Modular invariance in finite temperature Casimir effect
aa r X i v : . [ h e p - t h ] O c t Modular invariance in finite temperature Casimir effect
Francesco Alessio
Dipartimento di Fisica “E. Pancini” and INFNUniversit`a degli studi di Napoli “Federico II”, I-80125 Napoli, Italy
Glenn Barnich
Physique Th´eorique et Math´ematiqueUniversit´e libre de Bruxelles and International Solvay InstitutesCampus Plaine C.P. 231, B-1050 Bruxelles, Belgium A BSTRACT . The temperature inversion symmetry of the partition function of theelectromagnetic field in the set-up of the Casimir effect is extended to full modulartransformations by turning on a purely imaginary chemical potential for adapted spinangular momentum. The extended partition function is expressed in terms of a realanalytic Eisenstein series. These results become transparent after explicitly showingequivalence of the partition functions for Maxwell’s theory between perfectly con-ducting parallel plates and for a massless scalar with periodic boundary conditions.
F. A
LESSIO , G. B
ARNICH
When studying the Casimir effect [1] at finite temperature [2,3], one cannot fail to be intriguedby the temperature inversion symmetry of the partition function, originally derived using animage-source construction of the Green’s function [4]. This result has been re-discussed fromvarious points of view including a derivation in terms of Euclidean path integrals, Jacobi inte-grals and Epstein zeta functions, while the Casimir energy has been related to the thermody-namic potentials of the system [5–15].A natural question is then whether one may include another observable in the partitionfunction so as to enhance the temperature inversion symmetry to transformations under thefull modular group and complete the parallel to the massless free boson on the torus withmomentum operator included. Whereas for two-dimensional conformal field theories on thetorus, the central charge has been related to the Casimir energy of a field theory with boundaryconditions in one spatial dimension [16–19], we show here how techniques developed in thatcontext can be applied in the original setting of the Casmir effect in order to produce exactresults.In order to address this question, we construct an exact equivalence, at the level of finitetemperature partition functions, between Maxwell’s theory with perfectly conducting parallelplate boundary conditions on I d ˆ R (or I d ˆ T ), and a free massless scalar on S d ˆ R (or S d ˆ T ). The exact result for the extended partition function then readily follows from thatof this scalar field, which has been discussed in detail in [20], and is inline with the analysis ofmodular invariance in [21].That the spectrum of electromagnetism with perfectly conducting boundary conditions cor-responds to one scalar with Neumann and one scalar with Dirichlet conditions, is discussedimplicitly for instance in [22] section 2.4, and explicitly in [23] (see also [24], section 3.2.2).This implies that, when taking perfectly conducting boundary conditions into account, thecomputation cannot simply be done in terms of 2 polarizations with periodic or Dirichlet con-ditions as in empty space, but one has to base it on E and H modes. It is the E modes thatcontain the additional n “ modes because they satisfy Neumann conditions and have thus acosine expansion in the case of parallel plates. The scalar field theory with periodic boundaryconditions on the interval of double the length separating the plates is then constructed out ofthese E and H modes.Identifying the correct observable is straightforward in the scalar field formulation. Thecorresponding expression in terms of electromagnetic fields is somewhat harder to guess di-rectly. The main point here is that the explicit map between the scalar field and the originalelectromagnetic field is non-local in space. As a consequence, the momentum in this direc-tion in terms of the electromagnetic field, the space integral of T which is the observablediscussed in almost all other investigation of the Casimir effect, does not correspond to scalarfield