- Email: [email protected]
Confined Maxwell field and temperature inversion symmetry
1 June 2000
Physics Letters B 482 Ž2000. 323–328
Confined Maxwell field and temperature inversion symmetry F.C. Santos 1, A.C. Tort
2
´ Instituto de Fisica, UniÕersidade Federal do Rio de Janeiro, Cidade UniÕersitaria, ´ C.P. 68528, CEP 21945-970 Rio de Janeiro RJ, Brazil Received 29 March 2000; received in revised form 7 April 2000; accepted 12 April 2000 Editor: M. Cveticˇ
Abstract We evaluate the Casimir vacuum energy at finite temperature associated with the Maxwell field confined by a perfectly conducting rectangular cavity and show that an extended version of the temperature inversion symmetry is present in this system. q 2000 Elsevier Science B.V. All rights reserved. PACS: 11.10.-z; 11.10.Wx
1. Introduction Temperature inversion symmetry occurs in the free energy associated with the Casimir effect w1x at finite temperature and it is linked to the nature of the boundary conditions imposed on the quantum vacuum oscillations. As shown by Ravndal and Tollefsen w2x, temperature inversion symmetry holds for massless bosonic fields and symmetric boundary conditions and also for massless fermionic fields and antisymmetric boundary conditions. Temperature inversion symmetry was initially studied by Brown and Maclay w3x who wrote the scaled free energy density f Ž j ., where j [ Tdrp , associated with the Casimir effect at finite temperature as a sum of three contributions, namely: a zero temperature contribution, a Stefan–Boltzmann contribution, and a non-
1 2
E-mail: [email protected] E-mail: [email protected]
trivial contribution. These three contributions can be combined into the single double sum: f Ž j . sy
1 16p 2
`
`
Ý
Ý
nsy` msy`
Ž 2pj .
4
m 2 q Ž 2p n j .
2 2
,
Ž 1. where the term corresponding to n s m s 0 is excluded. Setting n s 0 and summing over m with m / 0, we obtain the Stefan–Boltzmann limit yp 6j 4r45. On the other hand, setting m s 0 and summing over n with n / 0, we obtain the zero temperature Casimir term yp 2r720. This function has the following property: 4 Ž 2pj . f Ž 1r4pj . s f Ž j . ,
Ž 2.
which is the mathematical statement of the temperature inversion symmetry. It was also shown by Gundersen and Ravndal w4x that the scaled free energy density associated with massless fermions fields at finite temperature submitted to MIT boundary condi-
0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 4 9 8 - 6
F.C. Santos, A.C. Tort r Physics Letters B 482 (2000) 323–328
324
tions satisfy the relation given by Eq. Ž2. and therefore exhibts temperature inversion symmetry. Tadaki and Takagi w5x calculated Casimir free energies for a massless scalar field obeying Dirichlet or Neumann boundary conditions on both plates and found this symmetry. For the parallel plate geometry and mixed boundary conditions it is possible to circunvent the restrictions found by w2x and discuss temperature inversion symmetry by recognizing that with respect to the evaluation of the free energy this arrangement is equivalent to the difference between two Dirichlet Žor Neumann. plates w6x. In the case of a massless scalar field at finite temperature and periodic boundary conditions, it is possible to show that the partition function, and consequently the free energy, can be written in a closed form such that the temperature inversion symmetry becomes explicit w7x. Here we shall show that for the case of the Maxwell field confined within a perfectly conduting rectangular cavity this symmetry is also present. We employ units such that Boltzmann constant k B , the speed of light c and " s hr2p are set equal to the unity.
Helmholtz free energy can be evaluated by means of the generalized zeta function technique w9,10x. The global zeta function is defined by 2s
m z Ž s. sm
2s
2p p
Ý Ý psy` n4
q v2 n 4
b
Ž 3.
where b s 1rT, the reciprocal of the temperature T, is the periodic length along the Euclidean time direction and a mass scale parameter m was introduced in 2 order to keep Eq. Ž3. dimensionless; vn4 are the eigenvalues associated with the Euclidean time-independent modal equation 2 y^ x wn4Ž x . s vn4 wn4Ž x . ,
Ž 4.
where x s Ž x, y, z . and ^ x is the Laplacian operator. Helmholtz free energy function F Ž b . for bosonic fields is given by F Ž b . sy
1
E
2b E s
m2 sz Ž s .
ss 0
.
lp
2
mp
2
np
ž / ž / ž / q
q
a
b
c
2
,
Ž 6.
where l,m,n g 1,2, . . . 4 . For l,n,m / 0 there are two possible polarization states and for l s 0 or m s 0 or n s 0 there is one polarization state only. Eigenfrequencies for which three or two of the integers l,m,n are simultaneously zero are not allowed. Taking Eq. Ž6. into Eq. Ž3. we end up with a sum of Epstein functions which can be recasted into a sum of generalized zeta functions Žsee Eqs. ŽA.1. and ŽA.2. in the Appendix. that reads 22
p
ž / m
2s
z Ž s.
s A 4 Ž s;1ra2 ,1rb 2 ,1rc 2 ,4rb 2 . y A 2 Ž s;1ra2 ,4rb 2 . y A 2 Ž s;1rb 2 ,4rb 2 . y A 2 Ž s;1rc 2 ,4rb 2 . q 4 E1 Ž s;4rb 2 . .
m2 sz Ž s . s
Ž 7.
m2 sabc b G Ž 2 y s . 8p 2
Ž 5.
