Casimir free energy for massive fermions and the applicability of the zeta function
aa r X i v : . [ h e p - t h ] S e p Casimir free energy for massive fermions and the applicabilityof the zeta function
M. Sasanpour ∗ , C. Ajilian † and S. S. Gousheh ‡ Department of Physics, Shahid Beheshti University, G.C., Evin, Tehran 19839,Iran
September 9, 2020
Abstract:
We compute the Casimir Helmholtz free energy, using its fundamentaldefinition, for a fermion field between two parallel plates with the MIT boundaryconditions. We show that the Casimir free energy and other Casimir thermody-namic quantities, including the pressure, energy, and entropy go to zero as thetemperature, the distance between the plates, or mass of the field increases. Wecompare our results with those of four different methods in common use, whichwe calculate explicitly. These include the zeta function method, the zero temper-ature subtraction method, and their renormalized versions. As is well known, thehigh temperature expansion of results of the former two contain the black-bodyterm T , the subtraction of which in the renormalized versions for the masslesscase yields the correct results based on the fundamental definition. However, forthe massive case, we show that these five methods yield five different results. Wethen explain the sources of the differences. Keywords: Casimir effects; finite temperature; massive fermion field; the generalized zetafunction.
In 1948, Casimir [1] predicted an attractive force between neutral parallel con-ducting plates due to the zero-point energy of the quantized electromagnetic field. ∗ m [email protected] † [email protected] ‡ [email protected] T .In a series of papers from 1976 to 1980, Dowker et al. [15–17] calculated thevacuum expectation value of the stress-energy tensor at finite temperature usingthe Green function formalism for a scalar field in curved space-time. They usedthree different renormalization schemes to obtain finite results. First, they sub-tracted the (0 , temperature-spatial mode. Second, they used a ‘Casimir renor-malization’ as the difference between free energies before and after constructingthe boundary, both being at the same temperature, to compute the heat kernel co-efficients. Third, they subtracted the contribution of the free Green function at thezero temperature, which they referred to as ‘the standard flat space renormaliza-tion prescription’. The high temperature limit of h T i in their first and third workhad terms proportional to T and T , while Casimir free energy in their secondwork had terms proportional to T , T and T ln T .In 1978 Balian and Duplantier [18] defined and used the fundamental defini-tion of the Casimir free energy for the electromagnetic field in a region boundedby thin perfect conductors with arbitrary smooth shapes. The high temperaturelimit of their results for parallel plates was proportional to T , while for the enclo-sures included an additional term proportional to T ln T .In 1983 Ambj ø rn and Wolfram [19], computed the Casimir energy and entropyfor scalar and electromagnetic fields in a hypercuboidal region, using the gener-alized zeta function along with the reflection formula as an analytic continuationtechnique. They showed that the high temperature limit of the Casimir energy forthe scalar field in a rectangular cavity in dimensions includes terms pro-portional to T , T and T ln T . In 1991 Kirsten [20] computed the heat kernelcoefficients for the grand thermodynamic potential for a massive bosonic field inhypercuboids in n-dimensions subject to the Dirichlet boundary condition, usingthe zeta function, and in four dimensions obtained terms proportional to T , T , T , T and T ln T .In 1992, Elizalde and Romeo [21], calculated the free energy for a massivebosonic field in hypercuboids of arbitrary dimensions, using multidimensional Ep-stein zeta functions. They indicated that, as stated in [22], to calculate the Casimirfree energy, one has to subtract the free energy of the unconstrained boson field,which would eliminate the T term. In 2008 Geyer et al. [23] suggested a renor-malization procedure to calculate the finite temperature free energy, which wouldsupplement the use of zeta function. They stated that the use of zeta function doesnot include all necessary subtractions, and the terms proportional to powers of T higher than the classical terms obtained in the high temperature limit from the heatkernel method, have to be subtracted. 3o far we have mentioned some of the controversies over the finite temperatureCasimir effects for bosons. Here, for illustrative purposes, we focus on the finitetemperature Casimir effects for the fermion fields. This subject started with thework of Gundersen and Ravndal in 1988 [24], where they explicitly calculated theCasimir free energy for a massless fermion field at finite temperature between twoparallel plates. They defined the Casimir free energy as the difference betweenthe free energy in the presence of the plates at temperature T and that of the freespace at T = 0 . The results that they obtained for the Casimir free energy andpressure included T terms, with force being attractive at low temperatures, andrepulsive and increasing without bound at high temperatures.In 2004, Santana et al. [25] calculated the Casimir pressure and energy by ageneralized Bogoliubov transformation for a massless fermion field in the cases oftwo parallel plates, square wave-guides, and cubic boxes, and they confirmed theresults of [24]. The high temperature limit of these results contains the T termwhich is equivalent to the black-body term. Since then, the zeta function techniquehas been employed in some papers, yielding T for the high temperature limit ofthe Casimir effects [26].In 2010 Cheng [27], calculated the Casimir force for a massless Majoranafermion field between two parallel plates using the piston approach. He used zetafunction analytic continuation for both parts of the piston, and upon subtractingthe two forces, he obtained a Casimir force which is always attractive and ap-proaches zero as temperature increases. In 2011 Khoo and Teo [28] presented asimilar analysis for massive fermions with extra compact dimensions, and foundthat the Casimir force is always attractive at any temperature. Also, they statedthat the high temperature limits of their results for the Casimir free energy andforce contain a term proportional to T , which they called the classical term. In2018 Mo and Jia [29] considered a massless fermion field confined in a rectangu-lar box and defined a renormalized free energy by subtracting the free black-bodyterm along with possible terms proportional to T and T so as to eliminate hightemperature divergences, with reference to Geyer’s work [23]. They used theSchl¨omilch formula which is based on the zeta function. They showed that, aftersubtracting these terms, both the Casimir free energy and force for parallel platesgo to zero at high temperatures.As is apparent from the historical outline presented, the zeta function has beenused extensively in the calculations of the Casimir effects to evaluate the sumsover the regular spatial or Matsubara modes, and often as an analytical contin-uation technique. In some methods, zeta function is used explicitly to calcu-late the Casimir thermodynamic quantities, e.g. in [21], or implicitly, e.g. in theSchl¨omilch’s formula, or as a supplementary part. Examples of the latter includethe heat kernel method and the Bogoliubov transformation where the zeta functionis used to evaluate the final summations. As we shall show, the use of the gener-4lized zeta function for the sum over spatial modes, which is possible only for themassless cases since they are regular, is equivalent to subtracting the case with noboundaries at zero temperature, and this yields the correct results for the zero tem-perature cases. Furthermore, as we shall show, the use of the zeta function for thesum over Matsubara frequencies, which are always regular, is roughly equivalentto subtracting the zero temperature case . However, these equivalences imply thatthey are both in principle contrary to the fundamental definition, and as we shallshow for the fermionic case, they yield, in principle, results which are not equiva-lent to that resulting from the fundamental definition. The former results containextra terms, whose high temperature limits include terms proportional to powersof T higher than the classical term, up to and including the black-body term T d in d space-time dimension. As we shall show, only for the case a massless fermionscan these terms be removed by the renormalization programs introduced [23, 29],or cancel out in the piston method [23, 27, 28].In this paper we solve the massive fermionic case between