-s} \left[2K_{\frac{1}{2}-s} \left(j \beta \omega(k)\right) -\sqrt{2^{(1-2s )}} K_{\frac{1}{2}-s} \left(j \beta \omega(k)\right) \right]\right].
\end{eqnarray}
Then, using the relationship between the Bessel functions, as given in the footnote of Appendix \ref{appendixB:The Argument Principle}, one can see that $F_{\mbox{\scriptsize Zeta}}^{\mbox{\scriptsize free}}(T,L)$, given by Eq.~(\ref{s36eew}), is equivalent to the sum of the last two terms of $F_{\mbox{\scriptsize Zeta}}(T,L)$, given by Eq.~(\ref{s34b}). This implies that 
\begin{equation}
	F_{\mbox{\scriptsize Casimir}}(T,L)= F_{\mbox{\scriptsize Zeta}}^{\mbox{\scriptsize bounded}}(T,L) - F_{\mbox{\scriptsize Zeta}}^{\mbox{\scriptsize free}}(T,L),
\end{equation}
where we have denoted $F_{\mbox{\scriptsize Zeta}}(T,L)$ by $F_{\mbox{\scriptsize Zeta}}^{\mbox{\scriptsize bounded}}(T,L)$ to emphasis that this is in accord with the fundamental definition. This expression can be looked upon as the correct renormalization scheme, but is nothing more than an, albeit useful, expression for the fundamental definition.
\begin{center}
\begin{figure}[!h] 
\includegraphics[width=13.5cm]{fp21.eps}
\caption{\label{fp21} \small
The Casimir free energies per unit volume for a massive fermion field between two parallel plates as a function of temperature with fixed plate separation $L = 1.0$, and mass $m = 0.5$, obtained using five methods within three approaches.}
\end{figure}
\end{center}

One can now obtain other thermodynamic quantities based on the expressions we have obtained for the free energy using the ZFA given in Eq.~(\ref{s35d}), the ZTSA given in Eq.~(\ref{s31}), and their renormalized versions given in Eqs.~(\ref{s36}, \ref{s36b}). For example, we calculate the pressure for a massive fermion field using the free energy obtained via the zeta function, in analogy with the massless case shown in Eq.~(\ref{s19}). We express the result in terms of $P_{\mbox{\scriptsize Casimir}}(T,L)$, given in Eq.~(\ref{s29}), as follows
\begin{eqnarray}\label{s35}
 P_{\mbox{\scriptsize Zeta}}(T,L) &=&  P_{\mbox{\scriptsize Casimir}}(T,L) -\frac{ m^4}{32 \pi^2}  \left[ 3 - 4 \ln(m) \right]+\Delta P_{\mbox{\scriptsize free}}(T,L), \mbox{where} \nonumber \\
\Delta P_{\mbox{\scriptsize free}}(T,L)&=&-\frac{2 T^2 m^2}{\pi^2} \sum\limits_{j = 1}^{\infty} \frac{(-1)^j}{j^2}  K_{2}\left( j \beta m\right) .
\end{eqnarray}
As before, the second term is a constant term which is a leftover from the use of the zeta function, and the third term is the thermal correction to the pressure of the free case, which the zeta function fails to subtract. Next, we calculate the pressure using the free energy obtained via the ZTSA. We express the result in terms of $P_{\mbox{\scriptsize Casimir}}(T,L)$ as follows,
\begin{equation}\label{s61}
P_{\mbox{\scriptsize ZTSA}}(T,L) =  P_{\mbox{\scriptsize Casimir}}(T,L) +\Delta P_{\mbox{\scriptsize free}}(T,L) .
\end{equation}
Next, we calculate the pressure obtained via the renormalized zeta function, {\it i.e.}, $F_{\mbox{\scriptsize Zeta}}^{\mbox{\scriptsize ren}}(T,L)$,  and the renormalized  ZTSA, {\it i.e.}, $F_{\mbox{\scriptsize ZTSA}}^{\mbox{\scriptsize ren}}(T,L)$. The results are,
\begin{eqnarray}\label{s62}
  P_{\mbox{\scriptsize ZTSA}}^{\mbox{\scriptsize ren}}(T,L) &=&  P_{\mbox{\scriptsize ZTSA}}(T,L)  - \left( \frac{7 \pi^2}{180 } \right) T^4 + \left( \frac{m^2}{12 } \right) T^2 \\
 P_{\mbox{\scriptsize Zeta}}^{\mbox{\scriptsize ren}}(T,L) &=&  P_{\mbox{\scriptsize Zeta}}(T,L)  - \left( \frac{7 \pi^2}{180 } \right) T^4 + \left( \frac{m^2}{12 } \right) T^2   .
 \end{eqnarray}
\begin{center}
	\begin{figure}[h!] 
		\includegraphics[width=13.5cm]{fp50.eps}
		\caption{\label{fp50} \small
		The Casimir pressure for a massive fermion field between two parallel plates as a function of temperature with fixed plate separation $L = 1.0$, and mass $m = 0.5$, obtained using five methods within three approaches.}
	\end{figure}
\end{center}
In Fig.~(\ref{fp50}), we compare these results with the Casimir pressure obtained based on the fundamental approach, given by Eq.~(\ref{s29}). As can be seen, the pressures obtained using the ZFA, and the ZTSA are negative at low temperatures and positive at high temperature, while the Casimir pressure is always negative and goes to zero as temperature increases. The differences between these results, besides the constant term present in $P_{\mbox{\scriptsize Zeta}}(T,L)$, are due to the thermal correction of pressure of free case $\Delta P_{\mbox{\scriptsize free}}(T,L)$ which is a non-polynomial function of $T$ for the massive fermion field. The pressures obtained using the ZFA, and the ZTSA all diverge as $T^4$ at high temperatures, and their renormalized versions as $\ln (T/m)$. At $T=0$, only the ZTSA results match the $P_{\mbox{\scriptsize Casimir}}(0,L)$.