G Ž s.
=A 4 Ž 2 y s;a 2 ,b 2 ,c 2 , b 2r4 .
ys
2
ž /
v l2m n s
We can regularize Eq. Ž7. by applying the reflection formula, Eq. ŽA.3. thus obtaining
2. Evaluation of Helmholtz free energy
`
The eigenfrequencies for a perfectly conducting rectangular box of linear dimensions a,b,c are
y
y
y
m2 sa b G Ž 1 y s . 8p 2
m 2 s bb G Ž 1 y s . 8p 2
G Ž s.
m2 s c b G Ž 1 y s . 8p 2
b
q
G Ž s.
ž / 2
G Ž s.
A 2 Ž 1 y s;a 2 , b 2r4 . A 2 Ž 1 y s;b 2 , b 2r4 . A 2 Ž 1 y s;c 2 , b 2r4 .
2s
zRŽ 2 z . ,
Ž 8.
where z RŽ2 z . is the Riemann zeta function and we have also made use of the fact that E1 Ž z ;a . s ayzz RŽ2 z .. Notice that all terms on the R.H.S. of Eq. Ž8. are zero when s 0 except the last one. This gives rise to a scale-dependent term absent in previ-
™
F.C. Santos, A.C. Tort r Physics Letters B 482 (2000) 323–328
ous calculations related to this problem w15x. Making use of Eq. Ž5. we obtain F Ž a,b,c, b . abc sy 16p 2 `
=
1
Ý l, m , n , psy`
a q 16p b q 16p c q 16p 1 q
b
ž
2 2
2
2
2
2
a l qb m qc n q
`
p 2b 2 4
2
/
1
Ý l, psy`
ž
p 2b 2
a2 l 2 q
`
4
/
1
Ý m, psy`
ž
`
b 2 m2 q
p 2b 2 4
/
1
Ý n, psy`
ž
c 2 n2 q
p 2b 2 4
/
mb
ln
ž' / 2p
.
Ž 9.
The presence of a scale-dependent term is a leftover from the renormalization procedure carried on via the generalized zeta method and in our case introduces an ambiguity in the Helmholtz free energy. This ambiguity is introduced whenever z Ž s . is not zero at s s 0, see for instance w10x. It was shown by Myers w11x that this is a legitimate feature of the generalized zeta function method, whenever the theory is massive orrand interacting andror some dimensions are compactified. The mode sum method and the generalized zeta function method are not always equivalent. However, the mode sum method can be modified so as to include a scale dependence w12x. At zero temperature and the simple cavity that we are considering here scale-dependent terms are not expected. This is confirmed by the independent calculations due to Lukosz w13x, Ruggiero et al. w14x, Ambjorn and Wolfram w15x and the present authors. For our purposes this ambiguity can be solved by the physical requirement that in unconstrained space, that is, for a very large rectangular box, the only surviving term should be the Stefan–Boltzmann term, then m s '2p rb .
325
3. Temperature inversion symmetry The explicit verification of temperature inversion symmetry in the case in question is complicated by the fact that in addition to the inverse temperature parameter b we have to deal with three other length parameters, namely, a,b and c y. At finite temperature and periodic or antiperiodic conditions along one spatial direction, or Dirichlet, or Neumann plane surfaces located perpendicularly to one spatial direction only two characteristic lengths are involved, a feature that rends this verification easier. Nevertheless, by making recourse to a simple trick we shall show that temperature inversion symmetry is also present here. First notice that the denominator of the first term on the R.H.S of Eq. Ž9. can be rewritten as a 2 l 2 q b 2 m 2 q c 2 n 2 s k 2 Ž a 2 q 2 q b 2 r 2 q c 2 t 2 . , where q,r ,t 4 is a sequence of three integers with no common factor and k is the common factor of l,m.n4 . For the sequence q,r ,t 4 we define a characteristic length L q , r ,t 4 by d q , r ,t 4 2 [ a 2 q 2 q b 2 r 2 q c 2 t 2 . Let us also define the dimensionless variable T q , r ,t 4 [ 2 dq, r,t4rb . The remaining terms on the R.H.S. of Eq. Ž9. can be treated in a simpler way. For example, for the second term we define Ta [ 2 arb . Then making the replacement: Ý l, m, nsy` Ýq, r,t4 Ýq` k sy` , the free energy density corresponding to Eq. Ž9. can be written as F Ž a,b,c, b .