two parallel plates,as the simplest nontrivial example, by five different methods, all yielding differentresults. These methods are based on the following: 1. the fundamental definition,2. the Epstein zeta function, 3. the zero temperature subtraction, 4. the Epstein zetafunction with the renormalization program of [23,29], and 5. the zero temperaturesubtraction with the renormalization program of [23, 29]. There are three limitsin which one might naturally expect the Casimir effects to vanish: large plateseparation, large mass limit, and the high temperature limit. Here we show thatout of these five methods, only the one based on the fundamental definition has allthree of the desired properties, i.e., all of the Casimir thermodynamics quantitiesincluding the free energy, the energy, the entropy, and the pressure go to zero atthose limits.Throughout this paper, we display or outline multiple ways of computing thesame physical quantities, in order to ascertain the delicate cancellations of diver-gent sums and integrals have been done correctly. In Sec. 2, we present two formsfor the free energy for a fermion field at finite temperature, which is subject to theMIT boundary conditions at two plates but otherwise free, starting with the pathintegral formalism. In Sec. 3, we calculate the Casimir free energy for a masslessfermion field, based on fundamental definition and using the Poisson summationformula, and show that the Casimir free energy and pressure go to zero at hightemperatures and large distances. In Sec. 4, we calculate the Casimir free en-ergy and pressure of a massless fermion field using the generalized zeta functionand the zero temperature subtraction methods, obtaining identical results whichhave extra black-body terms as compared to the results based on the fundamentaldefinition. We then show how the renormalization program subtracts these extra We shall clarify this statement in Secs. (4,6) T , and the last twoas ln T . Finally, in Sec. 7 we present our conclusion. Historically, the oldest and most often used approach to thermal field the-ory is the imaginary-time formalism. This approach started with the work ofFelix Bloch in 1932, who noticed the analogy between the inverse temperatureand imaginary-time [30], which led to the so-called temperature Green functionswith purely imaginary-time arguments. In 1955, Matsubara presented the firstsystematic approach to investigate quantum field theory at finite temperature byimaginary-time formalism, using the Wick rotation [31]. The discrete frequen-cies in this formalism are known as Matsubara frequencies. In 1957, Ezawa et al.extended the Matsubara’s work to the relativistic quantum field theory [32]. Theydiscovered the periodicity (anti-periodicity) conditions for the Green function ofboson (fermion) fields, the generalization of which became known as the KMS(Kubo [33] (1957), Martin and Schwinger [34] (1959)) condition. In the 1960s,Schwinger [35], Keldysh [36] and others [37] developed the real time formalismfor the finite temperature field theory. The latest developement of this formalismwas presented by Takahashi and Umezawa [38] based on an operator formulationof the field theory at finite temperature which is called Thermofield Dynamics(TFD). Since then, many subjects in finite temperature field theory, e.g. thermalWard-Takahashi relations, KMS relations, renormalization procedure, have beenstudied and reported in for example [39–43].In this paper we use the Matsubara formalism to study the Casimir effect fora free fermion field confined between two parallel plates at finite temperature.In this formalism, a Euclidean field theory is considered by a wick rotation onthe time coordinate t → − iτ such that, the Euclidean time τ is confined to theinterval, τ ∈ [0 , β ] , where β = ( kT ) − [31, 40]. The partition function in the path6ntegral representation becomes: Z = Z ψ ( β, −→ r )= − ψ (0 , −→ r ) ψ ( β, −→ r )= − ψ (0 , −→ r ) DψDψ exp − β Z dτ Z d xL E . (2.1)For a free fermion field, the expression simplifies as follows, Z = Det ( γ µE ∂ µE + m ) . (2.2)Using the partition function given by Eq. (2.2), the free energy is obtained as, F = − Ln( Z ) β = − T Ln [Det ( γ µE ∂ µE + m )] = − T Tr (cid:2) Ln (cid:0) P E + m (cid:1)(cid:3) . (2.3)The trace in Eq. (2.3) indicates the summation over eigenvalues of Dirac opera-tor in momentum space representation. Moreover, the modes of zero-componentof momentum or the Matsubara frequencies are discrete, due to the KMS anti-periodicity condition on the finite τ interval, ω n = n πβ , where ( n = ± , ± , ± , ... ) . (2.4)We impose the MIT boundary condition on the plates, which prevents the flowof fermion current out of the plates, as follows (cid:0) iγ µ n jµ (cid:1) Ψ( x ) | ( z = z j ) = 0 , (2.5)where n jµ is the unit vector perpendicular to the plate located at z j . We considerthe plates to be located at z = − L and z = L , and solve the free Dirac equationin three spatial dimensions, subject to the above boundary conditions. We obtainthe following condition for the discrete spatial modes in z direction, f ( k n ) = k n cos( k n L ) + m sin( k n L ) = 0 . (2.6)Note that the modes for the massive case are irregular, i.e., not equally spaced.However, for the massless case the modes are regular and given by, k n = n π L ( n = 1 , , , .... ) . (2.7)Using Eqs. (2.3,2.4), the expression for the free energy becomes F bounded ( T, L ) = − T A Z d K T (2 π ) ∞ ′ X n = −∞ X n Ln "(cid:18) n πβ (cid:19) + ω n ,K T , (2.8)7here ω n ,K T = p k n + K T + m , the prime on the summation denotes re-striction of the sum to odd integers, and A denotes the area of the plates. Weshall refer to this expression for free energy as the first form. One can perform thesum over the Matsubara frequencies, using the Principle of the Argument [44], toobtain the usual form of the free energy in statistical mechanics [11], F = − A R d K T (2 π ) P n (cid:2) ω n ,K T + 2 T Ln (cid:0) e − βω n ,KT (cid:1)(cid:3) . (2.9)One advantage of this second form of the free energy, given by Eq. (2.9), is that thecontribution of the zero temperature part is separated from the thermal correctionpart. In this section we calculate the Casimir free energy, using its fundamental defi-nition, for a free massless Dirac field between two parallel plates, separated bya distance L , with the MIT boundary conditions. In Sec. 5, we generalize to themassive case, and verify that, as expected, its massless limit coincide with the re-sults of this section. As mentioned above, the fundamental definition of F Casimir isthe difference between the free energy of the system in the presence of nonpertur-bative conditions or constraints, and the one with no constraints, both being at the same temperature T and having the same volume. The nonperturbative conditionsor constraints include boundary conditions, background fields such as solitons,and nontrivial space-time backgrounds. In cases where the constraints are in theform of non-trivial boundary conditions, the free case can be defined as the casein which the boundaries have been placed at spatial infinities. For the latter cases,the fundamental definition can be written as, F Casimir ( T, L ) = F bounded ( T, L ) − F free ( T, L ) , (3.1)where the dependence of F free on L simply denotes the restriction of the volumeof space considered.To calculate the free energy for a massless fermion we use the first form pre-sented in Eq. (2.8), along with Eq. (2.7), and obtain F bounded ( T, L ) = 2
T A Z d K T (2 π ) ∞ ′ X n = −∞ ∞ ′ X n =1 lim s → ∂∂s ∞ Z e − t h ( n πβ ) + ( n π L ) + K T i Γ( s ) t − s dt, (3.2) We have used the following identity: Log ( x ) = lim s → (cid:20) − ∂∂s ∞ R e − tx Γ( s ) t − s dt (cid:21) . n and n . First,we express the sums in the following symmetrized forms ∞ ′ X n = −∞ e − t ( n πβ ) = 12 " ∞ X n = −∞ e − t (2 n π β + ∞ X n = −∞ e − t (2 n − π β , (3.3) ∞ ′ X n =1 e − t ( n π L ) = 14 " ∞ X n = −∞ e − t ( n
1+ 12 ) π L + ∞ X n = −∞ e − t ( n − ) π L . (3.4)Using the Poisson summation formula for the sums over Matsubara frequenciesand the spatial modes on the right hand side of Eqs. (3.3, 3.4), we obtain ∞ ′ X n = −∞ e − t ( n πβ ) = β √ πt + β √ πt ∞ X n =1 ( − n e − n β t , (3.5) ∞ ′ X n =1 e − t ( n π L ) = L √ πt + L √ πt ∞ X n =1 ( − n e − n L t . (3.6)Next, we evaluate all of the integrals and limits, except for the divergent termresulting from the multiplication of the first terms on the right hand sides ofEqs. (3.5, 3.6), to obtain the free energy between the two plates. Below we ex-press the results in a form in which the zero temperature part is separated from thethermal correction part, as follows F bounded ( T, L ) = F bounded (0 , L ) + ∆ F bounded ( T, L ) , where F bounded (0 , L ) = AL π lim s → ∂∂s ∞ Z dt Γ( s ) t − s − Aπ L , and ∆ F bounded ( T, L ) = − ALπ T + 8 π ∞ X n =1 ∞ X n =0 ALT ( − n + n (cid:2) n + (2 n T L ) (cid:3) . (3.7)On the other hand, the contribution of the free energy of the unconstrained case, The Poisson summation formula (see, for example, [45, 46]) for a continous and boundedfunction f on R can be expressed as, ∞ X n = −∞ f ( n ) = ∞ X m = −∞ ∞ Z −∞ dxf ( x ) e − i πmx T and same volume V = AL , is F free ( T, L ) = 2