\section{Summary and Discussion}\label{summary}
\indent

In this paper, we have explored the implications and results of the fundamental definition of the Casimir free energy, and how they compare with the results based on two general approaches in common use, {\it i.e.}, the zeta function approach (ZFA), and the zero temperature subtraction approach (ZTSA), including their renormalized versions. We have used the zeta function approach as a representative of the analytic continuation techniques, due to uniqueness of their results. For a concrete example which would illustrate the similarity and differences, we have chosen the massless and massive fermion fields confined between two parallel plates, separated by a distance $L$, with the MIT bag boundary conditions at finite temperature. The fundamental definition of $F_{\mbox{\scriptsize Casimir}}$ is the difference between the free energy of the system in the presence of nonperturbative conditions or constraints, and the one with no constraints, which we have referred to as the free case, both being at the {\em same} temperature $T$ and having the same volume. First, we have calculated the Casimir free energy based on the fundamental definition, and have used it to calculate other Casimir thermodynamic quantities, including the pressure, energy, and entropy, and have shown that all of them are finite and go to zero as the temperature, $L$, or mass increases. This occurs due to the subtraction of the free case at the same temperature, which amounts to the cancellation of both the zero temperature and the thermal correction parts of the bounded case which are equivalent to those of the free case. We have also shown that the massless limits of the Casimir thermodynamic quantities obtained for the massive fermion field are identical to the ones obtained for the massless case. 

We have then computed the Casimir thermodynamic quantities using the two other general approaches in common use and compared their results to the ones obtained using the fundamental approach. First, we concentrated on computing the Casimir free energy for a massless fermion field. We first used the zeta function, implemented in three different ways, to evaluate the analytic continuation of the double sums of the spatial and Matsubara modes, and have shown that they all yield equivalent results. Moreover, we have calculated this Casimir free energy using the ZTSA, and have shown that the results are equivalent to those of ZFA. Therefore, we have shown that the use of the ZFA for computing the Casimir free energy of the massless case at finite temperature is equivalent to subtracting the free energy of the free case at zero temperature. However, these results are not equivalent to the ones obtained using the fundamental definition of the Casimir free energy. The difference is the $T^4$ term which is the equivalent of the black-body radiation term. This difference can be traced back to the fact that the free energy of the free case at finite temperature contains the black-body term, the subtraction of which is included in the fundamental definition of the Casimir free energy, but it is not included in the ZFA or the ZTSA. This difference also exists in all other Casimir thermodynamic quantities. For example, $P_{\mbox{\scriptsize Zeta}}$ or $P_{\mbox{\scriptsize ZTSA}}$, contrary to the Casimir pressure obtained by the fundamental approach, are positive at high temperatures, due to the black-body term. A renormalization program has been devised to subtract the high temperature expansions as polynomials in $T$, the use of which yields the correct results for the massless case. 