™
abc 1 sy
16p 2 `
=
Ý
1
Ý
4 q , r ,t 4 k , psy` dq , r ,t4
T4 q , r ,t 4
=
Ž k 2T2q , r ,t 4 q p 2 . 1 q
16p a 2 bc 1
q
16p a b 2 c 1
q
16p abc 2
2
`
Ý l, psy` `
Ý m, psy` `
Ý n, psy`
Ta2
Ž l a2 Ta2 q p 2 . Tb2
Ž m2b Tb2 q p 2 . Tc2
Ž n2Tc2 q p 2 .
Ž 10 .
F.C. Santos, A.C. Tort r Physics Letters B 482 (2000) 323–328
326
™™
The zero temperature limit of Eq. Ž10. is obtained by recognizing that when b ` the dimensionless paTa ,T Tb ,T Tc 0 and the surviving rameters T q , r ,t 4 ,T terms in Eq. Ž10. correspond to p s 0, as the reader can easily verify. Going back to our initial set of indexes l,m,n we obtain E0 ss y
q`
abc 16p
p
2 2
w a l q b m2 q c 2 n2 x
l, m , nsy`
1
ž
q
1
Ý
2
1 q
48 a
1 q
b
c
/
2
,
2
Ž 11 .
which is in perfect agreement with the result obtained by Lukosz w13x and also by w14x. In order to introduce temperature inversion symmetry let us define the dimensionless functions Fq , r ,t4 Ž T q , r ,t 4 . [y
Tq4 , r ,t4
`
1
Ý
16p 2
Ž k 2T2q , r ,t 4 q p 2 .
k , psy`
2
,
Ž 12 .
Eqs. Ž15. and Ž16. plus two similar equations for the functions Fb Ž Tb . and Fc Ž Tc . describe temperature inversion symmetry for the case in question, that is, all terms in Eq. Ž14. can be inverted by making use of these formulae. In the very high temperature limit we expect the leading contribution to be the Stefan–Boltzmann term, p 2r Ž 45 b 4 . . Now we argue that our transformations are applied in order to generate from the unique Stefan–Boltzmann density term, an infinite number of terms which must be added at zero temperature, and inversely, each term at zero temperature goes to the unique Stefan–Boltzmann term. In this form, that is, as far as the energy density is concerned, we can apply the temperature inversion symmetry transformations to several Casimir systems, including of course the ones previously analyzed in the literature. The specific form of the transformations depends on the particular system at hand, but the procedure is the same. We can easily check that Fq, r,t4Ž0. s yp 2r720, and that Fa Ž Ta . s Fb Ž Tb . s Fc Ž Tc . s pr48. Then, making use of Eqs. Ž15. and Ž16. we write
and Fa Ž Ta . [
Ta2
`
1 16p
Ý
Ž l a2 Ta2 q p 2 .
l, psy`
,
Ž 13 .
with similar definitions for Fb Ž Tb . and Fc Ž Tc . . In terms of these functions we can write the free energy density as F s V
Fq , r ,t4 Ž T q , r ,t 4 .
Ý
4 dq , r ,t4
q , r ,t 4
q
Fb Ž Tb .
q
cV
T4 q , r ,t 4 Fq , r ,t4
ž
T q , r ,t 4
1
ž / Ta
,
Ž 17 .
and Ta2 Fa Ž ` . s Tb2 Fb Ž ` . s Tc2 Fc Ž ` . s
p 48
/
Taking these results into Eq. Ž14. we obtain ,
s Fq , r ,t4 Ž T q , r ,t 4 . .
s Fa Ž Ta . ,
.
Ž 18 .
Ž 14 .
Ž 15 .
In same way Ta2 Fa
720
Fa Ž Ta .
where V s abc. It is easily seen that the functions Fq, r,t4 Ž Tq , r ,t4 . exhibit the following property 1
p2
aV
Fc Ž Tc .
bV
q
T4 q , r ,t 4 Fq , r ,t4Ž ` . s y
Ž 16 .
F Ž a,b,c, b
™ 0. f y abc45bp
2
4
p q
12 b 2
Ž a qb qc. , Ž 19 .
which is in agreement with w15x with respect to the leading terms in the high temperature approximation. As expected, the Stefan–Boltzmann term is the leading term in this limit. It is convenient to rewrite Eq. Ž5., or its equivalent Eq. Ž14., with the zero temperature terms and
F.C. Santos, A.C. Tort r Physics Letters B 482 (2000) 323–328
the Stefan–Boltzmann term separated from the nontrivial temperature corrections,
p2
F
p
sy 45 b
V
q
4
12 b
16p
ž
1
48 abc a
Ý
2
ac
bc
`
/
w a l q b m2 q c 2 n2 x
1 q
1 q
b
Ý
q, r ,t 4 k , ps1
1
c
2
2
/ 1
Žk
2 2
d q , r ,t 4 q b 2 p 2 .
2
1
the recognition that the concept of temperature inversion symmetry can be extended in the sense that it must be applied separately to the different terms comprising the non-trivial part of the free energy density, with each term behaving differently under this symmetry. This is the content of Eqs. Ž15. and Ž16.. The most important feature of temperature inversion symmetry is that it allows us to establish a relationship between the zero and low-temperature sector of the Helmholtz free energy and the hightemperature one. This can be very useful since in general it is easier to obtain low-temperature expansions. In particular, it is possible to relate the zero temperature Casimir effect to the Stefan–Boltzmann term in a straightforward way.