momentum in the x direction under the map. Rather, as a direct computation using the ODULAR INVARIANCE IN C ASIMIR EFFECT x -component of a suitably modifiedversion of spin angular momentum of light. This observable is usually not considered in thecontext of the Casimir effect and that is the reason modular invariance beyond temperatureinversion symmetry cannot be discussed either.Our motivation for studying this question originated from an attempt to understand thecontribution of specific degrees of freedom related to non-trivial boundary conditions to par-tition functions. More precisely, a pair of perfectly conducting plates at constant x requiresNeumann conditions for the third component of the electric field and the vector potential. Thisgives rise to an additional polarization at zero value of the quantized transverse momentumwhose dynamics is governed by a free massless scalar field in 2+1 dimensions. Its contributionto the partition function scales with the area of the plates and provides the leading correctionat low temperature to the zero temperature Casimir result for the free energy [25, 26]. Just asthe contribution of the mode at n “ in the Fourier expansions of a free boson on a torus isessential to achieve modular invariance of the partition function [27, 28], so is the contributionof this lower-dimensional massless scalar field in the current context. The partition function for perfectly conducting parallel plates of area L separated by a distance d , such that L " d , is given by [2–4] (see e.g. [24, 29] for reviews) ln Z p t q “ L d ” π t ´ tf p { t q ` π t ı , t “ β d , (2.1)and f p { t q “ ´ π ÿ l,m “ p t l ` m q , (2.2)or, equivalently, when using a Sommerfeld-Watson transformation, f p { t q “ ´ ÿ l “ ” πt l coth p πlt q ` t l sinh p πlt q ı ` π t . (2.3)The next step [11, 13] is to extend the sum over all integers p l, m q except for p , q and torecognize the relevant Epstein zeta function, Z p t , q “ ÿ p l,m qP Z {p , q p l t ` m q , (2.4)in order to write the result as ln Z p t q “ L td π Z p t , q , (2.5) F. A
LESSIO , G. B
ARNICH or, in terms of the free energy, F p t q “ ´ L d π Z p t , q . (2.6)The expression for the partition function in (2.1) is in line with the discussion in [4], but differsfrom the corresponding result for the free energy in [24, 29] by the last term in (2.1), which isabsent in the latter references. The reason is that the latter approach includes the subtractionof the full free energy of the black body in empty space, whereas this subtraction is limitedto the (divergent) zero temperature part in the former approach. This can be done becausethe thermal part of the free energy is convergent. The normalization condition chosen here isthe one where the standard black body result is recovered at large plate separation (see alsoe.g. [30, 31] for related discussions).As pointed out in [13], an advantage of the expression as given in (2.1) is that the inver-sion symmetry, f p { t q “ t ´ f p t q , established in [4], extends to the Epstein zeta function, Z p
2; 1 { t , q “ t Z p t , q , and thus turns into a symmetry of the full partition function andthe free energy, ln Z p { t q “ t ln Z p t q , F p { t q “ t F p t q . (2.7)Up to exponentially suppressed terms, the high temperature expansion t ! is ln Z p t q « L d r π t ` π ζ p qs , (2.8)where the leading piece is the black body contribution V π β ´ while the sub-leading temperature-independent contribution scales like the area; the low temperature expansion t " in turn isgiven by ln Z p t q « L d r π t ` πt ζ p qs , (2.9)where the second term is due to the lower-dimensional scalar as described above. Let us now provide some details on electromagnetism with Casimir boundary conditionsneeded for our purpose. We work in radiation gauge A “ “ ~ ∇ ¨ ~A and implement from theoutset the constraints π “ “ ~ ∇ ¨ ~π . Perfectly conducting large parallel plates of sides L at x “ and x “ d with unit normal ~n require ~E ˆ ~n “ “ ~B ¨ ~n on the plates, and periodicboundary conditions in the x a directions. Let i “ , , , a “ , , k a “ πL n a , n a P Z , k “ πd n , n P N , k “ a k i k i , k K “ a k a k a , V “ dL ,ψ Hk “ c V e ik a x a sin k x , ψ Ek a , “ ? V e ik a x a , ψ Ek “ c V e ik a x a cos k x , n ‰ . (3.1) ODULAR INVARIANCE IN C ASIMIR EFFECT φ E “ ÿ n i ? kkk K r a Ek ψ Ek ` c . c . s , π E “ ´ i ÿ n i ? kk K r a Ek ψ Ek ´ c . c . s ,φ H “ ÿ n i ? kk K r a Hk ψ Hk ` c . c . s , π H “ ´ i ÿ n i ? k ? k K r a Hk ψ Hk ´ c . c . s , (3.2)satisfying Neumann, respectively Dirichlet, conditions as well as the Helmholtz equations p ∆ ` k q φ λ “ “ p ∆ ` k q π λ , (3.3)with λ “ p E, H q . In these terms, the mode expansion of the canonical pair p ~A, ~π q and theassociated electric and magnetic fields ~E “ ´ ~π , ~B “ ~ ∇ ˆ ~A is given by the sum of E ortransverse magnetic modes, A Ea “ B a B φ E , A E “ p´ ∆ ` B q φ E ,π Ea “ B a B π E , π E “ p´ ∆ ` B q π E ,B Ea “ ǫ ab B b p´ ∆ q φ E , B E “ , (3.4)where ǫ ab is skew-symmetric with ǫ “ , and H or transverse electric modes, A Ha “ ǫ ab B b φ H , A H “ ,π Ha “ ǫ ab B b π H , π H “ ,B Ha “ B a B φ H , B H “ p´ ∆ ` B q φ H , (3.5)with the understanding that a Hk a , “ . In these variables, the first order electromagnetic action S “ ż d x “ ż V d x B ~A ¨ ~π ´ H ‰ , H “ ż V d x p ~E ¨ ~E ` ~B ¨ ~B q , (3.6)is given by S “ ż d x ÿ n i ,λ “ i pB a ˚ λk a λk ´ a ˚ λk B a λk q ´ ka ˚ λk a λk ‰ , (3.7)with λ “ p E, H q . In particular, Poisson brackets are read off from the kinetic term andoscillators do have the usual time dependence.Action (3.7) coincides with the mode expansion of an action for two massless scalar fields,one with Neumann and one with Dirichlet conditions. For later purposes, it is useful to intro-duce an equivalent formulation in terms of a single free massless scalar field, which satisfiesperiodic boundary conditions in x, y in intervals of length L and in x of length d , φ “ ? V P ÿ n i ? k r a k e ik j x j ` c . c . s , π “ ´ i ? V P ÿ n i c k r a k e ik j x j ´ c . c . s , (3.8) F. A
LESSIO , G. B
ARNICH where V P “ dL with n P Z as well, and a Ek a , ´ k “ a Ek a ,k , a Hk a , ´ k “ ´ a Hk a ,k , a k “ a Ek ´ ia Hk ? , n ‰ , a k a , “ a Ek a , . (3.9)With these definitions, the first order scalar field action S S “ ż d x “ ż V P d x “ B φπ ´ H S ‰ , H S “ ż V P d x p π ` ~ ∇ φ ¨ ~ ∇ φ q . (3.10)agrees with the first order electromagnetic action (3.6) because its expression in terms of modesis given by the RHS of (3.7). In the equivalent scalar field formulation, we will show below that the correct observable tobe turned on in order to produce a real part for the modular parameter τ and to consider fullmodular transformations is linear momentum in the x direction, P “ ´ ż V P d x π B φ. (4.1)which, in terms of oscillators, is given by P “ ÿ n i k a ˚ k a k “ ÿ n a ,n ą ik p a ˚ Hk a Ek ´ a ˚ Ek a Hk q . (4.2)The action of this observable in electromagnetic terms can be inferred from t φ E , P u “ p´B qp´ ∆ q ´ φ H , t φ H , P u “ p´B qp´ ∆ q φ E , (4.3)with similar relations holding for π λ by using (3.4) and (3.5). On the electromagnetic E and Hvector potentials, electric and magnetic fields, it acts like the curl followed by an applicationof p´B qp´ ∆ q ´ and an exchange of E and H : t ~A E , P u “ p´B qp´ ∆ q ´ ~B H , t ~B E , P u “ p´B qp´ ∆ q ´ B ~E H , t ~E E , P u “ p´B qp´ ∆ q ´ p´B q ~B H , (4.4)with the transformations of the H fields obtained by exchanging E and H in the above. Simi-larly, one may show by direct computation that the observable is given by P “ ż V d x ǫ ab p?´ ∆ A Ea π Hb ` ?