T AL lim s → ∂∂s ∞ Z dt Γ( s ) t − s ∞ ′ X n = −∞ e − t ( n πβ ) Z d K T (2 π ) Z dk π e − t ( k + K T ) . (3.8)Performing the same procedure as for the bounded case, we obtain the free energyof the free case. Below, we again express the results in a form in which the zerotemperature part is separated from the thermal correction part, as follows F free ( T, L ) = F free (0 , L ) + ∆ F free (0 , L ) , where F free (0 , L ) = AL π lim s → ∂∂s ∞ Z dt Γ( s ) t − s , and ∆ F free ( T, L ) = − ALπ T . (3.9)As can be seen from the Eq. (3.9), F free (0 , L ) is divergent and is precisely thesame as the divergent part of F bounded (0 , L ) given in Eq. (3.7), while the thermalcorrection term of the free case, i.e. black-body term, is finite. We can also startwith the second form of the free energy formula given by Eq. (2.9) to calculatethe free energy of the free case and obtain the same result as given by Eq. (3.9).When calculating the Casimir free energy, based on its fundamental definition,i.e., Eq. (3.1), the divergent terms and the black-body terms of F bounded ( T, L ) and F free ( T, L ) completely cancel, yielding, F Casimir ( T, L ) = − Aπ L + 8 ALT π ∞ X n =1 ∞ X n =0 ( − n + n (cid:2) n + (2 n T L ) (cid:3) . (3.10)One can now compute the sum over n to obtain, F Casimir ( T, L ) = AT πL ∞ X n =1 ( − n πn T L ) coth (2 πn T L ) n sinh (2 πn T L ) . (3.11)The zero temperature limits of Eqs. (3.10, 3.11) yield the following well knownresult F Casimir (0 , L ) = E Casimir (0 , L ) = − π A/ (2880 L ) . One can also start withthe second form of the free energy formula (2.9) for this case and obtain exactlythe same expression for the Casimir free energy as given by Eq. (3.11).In Fig. (1), the Casimir free energy is plotted as a function of temperaturefor various values of L . As can be seen from this figure, F Casimir ( T, L ) is alwaysnegative, and goes to zero as the temperature or L increases. Note that the van-ishing of F Casimir ( T, L ) as T goes to infinity, occurs due to the subtraction of the10ree case at the same temperature which amounts to the complete cancellationof the black-body term, without the need for any extra renormalization program.Moreover, this shows that there is no classical term proportional to T for the mass-less fermions between plates, which, as we shall show, also holds for the massivefermions. L = = = = - - - - T C as i m i r H e l m ho l t z f r eee n e r g y p e r un i t v o l u m e Figure 1:
The Casimir free energy per unit volume for a massless fermion field betweentwo parallel plates as a function of temperature for various values of plate separations L = { . , . , . , . } . Having obtained the Casimir free energy, one can easily calculate all otherthermodynamic quantities such as the Casimir pressure, energy, and entropy. Forexample the Casimir pressure is given by, P Casimir ( T, L ) = − A ∂∂L F
Casimir ( T, L ) = T πL ∞ X n =1 ( − n n sinh (2 πn T L ) × (cid:26) T Lπn ) coth (2 πn T L ) + 2( πn T L ) (cid:2) (2 πn T L ) − (cid:3) (cid:27) . (3.12)Moreover, one can calculate directly the Casimir pressure based on its fundamen-tal definition, as given by [12, 13], which is the differences between the pressure11nside the two plates and outside the plates. To this end, we have considered twoinner plates enclosed within two outer plates, as the distance of the latter goes toinfinity, and have obtained the same result as given by Eq. (3.12). By integratingover the distance between two plates, at fixed temperature, the Casimir free energycan be calculated, yielding the same result as given by Eq. (3.11), without any ex-tra terms. In Fig. 2, the Casimir pressure is plotted as a function of temperaturefor various values of L . As can be seen from this figure, P Casimir ( T, L ) is alwaysnegative, and goes to zero as the temperature or L increases. L = = = = - - - - - - T C as i m i r P r ess u r e Figure 2:
The Casimir pressure for a massless fermion field between two paral-lel plates as a function of temperature for various values of plate separations L = { . , . , . , . } . Another method which can be used to calculate the Casimir free energy, isthe Abel-Plana formula (see, for example, [47]) , which we use to calculate the The simplest form that is needed here is the following ∞ X n =0 f (cid:18) n + 12 (cid:19) = ∞ Z f ( t ) dt − i ∞ Z f ( it ) − f ( − it ) e πt + 1 dt. β , and then use the Abel-Plana formula to obtain, F Casimir ( T, L ) = − π A L + 7 π T AL
180 + T Aπ ∞ X j =1 ( − j πj T L coth (cid:0) πj T L (cid:1) j sinh (cid:0) πj T L (cid:1) . (3.13)The first term is the zero temperature part and the rest constitute the thermal cor-rection part. This form is equivalent to the result obtained above, i.e. Eq. (3.11).However, to have an accurate plot using this form, one has keep a large numberof terms, otherwise the graph would show an increase at high values of T . This isdue to the high β expansion mentioned above. In this section, we consider a commonly used alternative method, called the gen-eralized zeta function method, for computing the Casimir free energy for masslessfermions at finite temperature. To fully explore this method, we consider three dif-ferent ways of using the zeta function and show that they yield identical results.Moreover, we shall compute the Casimir free energy using the zero temperaturesubtraction method [24], and show that its results are identical to those of the zetafunction methods. However, as we shall show, these results are not equivalentto the one obtained in the last section based on the fundamental definition of theCasimir free energy.For the computation of the Casimir free energy using the zeta function method,we use only the first form given by Eq. (2.8). We evaluate the integral over thetransverse momenta to obtain the following expression, F ( T, L ) =
T A π lim s → ∂∂s ∞ ′ X n = −∞ ∞ ′ X n = −∞ ∞ Z dt Γ( s ) t − s e − t h ( n πβ ) + ( n π L ) i . (4.1)As mentioned before, both sums are over odd integers, which can be written asthe difference between sums over all integers and even integers. By integratingover the parameter t we obtain, F ( T, L ) =
T A π lim s → ∂∂s Γ( s − s ) ∞ X n = −∞ ∞ X n = −∞ "(cid:18) n π β + n π L (cid:19) − s − (cid:18) n π β + n π L (cid:19) − s − (cid:18) n π β + n π L (cid:19) − s + (cid:18) n π β + n π L (cid:19) − s . (4.2)13o use the generalized zeta function, we have to impose the constraint that each ofthe double sums should not include the ( n = 0 , n = 0) mode. For cases in whichthe spatial modes do not include a zero mode, this constraint is automaticallysatisfied, otherwise this would amount to a renormalization. Our case is in thecategory of the former, and the (0,0) modes cancel between the four terms inEq. (4.2).The first method is to use the homogeneous generalized zeta function to dosimultaneously the double summations for each of the four terms in Eq. (4.2). Inthis case the analytic continuation is rendered by the reflection formula (see Ap-pendix A). Exactly the same method has been used in [48] to obtain an expressionfor the Casimir free energy for a massless field confined between two plates. Thefinal result of [48] has been presented as a finite fractional expression which in-cludes a double sum over n and n . Here, we simplify these summations (seeAppendix A), compute the sum over the Matsubara frequencies, and present thefinal result as follows, F Zeta ( T, L ) = ∆ F free ( T, L ) + AT πL ∞ X n =1 n (cid:26) csch (4 πT Ln ) − πT Ln ) + 4 T Lπn (cid:20) coth (4 πT Ln )sinh (4 πT Ln ) − πT Ln )sinh (2 πT Ln ) (cid:21)(cid:27) , (4.3)where ∆ F free ( T, L ) is the thermal correction term of the massless free case, whichis the black-body term proportional to T , given in Eq. (3.9). The zero temperaturelimit of this expression gives the correct result for F Casimir (0 , L ) = E Casimir (0 , L ) = − π A/ (2880 L ) . We have denoted the Casimir free energy obtained by thismethod as F Zeta , to distinguish it from the one obtained using the fundamentaldefinition, which we have simply denoted by F Casimir .The second method is to use the inhomogeneous form of the zeta function tosum over the spatial modes for each of the four terms in Eq. (4.2) yielding (seeAppendix A), F Zeta ( T, L ) = − T ALπ ∞ X n =1 n + T A πL ∞ X n =1 ( − n πT Ln ) coth(2 πT Ln ) n sinh(2 πT Ln ) . (4.4)The first term on the right hand side is a sum over temperature modes and isdivergent and the second one is identical to our result (3.11). If we use the zetafunction on this sum as an analytic continuation, i.e. ζ ( −