Ý p bc l, ps1 Ž 4 a2 l 2 q b 2 p 2 . 1
`
1
Acknowledgements
Ý p ac l, ps1 Ž 4 b 2 l 2 q b 2 p 2 . 1
q
ab
1 q
`
1
q
1 q
2 2
l, m , nsy`
p
q
1
1
Ý
2
q
4p
ž
q`
1 y
y
2
327
`
1
. Ý p ab l, ps1 Ž 4 c 2 l 2 q b 2 p 2 .
Ž 20 .
The last four terms of Eq. Ž20. represent the non-trivial temperature-dependent corrections. The first term of these is equivalent to a set of conducting plates with each pair of plates characterized by a distance dq, r,t4 , which means that we can take advantage of results already established in the literature regarding high and low temperature approximations for a pair of conducting plates.
The authors wish to acknowledge useful conversations with Dr. A.A. Actor.
Appendix A Multidimensional homogeneous Epstein function w8x Žsee also Elizalde et al. w10x. for a modern presentation. are defined by EN Ž z ;a1 ,a2 ,... a N . `
s
Ž a1 n12 q a2 n22 q PPP qaN n2N .
Ý
yz
,
n1 , n 2 ,.. n N s1
4. Conclusions
Ž A.1 . In this Letter we have employed generalized zeta function techniques in its global version to derive the finite temperature vacuum energy associated with an electromagnetic field confined within a perfectly conducting rectangular box. We have shown that Helmholtz free energy for the problem in question is scale dependent, a feature that was not present in previous calculations concerning this problem at zero temperature w13,14x and finite temperature w15x. We have also shown that the concept of temperature inversion symmetry can be extended so as to include the electromagnetic field confined by perfectly conducting rectangular cavity. The main point here is
for R z ) Nr2 and a1 ,a2 ,... a N ) 0. Generalized zeta functions. A N Ž z ;a1 ,a2 ,... a N . are defined by w8,10x A N Ž z ;a1 ,a2 ,... a N . q`
s
X
Ý
Ž a1 n12 q a2 n22 q PPP qaN n2N .
yz
,
n1 , n 2 ,.. n N sy`
Ž A.2 . for R z ) Nr2 and a1 ,a 2 ,... a N ) 0. The prime means that the term corresponding to n1 s n 2 s ...s n N s 0 must be excluded from the summation. A
F.C. Santos, A.C. Tort r Physics Letters B 482 (2000) 323–328
328
useful reflection formula reads involving these functions is w10x A N Ž z ;a1 ,a2 ,... a N . N N py 2 q2 z G 2 y z s G Ž z. a1 ... a N
ž
/
w7x w8x
(
=AN
ž
N 2
w3x w4x w5x w6x
w9x
/
y z ;1ra1 ,1ra 2 ,..1r.a N .
Ž A.3 .
Relationships between generalized zeta functions and Epstein functions can be derived from the definition given by Eq. ŽA.1..
w10x
w11x w12x
References w1x H.B.G. Casimir, Proc. K. Ned, Akad. Wet. 51 Ž1948. 793; Philips Res. Rep. 6. 162 Ž1951.. w2x F. Ravndal and D Tollefsen, Phys. Rev. D 40 Ž1989. 4191.
w13x w14x w15x
L.S. Brown, G.J. Maclay Phys. Rev. 184 Ž1969. 1272. S.A. Gundersen, F. Ravndal, Ann. of Phys. 182 Ž1988. 90. S. Tadaki, S. Takagi, Progr. Theor. Phys. 75 Ž1986. 262. F.C. Santos, A. Tenorio, A.C. Tort, Phys. Rev. D 60 Ž1999. ´ 105022. C. Wotzasek, J. Phys. A 23 Ž1990. 1627. P. Epstein, Math. Ann. 56 Ž1902. 615; Math. Ann. 63 Ž1907. 205. A. Salam, J. Strathdee, Nucl. Phys. B 90 Ž1975. 203; J.S. Dowker, R. Critchley, Phys. Rev. D 13 Ž1976. 3224; S.W. Hawking, Commun. Math. Phys. 55 Ž1977. 133; G.W. Gibbons, Phys. Lett. A 60 Ž1977. 385. E. Elizalde, S.D. Odintsov, A. Romeo, A.A. Bytsenko, S. Zerbini: Zeta Regularization Techniques with Applications, World Scientific, Singapore, 1994. E. Myers, Phys. Rev. Lett. 59 Ž1987. 165. S.K. Blau, M. Visser, A. Wipf, Nucl. Phys. B 310 Ž1988. 163; See also L.C. Albuquerque, Renormalization Ambiguities in Casimir Energy, hep-thr9803223. W. Lukosz, Physica 56 Ž1971. 109. R. Ruggiero, A.H. Zimerman, A. Villani, Rev. Bras. Fıs. ´ 7 Ž1977. 663. J. Ambjorn, S. Wolfram, Ann. of Phys. 147 Ž1983. 1.