´ ∆ A Ha π Eb q . (4.5)Up to multiplication of each term in momentum space by k , this observable is the componentin the x direction of spin angular momentum, ~J “ ż V d x ~A ˆ ~π, (4.6)since one may show that J “ ż V d x ǫ ab p A Ea π Hb ` A Ha π Eb q . (4.7) ODULAR INVARIANCE IN C ASIMIR EFFECT For the computation of the extended partition function, the fastet way for our purpose is tofollow [33] and to start from the Hamiltonian path integral representation Z p β, µ q “ Tr e ´ β p ˆ H ´ µ ˆ P q “ ż ź dφ ź dπ π e ´ S EH , (5.1)where the sum is over periodic phase space path of period β , and the Euclidean action is S EH “ ż β d x “ ż V P d x p´ i B φπ q ` p H S ´ µP q ‰ . (5.2)After integration over the momenta, this leads to Z p β, µ q “ ż ź dφ e ´ S EL (5.3)with S EL “ ż β d x ż V P d x rpB φ ` iµ B φ q ` B j φ B j φ s . (5.4)Following [34] (see also e.g. [22, 35] for earlier connected work and [36] for a review), theevaluation of this path integral is done by zeta function techniques.Except for the replacement B Ñ B ` iµ B , the operator in the action is the Laplacian in4 dimensions with periodic boundary conditions in all directions. Since the eigenfunctions are e i p k j x j ` πn β x q , the eigenvalues are λ n A “ p π q r ÿ a p n a L q ` p n d q ´ p i n β ´ µ n d q s , (5.5)where A “ , . . . , the zeta function of the capacitor is ζ C p s q “ ÿ n A λ ´ sn A (5.6)where the prime means that the term with n A “ is excluded. Note that the scalar fielddiscussed at the end of the introduction corresponds to the modes for which n “ with n , n , n not all zero.The next step is to take the limit of large plate size L , and hence to turn the sum over thetransverse directions turns into integrals. As in two-dimensional conformal field theories onthe torus and also in the context of QCD (see e.g. [37–39]), one now uses a purely imaginarychemical potential µ “ iν with ν real. After introducing the complex parameter τ “ νβ ` iβ d , (5.7) F. A
LESSIO , G. B
ARNICH and doing the integral in polar coordinates, the zeta function becomes ζ C p s q “ L β s ´ p π q s ´ ż d y ÿ n ,n y r y ` | τ n ` n | s s . (5.8)When n “ “ n the integral is regulated through an infrared cut-off ǫ , ş ǫ d yy ´ s ` “´ ǫ ´ s ´ s . This expression together with its derivative both vanish in the limit at s “ in thelimit ǫ Ñ and can thus be discarded. After performing the integral for the remaining terms,the result may be written in terms of a real analytic Eisenstein series (see e.g. [40] for a recentreview), f s p τ q “ ÿ p m,n qP Z {p , q r Im p τ qs s | mτ ` n | s (5.9)as ζ C p s q “ ´ L r Im p τ qs s ´ f s ´ p τ qp ´ s qp π q s ´ p d q ´ s . (5.10)Using the inversion formula, π ´ s Γ p s q f s p τ q “ π s ´ Γ p ´ s q f ´ s p τ q , (5.11)then yields ζ C p s q “ ´ L Γ p ´ s qr Im p τ qs s ´ f ´ s p τ qp ´ s q s π p d q ´ s Γ p s ´ q (5.12)Since Γ p s ´ q ´ “ ´ s ` O p s q , it follows that ζ C p q “ , in which case ln Z p τ q “ ζ C p q isexplicitly given by ln Z p τ q “ L π d f p τ q Im p τ q . (5.13)The behaviour of the partition function under modular transformations τ Ñ τ “ aτ ` bcτ ` d , ad ´ bc “ , (5.14)where a, b, c, d P Z , then follow directly from modular invariance of f p τ q , ln Z p τ q “ | cτ ` d | ln Z p τ q . (5.15)The previous result (2.5) corresponds to vanishing chemical potential ν “ “ Re p τ q in whichcase the temperature inversion formula is recovered for a “ “ d , b “ ´ c “ . The main result of the paper is the exact computation in equation (5.13) of the partition functionfor electromagnetism between two perfectly conducting parallel plates and the operator P of(4.2), or equivalently (4.5), turned on, in terms of the real analytic Eisenstein series f p τ q . ODULAR INVARIANCE IN C ASIMIR EFFECT Z p τ q “ a Im p τ q| η p τ q| , (6.1)which is exactly modular invariant.More details on a direct derivation in the operator formalism and on an underlying infinite-dimensional symmetry algebra will appear elsewhere. Other formulations making gauge in-variance manifest will also be explored. At this stage let us just point out that, after havingidentified the modular parameter, one may write in the standard way the contribution to thepartition function that corresponds to the vacuum energy, i.e., the first term in the RHS of(2.9), L π βd “ ln p q ¯ q q ´ c { “ π Im p τ q c , q “ e πiτ , (6.2)provided that the central charge of the planar capacitor is taken as c “ L πd . (6.3) Acknowledgements
The authors are grateful to M. Bonte, G. Giribet, A. Kleinschmidt and P. Niro for com-ments. This work is supported by the F.R.S.-FNRS Belgium through conventions FRFC PDRT.1025.14 and IISN 4.4503.15.
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