3) = 1 / , we obtain, F Zeta ( T, L ) = ∆ F free ( T, L ) +
T A πL ∞ X n =1 ( − n πT Ln ) coth(2 πT Ln ) n sinh(2 πT Ln ) . (4.5)14he first term is again the back-body term given by Eq. (3.9). The zero tempera-ture limit of this expression gives the correct result for F Casimir (0 , L ) = E Casimir (0 , L ) = − π A/ (2880 L ) .The third method is to use the inhomogeneous form of the zeta function to sumover the Matsubara frequencies for each of the four terms in Eq. (4.2) yielding (seeAppendix A), F Zeta ( T, L ) = − Aπ L ∞ X n =1 n + T Aπ ∞ X j =1 ( − j (cid:0) πj T L (cid:1) coth (cid:0) πj T L (cid:1) j sinh (cid:0) πj T L (cid:1) . (4.6)If we again use the zeta function on the first sum as an analytic continuation, weobtain, F Zeta ( T, L ) = − Aπ L + T Aπ ∞ X j =1 ( − j (cid:0) πj T L (cid:1) coth (cid:0) πj T L (cid:1) j sinh (cid:0) πj T L (cid:1) . (4.7)The first term is the zero temperature part, while the black-body T term is em-bedded in the high temperature limit of the second term. It can be easily shownthat the three expressions obtained by the three different methods of using the zetafunction, i.e. Eqs. (4.5, 4.7, 4.3), are equivalent. Although the last two methodshave not been used before in the literature, as far as we know, for obtaining theCasimir free energy, it is important to see that various ways of utilizing the zetafunction yield equivalent results.Next we calculate the Casimir free energy using the zero temperature subtrac-tion method (ZTSM) [24]. This quantity is defined as follows, F ZTSM ( T, L ) = F bounded ( T, L ) − F free (0 , L ) . (4.8)We present or outline five different methods of calculating this quantity. The firstfour methods are based on the first form of the free energy given by Eq. (2.8), andthe fifth is based on the second form given by Eq. (2.9). In the first method, werepresent the sum over spatial modes in the symmetrized form used in Eq. (3.4)and use the Poisson summation formula to obtain, F ZTSM ( T, L ) = 16
ALπ T " ∞ X n =0 (cid:18) n + 12 (cid:19) − Z ∞ dkk + T A πL ∞ X n =1 ( − n πT Ln ) coth(2 πT Ln ) n sinh(2 πT Ln ) . (4.9)Using the Abel-Plana formula for the first part of Eq. (4.9), the final result isidentical to F Zeta ( T, L ) given by Eq. (4.5). In the second method, we represent15he sum over the spatial modes as the difference between sums over all integersand even integers, as used in Eq. (4.2), and again use Poisson summation formula.The final result is identical to Eq. (4.3). The third method is similar to the first,except we perform the sum over the Matsubara frequencies using the poissonsummation formula given by Eq. (3.5), and the final result is identical to Eq. (4.7).In the fourth method, we use directly the definition given in Eq. (4.8), with itsterms explicitly calculated in Sec. 3 and given by Eqs.(3.7, 3.9). The final resultis identical to Eq. (4.5). For the fifth method, we use the second form of thefree energy given by Eq. (2.9) and use the Abel-Plana formula, and the result isidentical to Eq. (4.7). Exactly the same method has been used in [24] and theCasimir free energy obtained in [24] is equivalent to Eq. (4.7). They also obtainedthe Casimir free energy by calculating and using the Casimir pressure. Their finalresult is identical to Eq. (4.3).So far, in this section we have shown that for massless fermions, the resultsobtained using any zeta function method is identical to that of ZTSM. However,these results are not equivalent to the one obtained in the last section using thefundamental definition of the Casimir free energy which is given by Eq. (3.11).We can summarize our results for the massless case as follows, F Zeta ( T, L ) = F ZTSM ( T, L ) = F Casimir ( T, L ) + ∆ F free ( T, L ) , (4.10)where two equivalent expressions for F Casimir ( T, L ) are given by Eqs.(3.11, 3.13).The difference is the thermal correction to the free energy of the free case, i.e. ∆ F free ( T, L ) = − (7 ALπ / T as given in Eq. (3.9), which is equivalent to theblack-body term. This difference can be traced back to the fact that the free energyof the free case at finite temperature contains the black-body term, the subtractionof which is included in the fundamental definition of the Casimir free energy, but itis not included in the zeta function method or the ZTSM. We compare these resultsin Fig (3). As can be seen from this figure, the free energy obtained via the zetafunction or ZTSM decreases as T , while the one obtained via the fundamentaldefinition goes to zero at high temperatures.One can now easily calculate all other thermodynamic quantities using theexpressions obtained for the free energies by the zeta function or ZTSM. For ex-ample, calculation of pressure yields, P Zeta ( T, L ) = P ZTSM ( T, L ) = P Casimir ( T, L ) + ∆ P free ( T, L ) , (4.11)where an expression for P Casimir ( T, L ) is given by Eq. (3.12), and ∆ P free ( T, L ) =(7 π / T is the thermal correction to the pressure of the free case. In Fig. (4),16 undamental definitionZeta function or ZTSM - - - (cid:4)(cid:5)(cid:6)(cid:7) - - (cid:8)(cid:9)(cid:10)(cid:11) - - T C as i m i r H e l m ho l t z f r eee n e r g y p e r un i t v o l u m e Figure 3:
The Casimir free energy per unit volume for a massless fermion field betweentwo parallel plates as a function of temperature with fixed plate separations, L = 1 . . Thesolid line is for the one obtained via the fundamental definition, and the dashed line is forthe ones obtained by the zeta function, or the zero temperature subtraction method ZTSM. we compare the pressure obtained using the zeta function method or ZTSM, givenby Eq. (4.11), with the Casimir pressure obtained based on the fundamental def-inition given by Eq. (3.12). As can be seen, the pressure obtained using the zetafunction method is attractive at low temperatures and becomes repulsive at hightemperature, while the Casimir pressure vanishes at high temperatures. The differ-ence between the two results (4.11, 3.12) is due to the pressure of the black-bodyterm.As mentioned in the Introduction, it has been recognized that the zeta functionmethod might yield additional unphysical terms, and renormalization programshave been devised to subtract polynomials in T appearing in the large temper-ature limits [23, 29]. These are usually calculated using the heat kernel coeffi-cients. In this case, the only nonzero term is the mononomial T term, whichturns out to be identical the the black-body term and the subtraction of whichyields the correct results, based on the fundamental definition. Specifically, the re-moval of ∆ F free ( T, L ) = − (7 ALπ / T from the expression for F Zeta ( T, L ) = F ZTSM ( T, L ) in Eq. (4.10), and the removal of ∆ P free ( T, L ) = (7 π / T from17 undamental definitionZeta function or ZTSM (cid:12)(cid:13)(cid:14) (cid:15)(cid:16)(cid:17) - (cid:18)(cid:19)(cid:20)(cid:21) (cid:22)(cid:23)(cid:24)(cid:25) T C as i m i r P r ess u r e Figure 4:
The Casimir pressure for a massless fermion field between two parallel platesas a function of temperature with fixed plate separations, L = 1 . . The solid line is for thepressure obtained via the fundamental definition, and the dashed line is for the pressureobtained by the zeta function, or the zero temperature subtraction method ZTSM. the expression for P Zeta ( T, L ) = P ZTSM ( T, L ) in Eq.