Physics Letters B 482 Ž2000. 323–328
Confined Maxwell field and temperature inversion symmetry F.C. Santos 1, A.C. Tort
2
´ Instituto de Fisica, UniÕersidade Federal do Rio de Janeiro, Cidade UniÕersitaria, ´ C.P. 68528, CEP 21945-970 Rio de Janeiro RJ, Brazil Received 29 March 2000; received in revised form 7 April 2000; accepted 12 April 2000 Editor: M. Cveticˇ
Abstract We evaluate the Casimir vacuum energy at finite temperature associated with the Maxwell field confined by a perfectly conducting rectangular cavity and show that an extended version of the temperature inversion symmetry is present in this system. q 2000 Elsevier Science B.V. All rights reserved. PACS: 11.10.-z; 11.10.Wx
1. Introduction Temperature inversion symmetry occurs in the free energy associated with the Casimir effect w1x at finite temperature and it is linked to the nature of the boundary conditions imposed on the quantum vacuum oscillations. As shown by Ravndal and Tollefsen w2x, temperature inversion symmetry holds for massless bosonic fields and symmetric boundary conditions and also for massless fermionic fields and antisymmetric boundary conditions. Temperature inversion symmetry was initially studied by Brown and Maclay w3x who wrote the scaled free energy density f Ž j ., where j [ Tdrp , associated with the Casimir effect at finite temperature as a sum of three contributions, namely: a zero temperature contribution, a Stefan–Boltzmann contribution, and a non-
1 2
E-mail: [email protected] E-mail: [email protected]
trivial contribution. These three contributions can be combined into the single double sum: f Ž j . sy
1 16p 2
`
`
Ý
Ý
nsy` msy`
Ž 2pj .
4
m 2 q Ž 2p n j .
2 2
,
Ž 1. where the term corresponding to n s m s 0 is excluded. Setting n s 0 and summing over m with m / 0, we obtain the Stefan–Boltzmann limit yp 6j 4r45. On the other hand, setting m s 0 and summing over n with n / 0, we obtain the zero temperature Casimir term yp 2r720. This function has the following property: 4 Ž 2pj . f Ž 1r4pj . s f Ž j . ,
Ž 2.
which is the mathematical statement of the temperature inversion symmetry. It was also shown by Gundersen and Ravndal w4x that the scaled free energy density associated with massless fermions fields at finite temperature submitted to MIT boundary condi-
0370-2693r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 0 0 . 0 0 4 9 8 - 6
F.C. Santos, A.C. Tort r Physics Letters B 482 (2000) 323–328
324
tions satisfy the relation given by Eq. Ž2. and therefore exhibts temperature inversion symmetry. Tadaki and Takagi w5x calculated Casimir free energies for a massless scalar field obeying Dirichlet or Neumann boundary conditions on both plates and found this symmetry. For the parallel plate geometry and mixed boundary conditions it is possible to circunvent the restrictions found by w2x and discuss temperature inversion symmetry by recognizing that with respect to the evaluation of the free energy this arrangement is equivalent to the difference between two Dirichlet Žor Neumann. plates w6x. In the case of a massless scalar field at finite temperature and periodic boundary conditions, it is possible to show that the partition function, and consequently the free energy, can be written in a closed form such that the temperature inversion symmetry becomes explicit w7x. Here we shall show that for the case of the Maxwell field confined within a perfectly conduting rectangular cavity this symmetry is also present. We employ units such that Boltzmann constant k B , the speed of light c and " s hr2p are set equal to the unity.
Helmholtz free energy can be evaluated by means of the generalized zeta function technique w9,10x. The global zeta function is defined by 2s
m z Ž s. sm
2s
2p p
Ý Ý psy` n4
q v2 n 4
b
Ž 3.
where b s 1rT, the reciprocal of the temperature T, is the periodic length along the Euclidean time direction and a mass scale parameter m was introduced in 2 order to keep Eq. Ž3. dimensionless; vn4 are the eigenvalues associated with the Euclidean time-independent modal equation 2 y^ x wn4Ž x . s vn4 wn4Ž x . ,
Ž 4.
where x s Ž x, y, z . and ^ x is the Laplacian operator. Helmholtz free energy function F Ž b . for bosonic fields is given by F Ž b . sy
1
E
2b E s
m2 sz Ž s .
ss 0
.
lp
2
mp
2
np
ž / ž / ž / q
q
a
b
c
2
,
Ž 6.
where l,m,n g 1,2, . . . 4 . For l,n,m / 0 there are two possible polarization states and for l s 0 or m s 0 or n s 0 there is one polarization state only. Eigenfrequencies for which three or two of the integers l,m,n are simultaneously zero are not allowed. Taking Eq. Ž6. into Eq. Ž3. we end up with a sum of Epstein functions which can be recasted into a sum of generalized zeta functions Žsee Eqs. ŽA.1. and ŽA.2. in the Appendix. that reads 22
p
ž / m
2s
z Ž s.
s A 4 Ž s;1ra2 ,1rb 2 ,1rc 2 ,4rb 2 . y A 2 Ž s;1ra2 ,4rb 2 . y A 2 Ž s;1rb 2 ,4rb 2 . y A 2 Ž s;1rc 2 ,4rb 2 . q 4 E1 Ž s;4rb 2 . .
m2 sz Ž s . s
Ž 7.
m2 sabc b G Ž 2 y s . 8p 2
Ž 5.