Next, we have used the ZFA and the ZTSA to calculate the Casimir free energy for a massive fermion field and have shown that, contrary to the massless case, they yield different results. Moreover, similar to the massless case, neither of these results is equivalent to $F_{\mbox{\scriptsize Casimir}}$ obtained via the fundamental definition. The major difference is that both $F_{\mbox{\scriptsize Zeta}}$ and $F_{\mbox{\scriptsize ZTSA}}$ contain the thermal correction to the free case, denoted by $\Delta F_{\mbox{\scriptsize free}}(T,L)$, which they have failed to subtract and is a non-polynomial function of $T$. Moreover, $F_{\mbox{\scriptsize Zeta}}$, includes an extra unphysical temperature-independent term which is a leftover from the use of the inhomogeneous zeta function, is a non-polynomial function of the mass, and goes to zero as $m\to 0$. The high-temperature expansion of $\Delta F_{\mbox{\scriptsize free}}(T,L)$ includes $T^4$, $T^2$ and $\ln (T/m)$ terms. The renormalization program mentioned above removes the first two of these terms in this case, the sum of which does not equal $\Delta F_{\mbox{\scriptsize free}}(T,L)$ for any nonzero temperature. Consequently, $F_{\mbox{\scriptsize Zeta}}^{\mbox{\scriptsize ren}}$ and $F_{\mbox{\scriptsize ZTSA}}^{\mbox{\scriptsize ren}}$ are also not equal to $F_{\mbox{\scriptsize Casimir}}$ at any nonzero temperature. This is in contrast to the massless case, where $\Delta F_{\mbox{\scriptsize free}}(T,L)$ is a simple mononomial $T^4$, the subtraction of which leads to $F_{\mbox{\scriptsize Casimir} }=F_{\mbox{\scriptsize Zeta}}^{\mbox{\scriptsize ren}}=F_{\mbox{\scriptsize ZTSA}}^{\mbox{\scriptsize ren}}$. Therefore, as can be seen from Fig.~(\ref{fp21}), for the massive case, the five expressions for the Casimir free energy are not equivalent at any temperature, except at $T=0$, where the ZTSA results are equal to $F_{\mbox{\scriptsize Casimir}}(0,L)$. In particular, as $T\to \infty$, $F_{\mbox{\scriptsize Casimir}}\to 0$, $F_{\mbox{\scriptsize Zeta}}$ and $F_{\mbox{\scriptsize ZTSA}}$ $\sim T^4$, and $F_{\mbox{\scriptsize Zeta}}^{\mbox{\scriptsize ren}}$ and $F_{\mbox{\scriptsize ZTSA}}^{\mbox{\scriptsize ren}}$ $\sim \ln(T/m)$. These differences are also present in all other Casimir thermodynamic quantities, e.g., the Casimir pressures illustrated in Fig.~(\ref{fp50}). 

To summarize, it has long been recognized that the use of the ZFA yields, as does any other approach based on analytic continuation, extra unphysical terms. In particular, in 2008, Geyer et al.~\cite{r17Geyer.} stated that the use of zeta function does not include all necessary subtractions, and they devised a renormalization procedure in which the terms proportional to powers of $T$ higher than the classical terms obtained in the high temperature limit from the heat kernel method, are to be subtracted. Here, we have shown that this approach, as well as the ZTSA, yields the correct results only for the massless cases. We believe that the correct approach for computing the Casimir thermodynamic quantities is by the use of their fundamental definitions. Using this approach, the final results do not contain any extra unphysical terms, and there is no need to devise any new renormalization program, which would even seem impossible for general non-polynomial terms. However, as we have shown, we can utilize the facility of the zeta function in computing the Casimir quantities by their fundamental definitions, {\it e.g.} $F_{\mbox{\scriptsize Casimir} }=F_{\mbox{\scriptsize Zeta}}^{\mbox{\scriptsize bounded}}-F_{\mbox{\scriptsize Zeta}}^{\mbox{\scriptsize free}}$. Here we have denoted $F_{\mbox{\scriptsize Zeta}}$ by $F_{\mbox{\scriptsize Zeta}}^{\mbox{\scriptsize bounded}}$ to emphasize that the zeta function is applied to the bounded case, and this expression for $F_{\mbox{\scriptsize Casimir} }$ is in accordance with its fundamental definition. 
 

\section*{Acknowledgments}

We would like to thank the research office of the Shahid Beheshti University for financial support.