(4.11), will yield the correctthe results. We like to emphasize that these extra unphysical terms appear in theresults of zeta function method and the ZTSM for different reasons. In the formercase they are left out by the embedded analytic continuation, and in the latter casethey are left out by its definition. As we have shown, in the massless case theseterms can be easily removed by the renormalization programs that have been de-vised. However, in the next section, we explore the massive case for which theextra unphysical terms are not of simple polynomial forms, and hence the renor-malization programs thus defined cannot completely remove them. In this section, we calculate the Casimir free energy, using its fundamental def-inition as given by Eq. (3.1), for a massive fermion field confined between twoparallel plates with the MIT boundary conditions at finite temperature. Then, wecalculate other Casimir thermodynamic quantities, including pressure, energy, and18ntropy, and show that all of them are finite and vanish as the temperature, mass,or L increases. In the next section, we compute the Casimir free energy usingthe inhomogeneous zeta function and also using the zero temperature subtractionmethod [24] and compare the results.We start with the first form of the free energy given by Eq. (2.8) and integrateover the transverse momenta. Then we use the Poisson summation formula onthe Matsubara frequencies , evaluate the derivative with respect to s , and take thelimit s → , to obtain F bounded ( T, L ) = A π X n > " ω n + r T ω n π ∞ X n =1 ( − n p n K ( βn ω n ) , (5.1)where ω n = p k n + m . The spatial modes are the roots of f ( k n ) in Eq. (2.6),which for the massive case are irregular, i.e., they are not equally spaced. Toevaluate the sum over the spatial modes, we use the Principle of the Argumentand after simplifying (see Appendix B) we can express the free energy for thebounded region as F bounded ( T, L ) = − Aπ ( L Z ∞ p ω ( p ) dp − LT m ∞ X j =1 ( − j j K ( jβm ) + Z ∞ " p + 2 T ∞ X j =1 ( − j j p sin ( jβp ) Ln (cid:18) ω ( p ) − mω ( p ) + m e − Lω ( p ) (cid:19) dp ) , (5.2)where ω ( p ) = p p + m . Since we are going to use the fundamental definition ofthe Casimir free energy, we also need to calculate the free energy of the free mas-sive case at finite temperature. We start with the first form of the free energy, andfollowing steps analogous to those of the bounded case, we can express the freeenergy of free case as a zero temperature part and a finite temperature correctionpart as follows, F free ( T, L ) = F free (0 , L ) + ∆ F free ( T, L ) , where ,F free (0 , L ) = − ALπ Z ∞ k ω ( k ) dk , and , ∆ F free ( T, L ) = 2
ALT m π ∞ X j =1 ( − j j K ( jβm ) . (5.3) The form of the Poisson summation we have used is: ∞ ′ P n = −∞ e − t ( n πβ ) = βπ R ∞ e − ty dy + π ∞ P n =1 ( − n n R ∞ yte − ty sin( yn β ) dy F bounded ( T, L ) given by Eq. (5.2) are identical to thetwo terms of F free ( T, L ) given by Eq. (5.3). Now, using the fundamental definition,as expressed in Eq. (3.1), these terms cancel each other upon subtraction andthe Casimir free energy for a massive fermion field confined between two platesbecomes , F Casimir ( T, L ) = − Aπ Z ∞ ( p + 2 T p ∞ X j =1 ( − j j sin ( jβp ) ) × Ln (cid:18) ω ( p ) − mω ( p ) + m e − Lω ( p ) (cid:19) dp. (5.4)The zero temperature and finite temperature correction parts, i.e. F Casimir (0 , L ) and ∆ F Casimir ( T, L ) , are associated with the two terms in the curly bracket in Eq. (5.4),respectively.One can easily show that using the second form of the free energy given byEq. (2.9), one obtains exactly the same expression as in Eq. (5.4). In Fig. (5), weplot the Casimir free energy of a massive fermion field for various values of mass.As can be seen, the Casimir free energy goes to zero rapidly as the temperatureor mass of the fermion field increases. As can be seen directly in Eq. (5.4), theCasimir free energy goes to zero rapidly as L increases. Moreover, as can beseen from Fig. (5), and can be shown easily from Eq. (5.4), the massless limit ofour result for the massive case coincides exactly with the massless case given byEq. (3.11). The zero temperature limit of F Casimir ( T, L ) given by Eq. (5.4) yieldsthe following well known result, as reported in, for example, [49, 50] F Casimir (0 , L ) = E Casimir (0 , L ) = − Aπ Z ∞ p Ln (cid:20) ω ( p ) − mω ( p ) + m e − Lω ( p ) (cid:21) dp. (5.5)Now, one can obtain other thermodynamic quantities including, the Casimirpressure, Casimir energy, and Casimir entropy from the expression we have ob-tained for the Casimir free energy Eq. (5.4). We calculate the Casimir pressure fora massive fermion field, in analogy with the massless case shown in Eq. (3.12), The sum in Eq. (5.4) can be written in closed form: ∞ P j =1 ( − j j sin ( jβp ) = − tan − h tan (cid:16) pβ (cid:17)i = = = = = = = (cid:26)(cid:27)(cid:28) (cid:29)(cid:30)(cid:31) - !" - - %&’() - - *+,-. T C as i m i r H e l m ho l t z f r eee n e r g y p e r un i t v o l u m e Figure 5:
The Casimir free energy per unit volume for a massive fermion field betweentwo parallel plates as a function of temperature with fixed plate separations, L = 1 . , forvarious values of mass m = { . , . , . , . , . , . , . } . Note that the Casimir freeenergy goes to zero as the temperature or mass increases. and obtain, P Casimir ( T, L ) = − π ∞ Z (cid:2) ω ( p ) − mω ( p ) (cid:3) [1 − tanh ( Lω ( p ))] × p + (2 T p ) ∞ P j =1 ( − j j sin ( jβp ) ω ( p ) + m tanh ( Lω ( p )) dp. (5.6)The zero temperature and finite temperature correction parts, i.e. P Casimir (0 , L ) and ∆ P Casimir ( T, L ) , are associated with the two terms on the numerator of the fractionterm in Eq. (5.6), respectively. We plot P Casimir ( T, L ) for various values of massin Fig. (6). As can be seen, the Casimir pressure also goes to zero rapidly as thetemperature or the mass of fermion field increases. Moreover, as can be seen fromFig. (6), and can be shown easily from Eq. (5.6), the massless limit of our resultfor the massive case coincides exactly with the massless case given by Eq. (3.12).The Casimir energy can be calculated using either of the following two ex-21 = = = = = = = /23 - - - T C as i m i r P r ess u r e Figure 6:
The Casimir pressure for a massive fermion field between two parallel platesas a function of temperature with fixed plate separations, L = 1 . , for various values ofmass m = { . , . , . , . , . , . , . } . Note that the Casimir pressure goes to zeroas the temperature or mass increases. pressions, E Casimir ( T, L ) = E bounded ( T, L ) − E free ( T, L ) = ∂∂β [ βF Casimir ( T, L )] . (5.7)The first expression is its fundamental definition. We use the second expressionto obtain, E Casimir ( T, L ) = − A π Z ∞ dp (cid:0) Lp − m (cid:1) [1 − tanh ( Lω ( p ))] × p + 6 T ∞ P j =1 ( − j j p ( (2 jβp ) cos( jβp ) + [( jβp ) −
2] sin( jβp ) ) ω ( p ) ( ω ( p ) + m ) [ ω ( p ) + m tanh ( Lω ( p ))] . (5.8)Finally, we calculate the Casimir entropy and obtain, S Casimir ( T, L ) = − ∂∂T F Casimir ( T, L ) = − AT π Z ∞ dp (cid:0) Lp − m (cid:1) × [1 − tanh ( Lω ( p ))] ∞ X j =1 ( − j p [(3 jβp ) cos( jβp ) + [( jβp ) −
3] sin( jβp )] j ω ( p ) ( ω ( p ) + m ) [ ω ( p ) + m tanh ( Lω ( p ))] . (5.9)22n Fig. (7), we show all of these Casimir thermodynamic quantities. Note thatall of these quantities are finite and go to zero at high temperatures. In analogywith the case of Casimir free energy, one can easily show that all of the Casimirthermodynamic quantities also go to zero as m or L increases. F Casimir ( T78 ) / V P Casimir ( ) E Casimir ( <=> ) / ?@A Casimir ( BCD ) / E FGH
IJK - - LMNO - - T C as i m i r qPQRSUWXYZ Figure 7:
The Casimir thermodynamic quantities, including the free energy, pressure,energy, and entropy, obtained using the fundamental definition, for a massive fermionfield between two parallel plates as a function of temperature with fixed plate separation L = 1 . , and mass m = 0 . . The solid line is for the Casimir free energy per unitvolume F Casimir ( T, L ) /V , the large dashed line is for the Casimir pressure P Casimir ( T, L ) ,the small dashed line is for the Casimir energy per unit volume E Casimir ( T, L ) /V , anddot-dashed line is for the Casimir entropy per unit volume multiplied by the temperature T S
Casimir ( T, L ) /V . Note that all of the Casimir quantities go to zero as the temperatureincreases. The zeta function method has been used to calculate the Casimir free energy forthe massive fermion field between two plates and some solutions have been pre-sented (see for example in [51,52]). In this section, we compute explicitly the final23esults for the Casimir free energy and Casimir pressure for this problem using thezeta function method, and also using the zero temperature subtraction method, andshow that, contrary to the massless case, they yield different results. Most impor-tantly, we show that neither of these results are equivalent to the one obtained inSec.5 based on the fundamental definition. Moreover, we show that these dis-crepancies cannot be fixed completely by the renormalization program mentionedbefore (see e.g. [23, 29]), since the extra unphysical terms are non-polynomial.We start with the first form of the free energy given by Eq. (2.8) and integrateover the transverse momenta. Then we present the sum over Matsubara frequen-cies as the the difference between sum over all integers and even integers, as usedin Eq. (4.2), and use the inhomogeneous zeta function on the Matsubara frequen-cies (see Appendix A). The result is, F Zeta ( T, L ) = A √ π X n > lim s → ∂∂s s ) (cid:26) Γ (cid:18) s − (cid:19) (cid:0) ω n (cid:1) − s +8 ∞ X n =1 (cid:18) ω n βn (cid:19) − s h K − s (2 βn ω n ) − − s K − s ( βn ω n ) i) , (6.1)where ω n = p k n + m . To explore the mechanism of removal of divergencesfrom this point forward, it is useful to compare this expression with the analogousone that we have obtained for the massless case after using the inhomogeneouszeta function on the Matsubara