G Ž s.
=A 4 Ž 2 y s;a 2 ,b 2 ,c 2 , b 2r4 .
ys
2
ž /
v l2m n s
We can regularize Eq. Ž7. by applying the reflection formula, Eq. ŽA.3. thus obtaining
2. Evaluation of Helmholtz free energy
`
The eigenfrequencies for a perfectly conducting rectangular box of linear dimensions a,b,c are
y
y
y
m2 sa b G Ž 1 y s . 8p 2
m 2 s bb G Ž 1 y s . 8p 2
G Ž s.
m2 s c b G Ž 1 y s . 8p 2
b
q
G Ž s.
ž / 2
G Ž s.
A 2 Ž 1 y s;a 2 , b 2r4 . A 2 Ž 1 y s;b 2 , b 2r4 . A 2 Ž 1 y s;c 2 , b 2r4 .
2s
zRŽ 2 z . ,
Ž 8.
where z RŽ2 z . is the Riemann zeta function and we have also made use of the fact that E1 Ž z ;a . s ayzz RŽ2 z .. Notice that all terms on the R.H.S. of Eq. Ž8. are zero when s 0 except the last one. This gives rise to a scale-dependent term absent in previ-
™
F.C. Santos, A.C. Tort r Physics Letters B 482 (2000) 323–328
ous calculations related to this problem w15x. Making use of Eq. Ž5. we obtain F Ž a,b,c, b . abc sy 16p 2 `
=
1
Ý l, m , n , psy`
a q 16p b q 16p c q 16p 1 q
b
ž
2 2
2
2
2
2
a l qb m qc n q
`
p 2b 2 4
2
/
1
Ý l, psy`
ž
p 2b 2
a2 l 2 q
`
4
/
1
Ý m, psy`
ž
`
b 2 m2 q
p 2b 2 4
/
1
Ý n, psy`
ž
c 2 n2 q
p 2b 2 4
/
mb
ln
ž' / 2p
.
Ž 9.
The presence of a scale-dependent term is a leftover from the renormalization procedure carried on via the generalized zeta method and in our case introduces an ambiguity in the Helmholtz free energy. This ambiguity is introduced whenever z Ž s . is not zero at s s 0, see for instance w10x. It was shown by Myers w11x that this is a legitimate feature of the generalized zeta function method, whenever the theory is massive orrand interacting andror some dimensions are compactified. The mode sum method and the generalized zeta function method are not always equivalent. However, the mode sum method can be modified so as to include a scale dependence w12x. At zero temperature and the simple cavity that we are considering here scale-dependent terms are not expected. This is confirmed by the independent calculations due to Lukosz w13x, Ruggiero et al. w14x, Ambjorn and Wolfram w15x and the present authors. For our purposes this ambiguity can be solved by the physical requirement that in unconstrained space, that is, for a very large rectangular box, the only surviving term should be the Stefan–Boltzmann term, then m s '2p rb .
325
3. Temperature inversion symmetry The explicit verification of temperature inversion symmetry in the case in question is complicated by the fact that in addition to the inverse temperature parameter b we have to deal with three other length parameters, namely, a,b and c y. At finite temperature and periodic or antiperiodic conditions along one spatial direction, or Dirichlet, or Neumann plane surfaces located perpendicularly to one spatial direction only two characteristic lengths are involved, a feature that rends this verification easier. Nevertheless, by making recourse to a simple trick we shall show that temperature inversion symmetry is also present here. First notice that the denominator of the first term on the R.H.S of Eq. Ž9. can be rewritten as a 2 l 2 q b 2 m 2 q c 2 n 2 s k 2 Ž a 2 q 2 q b 2 r 2 q c 2 t 2 . , where q,r ,t 4 is a sequence of three integers with no common factor and k is the common factor of l,m.n4 . For the sequence q,r ,t 4 we define a characteristic length L q , r ,t 4 by d q , r ,t 4 2 [ a 2 q 2 q b 2 r 2 q c 2 t 2 . Let us also define the dimensionless variable T q , r ,t 4 [ 2 dq, r,t4rb . The remaining terms on the R.H.S. of Eq. Ž9. can be treated in a simpler way. For example, for the second term we define Ta [ 2 arb . Then making the replacement: Ý l, m, nsy` Ýq, r,t4 Ýq` k sy` , the free energy density corresponding to Eq. Ž9. can be written as F Ž a,b,c, b .
™
abc 1 sy
16p 2 `
=
Ý
1
Ý
4 q , r ,t 4 k , psy` dq , r ,t4
T4 q , r ,t 4
=
Ž k 2T2q , r ,t 4 q p 2 . 1 q
16p a 2 bc 1
q
16p a b 2 c 1
q
16p abc 2
2
`
Ý l, psy` `
Ý m, psy` `
Ý n, psy`
Ta2
Ž l a2 Ta2 q p 2 . Tb2
Ž m2b Tb2 q p 2 . Tc2
Ž n2Tc2 q p 2 .
Ž 10 .
F.C. Santos, A.C. Tort r Physics Letters B 482 (2000) 323–328
326
™™
The zero temperature limit of Eq. Ž10. is obtained by recognizing that when b ` the dimensionless paTa ,T Tb ,T Tc 0 and the surviving rameters T q , r ,t 4 ,T terms in Eq. Ž10. correspond to p s 0, as the reader can easily verify. Going back to our initial set of indexes l,m,n we obtain E0 ss y
q`
abc 16p
p
2 2
w a l q b m2 q c 2 n2 x
l, m , nsy`
1
ž
q
1
Ý
2
1 q
48 a
1 q
b
c
/
2
,
2
Ž 11 .
which is in perfect agreement with the result obtained by Lukosz w13x and also by w14x. In order to introduce temperature inversion symmetry let us define the dimensionless functions Fq , r ,t4 Ž T q , r ,t 4 . [y
Tq4 , r ,t4
`
1
Ý
16p 2
Ž k 2T2q , r ,t 4 q p 2 .
k , psy`
2
,
Ž 12 .
Eqs. Ž15. and Ž16. plus two similar equations for the functions Fb Ž Tb . and Fc Ž Tc . describe temperature inversion symmetry for the case in question, that is, all terms in Eq. Ž14. can be inverted by making use of these formulae. In the very high temperature limit we expect the leading contribution to be the Stefan–Boltzmann term, p 2r Ž 45 b 4 . . Now we argue that our transformations are applied in order to generate from the unique Stefan–Boltzmann density term, an infinite number of terms which must be added at zero temperature, and inversely, each term at zero temperature goes to the unique Stefan–Boltzmann term. In this form, that is, as far as the energy density is concerned, we can apply the temperature inversion symmetry transformations to several Casimir systems, including of course the ones previously analyzed in the literature. The specific form of the transformations depends on the particular system at hand, but the procedure is the same. We can easily check that Fq, r,t4Ž0. s yp 2r720, and that Fa Ž Ta . s Fb Ž Tb . s Fc Ž Tc . s pr48. Then, making use of Eqs. Ž15. and Ž16. we write
and Fa Ž Ta . [
Ta2
`
1 16p
Ý
Ž l a2 Ta2 q p 2 .
l, psy`
,
Ž 13 .
with similar definitions for Fb Ž Tb . and Fc Ž Tc . . In terms of these functions we can write the free energy density as F s V
Fq , r ,t4 Ž T q , r ,t 4 .
Ý
4 dq , r ,t4
q , r ,t 4
q
Fb Ž Tb .
q
cV
T4 q , r ,t 4 Fq , r ,t4
ž
T q , r ,t 4
1
ž / Ta
,
Ž 17 .
and Ta2 Fa Ž ` . s Tb2 Fb Ž ` . s Tc2 Fc Ž ` . s
p 48
/
Taking these results into Eq. Ž14. we obtain ,
s Fq , r ,t4 Ž T q , r ,t 4 . .
s Fa Ž Ta . ,
.
Ž 18 .
Ž 14 .
Ž 15 .
In same way Ta2 Fa
720
Fa Ž Ta .
where V s abc. It is easily seen that the functions Fq, r,t4 Ž Tq , r ,t4 . exhibit the following property 1
p2
aV
Fc Ž Tc .
bV
q
T4 q , r ,t 4 Fq , r ,t4Ž ` . s y
Ž 16 .
F Ž a,b,c, b
™ 0. f y abc45bp
2
4
p q
12 b 2
Ž a qb qc. , Ž 19 .
which is in agreement with w15x with respect to the leading terms in the high temperature approximation. As expected, the Stefan–Boltzmann term is the leading term in this limit. It is convenient to rewrite Eq. Ž5., or its equivalent Eq. Ž14., with the zero temperature terms and
F.C. Santos, A.C. Tort r Physics Letters B 482 (2000) 323–328
the Stefan–Boltzmann term separated from the nontrivial temperature corrections,
p2
F
p
sy 45 b
V
q
4
12 b
16p
ž
1
48 abc a
Ý
2
ac
bc
`
/
w a l q b m2 q c 2 n2 x
1 q
1 q
b
Ý
q, r ,t 4 k , ps1
1
c
2
2
/ 1
Žk
2 2
d q , r ,t 4 q b 2 p 2 .
2
1
the recognition that the concept of temperature inversion symmetry can be extended in the sense that it must be applied separately to the different terms comprising the non-trivial part of the free energy density, with each term behaving differently under this symmetry. This is the content of Eqs. Ž15. and Ž16.. The most important feature of temperature inversion symmetry is that it allows us to establish a relationship between the zero and low-temperature sector of the Helmholtz free energy and the hightemperature one. This can be very useful since in general it is easier to obtain low-temperature expansions. In particular, it is possible to relate the zero temperature Casimir effect to the Stefan–Boltzmann term in a straightforward way.