frequencies, i.e. Eq. (4.6). The first term in bothexpressions is a divergent term which is the sum over the spatial modes and isleftover from the use of inhomogeneous zeta function on the Matsubara frequen-cies. In the massless case, the spatial modes were regular and we could obtain theanalytic continuation of its divergent term using a supplementary zeta function.In the present case, the modes are irregular and, as before, we compute the sumover the spatial modes using the Principle of the Argument (see Appendix B), andobtain the following expression, F Zeta ( T, L ) = A √ π lim s → ∂∂s s ) (cid:26) Γ (cid:18) s − (cid:19) Z ∞ p − s (cid:20) Lω ( p )+Ln (cid:18) ω ( p ) − mω ( p ) + m e − Lω ( p ) (cid:19)(cid:21) dp − ∞ X n =1 r Tn Z ∞−∞ ( ip ) − s h K − s (2 iβn p ) − − s K − s ( iβn p ) i (cid:20) Lω ( p ) + Ln (cid:18) ω ( p ) − mω ( p ) + m e − Lω ( p ) (cid:19)(cid:21) dp (cid:27) . (6.2)where ω ( p ) = p p + m . The terms which include the logarithm function, arefinite in the domain of integration. So, for these terms we take the derivative with24espect to s , take the limit s → , and obtain a result which is identical to F Casimir as given by Eq. (5.4), F Zeta ( T, L ) = F Casimir + AL √ π lim s → ∂∂s s ) ( Γ (cid:18) s − (cid:19) Z ∞ p − s ω ( p ) dp − ∞ X n =1 r Tn × Z ∞−∞ ( ip ) − s h K − s (2 iβn p ) − √ (1 − s ) K − s ( iβn p ) i ω ( p ) dp ) . (6.3)The remaining terms are extra unphysical terms leftover by the inhomogeneouszeta function. To compute these two terms, we replace the square root term ω ( p ) = p p + m by the integral representation of generalized gamma func-tion . Next we evaluate the integral over p and t . Only the first of these termsis divergent, and its divergence appears as Γ( s − , which after dividing Γ( s ) ,taking the derivative with respect to s , and taking the limit s → , yields a finiteresult. We can express the final result as follows, F Zeta ( T, L ) = F Casimir ( T, L ) +
ALm π [3 − m )] + ∆ F free ( T, L ) . (6.4)The order of the terms presented above is the same as in Eq. (6.3), i.e., the secondterm is the finite term mentioned above, and the last term is the thermal correctionto the free energy of the massive free case, i.e. ∆ F free ( T, L ) given in Eq. (5.3),which the zeta function fails to subtract, just as in the massless case. As shownby Elizald [21] the procedure that we have used to remove the divergences isprecisely equivalent to isolating the poles of the inhomogeneous zeta function,which appear as the poles of the gamma function, and then removing them. Inthis respect, this procedure is an analytic continuation scheme.Next we calculate the Casimir free energy using the zero temperature sub-traction method (ZTSM) [24], as defined in Eq. (4.8). We use the results ob-tained in the Sec. 5 for the free energy of the bounded and free cases, given byEqs. (5.2,5.3). We can summarize the results as follows, F ZTSM ( T, L ) = F Casimir ( T, L ) + ∆ F free ( T, L ) , (6.5) = F Zeta ( T, L ) − ALm π [3 − m )] It is worth mentioning that the massless limit of F Zeta ( T, L ) and F ZTSM ( T, L ) forthe massive cases given by Eqs. (6.4, 6.5), coincide exactly with their masslesscases given by Eq. (4.5). p p + m = ∞ R dtt Γ ( − ) e − t [ p + m ] F Casimir given by Eq. (5.4), the zeta func-tion method, F Zeta given by Eq. (6.4), and the zero temperature subtraction method, F ZTSM given in Eq. (6.5). Comparing F Zeta with F ZTSM , we observe that, contraryto the massless case, the results are not equivalent: there is an extra term in F Zeta ,which is its second term in Eq. (6.4) and is temperature-independent. As men-tioned above, this extra term is an analytic continuation of the divergent termwhich appears after using the inhomogeneous Epstein zeta function. Next, wecompare these two results with F Casimir . First, as can be seen from the Eqs. (5.4,6.4,6.5), F Casimir does not include the extra temperature independent term in F Zeta ,mentioned above. Second, F Casimir does not include the thermal correction termof the free case, i.e. ∆ F free ( T, L ) given by Eq. (5.3), which appears in both F Zeta and F ZTSM . Note that this extra term is a non-polynomial function of T , the hightemperatures limit of which is, lim T →∞ ∆ F free ( T, L ) = − (cid:18) ALπ (cid:19) T + (cid:18) ALm (cid:19) T − ALm π [ln( πT ) − γ ] , (6.6)where γ = 0 . is the Euler-Maschernoi constant. This expansion can also beobtained by the heat kernel coefficients.As mentioned before, it has long been recognized that the use of the zetafunction method yields extra unphysical terms. To remedy this, Geyer et al. [23]defined a renormalization program in which the polynomial terms obtained usingthe heat kernel coefficients with powers greater or equal to two are subtracted. Intheir work on bosonic cases, they emphasized that all of the mentioned terms areof quantum character and do not include the classical term which is proportional totemperature. We now explore the results of this renormalization program. Belowwe state the renormalization program as presented in reference [23] F ren = E ren + ∆ T F − α ( k B T ) − α ( k B T ) − α ( k B T ) . (6.7)The coefficients of these terms depend on geometrical characteristics of the con-figuration and can be expressed in terms of heat kernel coefficients. Therefore,based on this renormalization program, the physical Casimir free energy for amassive fermion field confined between two parallel plates obtained using zetafunction is as follows, F renZeta ( T, L ) = F Zeta + (cid:18) ALπ (cid:19) T − (cid:18) ALm (cid:19) T , (6.8)where F Zeta is given by Eq. (6.4). One can analogously define a renormalizedZTSM free energy as follows, F renZTSM ( T, L ) = F ZTSM + (cid:18) ALπ (cid:19) T − (cid:18) ALm (cid:19) T . (6.9)26here F ZTSM is given by Eq. (6.5).To illustrate the differences between the five expressions for the Casimir freeenergy, we plot them in Fig. (8). As can be seen in this figure, the free energiesobtained via the zeta function and ZTSM decrease without bound as temperatureincreases, while the Casimir free energy goes to zero at high temperatures. More-over, the free energies obtained by applying the renormalization program, i.e., F renZeta , F renZTSM , do not go to zero as temperature increases, and in fact diverge due tothe subtraction of only the first two terms of Eq. (6.6) which are proportional to T and T . The divergence is due to the remaining ln( T ) term. The zero temperaturelimit of both ZTSM results are compatible with that of the F Casimir , while those ofthe zeta function methods are not. This is due to the extra constant term in the F Zeta , and F renZeta . Fundamental definitionZeta functionZTSM
Renormalized Zeta functionRenormalized ZTSM [\] ^_‘ - abcde - fghij - - klmno - - prstu T C as i m i r H e l m ho l t z f r eee n e r g y p e r un i t v o l u m e Figure 8:
The Casimir free energies per unit volume for a massive fermion field betweentwo parallel plates as a function of temperature with fixed plate separation L = 1 . andmass m = 0 . , obtained using five methods. The solid line is for the fundamental defini-tion, F Casimir ( T, L ) , the medium dashed line is for the zeta function method, F Zeta ( T, L ) ,the dash-dot line is for the ZTSM, F ZTSM ( T, L ) , the small dashed line is for the renormal-ized zeta function, F renZeta ( T, L ) , and the large dashed line is for the renormalized ZTSM, F renZTSM ( T, L ) . One can now obtain other thermodynamic quantities based on the expressionswe have obtained for the free energy using the zeta function method Eq. (6.4),27TSM Eq. (6.5), and their renormalized versions Eqs. (6.8,6.9). For example, wecalculate the pressure for a massive fermion field using the free energy obtainedvia the zeta function, in analogy with the massless case shown in Eq. (4.11). Weexpress the result in terms of P Casimir ( T, L ) as follows, P Zeta ( T, L ) = P Casimir ( T, L ) − m π [3 − m )] + ∆ P free ( T, L )∆ P free ( T, L ) = − T m π ∞ X j =1 ( − j j K ( jβm ) (6.10)As before, the second terms is a constant term leftover by the zeta function, andthe third term is the thermal correction to the pressure of the free case, which thezeta function fails to subtract. Next we calculate the pressure using the free energyobtained via the ZTSM. We express the result in terms of P Casimir ( T, L ) as follows, P ZTSM ( T, L ) = P Casimir ( T, L ) + ∆ P free ( T, L ) (6.11)Next, we calculate the pressure obtained via the renormalized zeta function, i.e., F renZeta ( T, L ) , and renormalized ZTSM, i.e., F renZTSM ( T, L ) . The results are, P renZTSM ( T, L ) = P ZTSM ( T, L ) − (cid:18) π (cid:19) T + (cid:18) m (cid:19) T (6.12) P renZeta ( T, L ) = P Zeta ( T, L ) − (cid:18) π (cid:19) T + (cid:18) m (cid:19) T . (6.13)In Fig. (9), we compare these results with the Casimir pressure obtained basedon the fundamental definition given by Eq. (5.6). As can be seen, the pressuresobtained using the zeta function method, and ZTSM are attractive at low tem-peratures and repulsive at high temperature, while the Casimir pressure is alwaysattractive and vanishes as temperature increases. The differences between theseresults, besides the constant term present in P Zeta ( T, L ) , are due to the thermalcorrection of pressure of free case which is a non-polynomial function of T forthe massive fermion field. The pressures obtained using the zeta function method,and ZTSM all diverge as T at high temperatures, and their renormalized versionsas ln( T ) . At T = 0 , only the ZTSM results match the P Casimir (0 , L ) . In this paper, we have explored the implications of the fundamental definitionof the Casimir free energy, and how they compare with some of the methods in28 undamental definitionZeta functionZTSM