Ý p bc l, ps1 Ž 4 a2 l 2 q b 2 p 2 . 1
`
1
Acknowledgements
Ý p ac l, ps1 Ž 4 b 2 l 2 q b 2 p 2 . 1
q
ab
1 q
`
1
q
1 q
2 2
l, m , nsy`
p
q
1
1
Ý
2
q
4p
ž
q`
1 y
y
2
327
`
1
. Ý p ab l, ps1 Ž 4 c 2 l 2 q b 2 p 2 .
Ž 20 .
The last four terms of Eq. Ž20. represent the non-trivial temperature-dependent corrections. The first term of these is equivalent to a set of conducting plates with each pair of plates characterized by a distance dq, r,t4 , which means that we can take advantage of results already established in the literature regarding high and low temperature approximations for a pair of conducting plates.
The authors wish to acknowledge useful conversations with Dr. A.A. Actor.
Appendix A Multidimensional homogeneous Epstein function w8x Žsee also Elizalde et al. w10x. for a modern presentation. are defined by EN Ž z ;a1 ,a2 ,... a N . `
s
Ž a1 n12 q a2 n22 q PPP qaN n2N .
Ý
yz
,
n1 , n 2 ,.. n N s1
4. Conclusions
Ž A.1 . In this Letter we have employed generalized zeta function techniques in its global version to derive the finite temperature vacuum energy associated with an electromagnetic field confined within a perfectly conducting rectangular box. We have shown that Helmholtz free energy for the problem in question is scale dependent, a feature that was not present in previous calculations concerning this problem at zero temperature w13,14x and finite temperature w15x. We have also shown that the concept of temperature inversion symmetry can be extended so as to include the electromagnetic field confined by perfectly conducting rectangular cavity. The main point here is
for R z ) Nr2 and a1 ,a2 ,... a N ) 0. Generalized zeta functions. A N Ž z ;a1 ,a2 ,... a N . are defined by w8,10x A N Ž z ;a1 ,a2 ,... a N . q`
s
X
Ý
Ž a1 n12 q a2 n22 q PPP qaN n2N .
yz
,
n1 , n 2 ,.. n N sy`
Ž A.2 . for R z ) Nr2 and a1 ,a 2 ,... a N ) 0. The prime means that the term corresponding to n1 s n 2 s ...s n N s 0 must be excluded from the summation. A
F.C. Santos, A.C. Tort r Physics Letters B 482 (2000) 323–328
328
useful reflection formula reads involving these functions is w10x A N Ž z ;a1 ,a2 ,... a N . N N py 2 q2 z G 2 y z s G Ž z. a1 ... a N
ž
/
w7x w8x
(
=AN
ž
N 2
w3x w4x w5x w6x
w9x
/
y z ;1ra1 ,1ra 2 ,..1r.a N .
Ž A.3 .
Relationships between generalized zeta functions and Epstein functions can be derived from the definition given by Eq. ŽA.1..
w10x
w11x w12x
References w1x H.B.G. Casimir, Proc. K. Ned, Akad. Wet. 51 Ž1948. 793; Philips Res. Rep. 6. 162 Ž1951.. w2x F. Ravndal and D Tollefsen, Phys. Rev. D 40 Ž1989. 4191.
w13x w14x w15x
L.S. Brown, G.J. Maclay Phys. Rev. 184 Ž1969. 1272. S.A. Gundersen, F. Ravndal, Ann. of Phys. 182 Ž1988. 90. S. Tadaki, S. Takagi, Progr. Theor. Phys. 75 Ž1986. 262. F.C. Santos, A. Tenorio, A.C. Tort, Phys. Rev. D 60 Ž1999. ´ 105022. C. Wotzasek, J. Phys. A 23 Ž1990. 1627. P. Epstein, Math. Ann. 56 Ž1902. 615; Math. Ann. 63 Ž1907. 205. A. Salam, J. Strathdee, Nucl. Phys. B 90 Ž1975. 203; J.S. Dowker, R. Critchley, Phys. Rev. D 13 Ž1976. 3224; S.W. Hawking, Commun. Math. Phys. 55 Ž1977. 133; G.W. Gibbons, Phys. Lett. A 60 Ž1977. 385. E. Elizalde, S.D. Odintsov, A. Romeo, A.A. Bytsenko, S. Zerbini: Zeta Regularization Techniques with Applications, World Scientific, Singapore, 1994. E. Myers, Phys. Rev. Lett. 59 Ž1987. 165. S.K. Blau, M. Visser, A. Wipf, Nucl. Phys. B 310 Ž1988. 163; See also L.C. Albuquerque, Renormalization Ambiguities in Casimir Energy, hep-thr9803223. W. Lukosz, Physica 56 Ž1971. 109. R. Ruggiero, A.H. Zimerman, A. Villani, Rev. Bras. Fıs. ´ 7 Ž1977. 663. J. Ambjorn, S. Wolfram, Ann. of Phys. 147 Ž1983. 1.