Renormalized Zeta functionRenormalized ZTSM vwx yz{ - - |}~(cid:127) - - T C as i m i r P r ess u r e Figure 9:
The Casimir pressure for a massive fermion field between two parallel plates asa function of temperature with fixed plate separation L = 1 . and mass m = 0 . , obtainedusing five methods. The solid line is for the fundamental definition, P Casimir ( T, L ) , themedium dashed line is for the zeta function method, P Zeta ( T, L ) , the dash-dot line isfor the ZTSM, P ZTSM ( T, L ) , the small dashed line is for the renormalized zeta function, P renZeta ( T, L ) , and the large dashed line is for the renormalized ZTSM, P renZTSM ( T, L ) . common use, i.e., the generalized zeta function method and the zero subtractionmethod, along with their supplementary renormalization program. For a concreteexample which would illustrate the similarity and differences, we have chosenmassless and massive fermion fields confined between two parallel plates, sepa-rated by a distance L , with MIT bag boundary conditions at finite temperature.The fundamental definition of F Casimir is the difference between the free energyof the system in the presence of nonperturbative conditions or constraints, andthe one with no constraints, which we have referred to as the free case, both be-ing at the same temperature T and having the same volume. We have calculatedthe Casimir free energy based on the fundamental definition, and have used it tocalculate other Casimir thermodynamic quantities, including the pressure, energy,and entropy, and have shown that all of them are finite and vanish as the temper-ature, L , or mass increases. This occurs due to the subtraction of the free caseat the same temperature which amounts to the cancellation of the part of thermalcorrection of the bounded case which is equivalent to that of the free case. Wehave also shown that the massless limits of the Casimir thermodynamic quantities29btained for the massive fermion field are identical to the ones obtained for themassless case.We have then computed the Casimir thermodynamic quantities using someother methods in common use and compared the results to the ones obtained usingthe fundamental definition. First, we concentrated on computing the Casimir freeenergy for a massless fermion field. We first used the zeta function, implementedin three different ways, to evaluate the analytic continuation of the double sumsof the spatial and Matsubara modes, and have shown that they all yield equivalentresults. Moreover, we have calculated this Casimir free energy using the zerotemperature subtraction method, and have shown that the results are equivalentto those of zeta function methods. However, these results are not equivalent tothe ones obtained using the fundamental definition of the Casimir free energy.The difference is the T term which is the equivalent of the black-body radiationterm. This difference can be traced back to the fact that the free energy of the freecase at finite temperature contains the black-body term, the subtraction of whichis included in the fundamental definition of the Casimir free energy, but it is notincluded in the zeta function method or the ZTSM. This difference also exists in allother Casimir thermodynamic quantities. For example, P Zeta or P ZTSM , contrary tothe Casimir pressure obtained by the fundamental definition, are repulsive at hightemperatures, due to the black-body term. A renormalization program has beendevised to subtract the high temperature expansions as polynomials in T , the useof which yield the correct results for the massless case.Next, we have used the inhomogeneous zeta function method and the ZTSM tocalculate the Casimir free energy for a massive fermion field and have shown that,contrary to the massless case, they yield different results. Moreover, similar tothe massless case, neither of these results is equivalent to F Casimir obtained via thefundamental definition. The major difference is that both F Zeta and F ZTSM containthe thermal correction to the free case, denoted by ∆ F free ( T, L ) , which they havefailed to subtract and is a non-polynomial function of T . Moreover, F Zeta , includesan extra unphysical temperature-independent term leftover by the inhomogeneouszeta function, which is a non-polynomial function of the mass which vanishesas m → . The high-temperature expansion of ∆ F free ( T, L ) includes T , T and ln( T ) terms. The renormalization program mentioned above removes the first twoof these terms in this case which do not equal ∆ F free ( T, L ) . Consequently, F renZeta and F renZTSM are also not equal to F Casimir at any nonzero temperature. This is incontrast to the massless case, where ∆ F free ( T, L ) was a simple mononomial T ,the subtraction of which lead to F Casimir = F renZeta = F renZTSM . Therefore, as can beseen from Fig. (8), for the massive case the five expressions for the Casimir freeenergy are not equivalent at any temperature, except at T = 0 where the ZTSMresults are equal to F Casimir (0 , L ) . In particular, as T → ∞ , F Casimir → , F Zeta and F ZTSM ∼ T , and F renZeta and F renZTSM ∼ ln( T ) . These differences are also present in30ll other Casimir thermodynamic quantities, e.g., the Casimir pressures illustratedin Fig. (9).We believe that the correct method for computing the Casimir free energy isby the use of its fundamental definition. 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Calculation of the free energy using generalizedzeta function
The most commonly used method for calculating the Casimir effects is the zetafunction method. The generalized zeta function [53] is given dy the followingexpression, Z M p ( s ; a , ..., a p ; c , ..., c p ) = ∞ X n = −∞ ... ∞ X n p = −∞ (cid:2) a ( n − c ) + ... + a p ( n p − c p ) + M (cid:3) − s . (A.1)The above expression yields finite results for Re( s ) > p , and admits an analyticcontinuation for Re( s ) < p , [19, 53]. This form is also referred to as the inho-mogeneous generalized zeta function. If we set the parameters c , ..., c p to zero,we obtain a special form of the inhomogeneous generalized zeta function. An im-portant special form called the homogeneous zeta function is obtained when theparameters M and c , ..., c p are set to zero. For this case, there is a constraint thatthe sums should not include the ( n = 0 , ..., n p = 0) mode. Obviously, for themassive case we have to use the inhomogeneous form, while, as shown in the text,both forms can be used for the massless case.In the first part of this appendix, we show explicitly three different ways of us-ing the zeta function for obtaining the free energy of the massless case, as outlinedin Sec. 4, starting with Eq. (4.2) and obtaining the three equivalent expressionsgiven in Eqs. (4.3,4.5,4.7). In the first method, we do the double sums simulta-neously, so as to obtain the final result shown in Eq. (4.3). The expression thatwe have obtained for F Zeta ( T, L ) , given by Eq. (4.2), can be expressed in terms ofhomogeneous generalized zeta functions as follows, F Zeta ( T, L ) =
T A π lim s → ∂∂s Γ( s − s ) (cid:26) Z (cid:18) s − π β , π L (cid:19) − Z (cid:18) s − π β , π L (cid:19) − Z (cid:18) s −
1; 4 π β , π L (cid:19) + Z (cid:18) s −
1; 4 π β , π L (cid:19)(cid:27) . (A.2)Here s − s ′ < , and an analytic continuation may be implemented byapplication of the following zeta function reflection formula [20, 54], π − s ′ Γ( s ′ ) Z p ( s ′ ; a , ..., a p ) = π − p + s ′ √ a a ...a p Γ( p − s ′ ) Z p (cid:18)(cid:16) p − s ′ (cid:17) ; 1 a , ..., a p (cid:19) . (A.3)37sing this for the first term of Eq. (A.2), as an example, we obtain, Z (cid:18) s − π β , π L (cid:19) = 2 βL Γ(2 − s )Γ( s − π − s Z (cid:18) − s ; β π , L π (cid:19) = 2 βL Γ(2 − s )Γ( s − π − s ∞ X n = −∞ ∞ ′ X n = −∞ "(cid:18) n βπ (cid:19) + (cid:18) n Lπ (cid:19) ( s − . (A.4)Using the reflection formula for all four terms of Eq. (A.2), taking the derivativewith respect to s , and taking the limit s → , the expression for the free energybecomes F Zeta ( T, L ) = − AL π ∞ X n =1 n β ) − AL π ∞ X n =1 n L ) + AL π ∞ X n =1 ∞ X n =1 " (cid:0) n β + 4 n L (cid:1) − − (cid:0) n β + n L (cid:1) − − (cid:18) n β n L (cid:19) − + (cid:18) n β n L (cid:19) − . (A.5)Since the summations in Eq. (A.5) are over only positive definite integers, weuse the homogeneous form of the generalized inhomogeneous Epstein zeta func-tion [55] given by E M p ( s ; a , ..., a p ) = ∞ X n =1 ... ∞ X n p =1 (cid:2) a n + a n + ... + a p n p + M (cid:3) − s . (A.6)That is, we use E p which is usually denoted by E p . Before we apply this to thefour terms in Eq. (A.5), we use the following relation for the Epstein zeta function, E , E ( s ; a , a ) = − ζ (2 s )2 a s + r πa Γ( s − ) ζ (2 s − s ) a ( s − ) +2 π s q a ( s + )Γ( s ) q a ( s − ) ∞ X m =1 ∞ X m =1 (cid:18) m m (cid:19) ( s − ) K − s (cid:18) πm m r a a (cid:19) . (A.7)Using this for each term in Eq. (A.5), we obtain F Zeta ( T, L ) = − AL π ∞ X n =1 n β ) + A r T L ∞ X n =1 ∞ X n =1 s n n × h √ K − (4 πn n LT ) + 2 K − (2 πn n LT ) − K − (8 πn n LT ) i . (A.8) We have used lim s → ∂∂s f ( s )Γ( s ) = f (0) , since f ( s ) is an analytic function for s < . n modes and obtain, F Zeta ( T, L ) = − ALT π
180 + AT πL ∞ X n =1 n (cid:20) πn LT + 1) e πn LT − e πn LT − − πn LT + 1) e πn LT − e πn LT − − (8 πn LT + 1) e πn LT − e πn LT − (cid:21) . (A.9)Then we simplify the above expression and obtain the free energy given by Eq. (4.3).Next, we compute the free energy of the massless case using the zeta functionto do the sums separately. To do this, first we note that there are partial cancel-lations in the sums of Eq. (4.2), i.e., the terms with n = 0 or n = 0 canceleach other. Next, we express the remaining sums as sums over positive integersas follows, F ( T, L ) =
T Aπ lim s → ∂∂s Γ( s − s ) ∞ X n =1 ∞ X n =1 "(cid:18) n π β + n π L (cid:19) − s − (cid:18) n π β + n π L (cid:19) − s − (cid:18) n π β + n π L (cid:19) − s + (cid:18) n π β + n π L (cid:19) − s . (A.10)For our first case, which constitutes our second method, we first calculate thesum over spatial modes and then the sum over the remaining Matsubara modes. Todo this, we consider the Matsubara modes in Eq. (A.10), i.e., n π/β and n π/β ,as the constant term of Eq. (A.6). Then we use the following expression for E M ( s ; a ) [55], E M ( s ; a ) = − M s + r πa s ) M s − " Γ( s −
12 ) +4 ∞ X j =1 (cid:18) √ aπjM (cid:19) ( − s ) K − s (cid:18) πjM √ a (cid:19) , (A.11) We have used the following identities, ∞ P m =1 √ m K ( ma ) = p π a ( a +1) e a − e a − = ∞ P m =1 √ m K − ( ma )
39o compute the free energy. Then we obtain, F Zeta ( T, L ) =
T AL √ π lim s → ∂∂s s ) ∞ X n =1 (cid:18) n πβ (cid:19) − s ( Γ (cid:18) s − (cid:19) (cid:18) − − s (cid:19) +2 ∞ X n =1 (cid:18) βn n Lπ (cid:19) − s h(cid:16) s − + 2 − s (cid:17) K − s (4 πn n LT ) − K − s (2 πn n LT ) − K − s (8 πn n LT ) i ) . (A.12)Taking the derivative with respect to s and the limit s → , the free energy be-comes F Zeta ( T, L ) = − ALT π ∞ X n =1 n + A r T L ∞ X n =1 ∞ X n =1 s n n ( √ K − (4 πn n LT ) − K − (2 πn n LT ) − K − (8 πn n LT ) ) . (A.13)This expression is equivalent to Eq. (4.4) and, as mentioned in Sec. 4, we calcu-late the divergent sum over the Matsubara frequencies using the analytic continu-ation obtained via ζ ( − . This results in the explicit appearance of the unphysicalblack-body radiation term in the expression for our final result given by Eq. (4.5).For our second case, which constitutes our third method, we first calculate thesum over Matsubara modes and then the sum over the remaining spatial modes.To do this, we consider the spatial modes in Eq. (A.10), i.e., n π/ L and n π/L ,as the constant term of Eq. (A.6). Then we use Eq. (A.11) and compute the freeenergy, obtaining the following expression, F Zeta ( T, L ) = − Aπ L ∞ X n =1 n + A r T L ∞ X n =1 ∞ X n =1 s n n (cid:26) √ K − (cid:16) πn n LT (cid:17) + − K − (cid:18) πn n LT (cid:19) − K − (cid:16) πn n LT (cid:17)(cid:27) . (A.14)This expression is equivalent to Eq. (4.6) and, as mentioned in Sec. 4, we calculatethe divergent sum over the spatial modes using the analytic continuation renderedby ζ ( − . This yields the correct zero temperature part present in the expressionfor our final result given by Eq. (4.7), where the unphysical black-body radiationterm is embedded in the other three terms.In the last part of this appendix, we use the inhomogeneous generalized zetafunctions to compute the free energy of the massive case. As mentioned in Sec. 6,40e start with the first form of the free energy given by Eq. (2.8) and integrate overthe transverse momenta. Then we present the sum over Matsubara frequenciesas the the difference between sum over all integers and even integers, as used inEq. (4.2), to obtain F Zeta ( T, L ) =
T Aπ lim s → ∂∂s Γ( s − s ) ∞ X n =1 X n > "(cid:18) n πβ (cid:19) + k n + m − s − "(cid:18) n πβ (cid:19) + k n + m − s . (A.15)To calculate the sum over Matsubara modes, we consider the mass term and theirregular spatial modes in Eq. (A.15), i.e., k n + m , as the constant term ofEq. (A.6), i.e., M . Then we use Eq. (A.11) to obtain the free energy given byEq. (6.1). B Calculation of the summation over irregular modesusing the Principle of the Argument
The Principle of the Argument relates the difference between the number of zerosand poles of a meromorphic function f ( z ) , to a contour integral of the logarithmicderivative of the function [44]. In this paper, we use the generalized form of thePrinciple of the Argument which is as follows [44], X n g ( a n ) − X m g ( b m ) = 12 πi I C g ( z ) d [ln( f ( z ))] , (B.1)where a n and b m are the zeroes and poles of f ( z ) inside the closed contour C ,respectively, and g ( z ) is assumed to be an analytic function in the region enclosedby the contour C . In applying this theorem to our problem, we find it convenientto use the following generalization of Eq. (B.1) X n g ( a n ) − X m g ( b m ) = 12 πi I C g ( z ) d [ln( f ( z ) h ( z ))] , (B.2)with the condition that the function h ( z ) should be analytic and have no zeros inthe region enclosed by the contour C .The expression that we have obtained for the free energy of the massive casebetween two plates in Sec. 5, given by Eq. (5.1), contains a sum over the irregular41patial modes which are the roots of f ( k n ) in Eq.(2.6). We use the Principle ofthe Argument, as expressed in Eq. (B.2), to compute this sum and obtain, F bounded ( T, L ) = A π πi I C " q + r T π ∞ X n =1 ( − n √ n p q K ( βn q ) × d Ln p q − m cos (cid:16)p q − m L (cid:17) + 2 m sin (cid:16)p q − m L (cid:17)p q − m + im , (B.3)where g ( q n ) = g ( k n + m ) is the summand in Eq. (5.1), while g ( q ) is the inte-grand defined in Eq. (B.2). We have chosen h ( q ) = 2 / ( p q − m + im ) . Theclosed contour C in the complex q -plane should enclose all of the roots of f ( k n ) .As can be seen in Fig. (10), the closed contour C is composed of two arcs, C R and C r , and also two straight line segments L , and L . To compute this contourintegral over q , we replace the first term in the integrand in Eq. (B.3), i.e., the q term, by the following integral representation q = Z ∞ e − tq dt √ t Γ (cid:0) − (cid:1) . (B.4)Next we integrate by parts. In the limit R → ∞ and r → , only L and L givenonzero contributions, which can be written as follows, F bounded ( T, L ) = iAπ Z i ∞− i ∞ dq "Z ∞ dte − tq q √ t Γ (cid:0) − (cid:1) + r βπ ∞ X n =1 ( − n √ n p q × K ( βn q ) i Ln p q − m cos (cid:16)p q − m L (cid:17) + 2 m sin (cid:16)p q − m L (cid:17)p q − m + im . (B.5)After changing variable q = ip , and evaluating the integral over t , we express theresults as follows F bounded ( T, L ) = A π Z ∞ dp ( (cid:2) ( ip ) + ( − ip ) (cid:3) + − r Tπ ∞ X n =1 ( − n √ n h ( ip ) K ( ipβn ) + ( − ip ) K ( − ipβn ) i ) × ( Ln " e L √ p + m p p + m − m p p + m + m e − L √ p + m ! . (B.6)42igure 10: The closed integration contour C , referred to in Eq. (B.3), for evaluating thefree energy for the massive fermion between two parallel plates, where q i is related to theirregular spatial mode by q i = k i + m . As R → ∞ and r → , only L and L givenonzero contributions. We can simplify this expression to obtain the free energy for the bounded casegiven by Eq. (5.2).In Sec. 6, we have calculated the free energy of a massive fermion using theinhomogeneous zeta function and have displayed the result in Eq. (6.2). The de-tails of calculations are as follows. We start with Eq. (6.1), follow the same steps Using q ( ip ) K ( ipa ) + q ( − ip ) K ( − ipa ) = π p p J ( pa ) = p q πa sin ( pa )