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Casimir Effect for Gauge Fields in Spaces with Negative Constant Curvature A. A. Bytsenko Departamento de F´ısica, Universidade Estadual de Londrina, Caixa Postal 6001, Londrina-Paraná, Brazil E-mail address: [email protected] M. E. X. Guimarães Departamento de Matemática, Universidade de Bras´ılia, Bras´ılia, DF, Brazil E-mail address: [email protected] V. S. Mendes Departamento de F´ısica, Universidade Estadual de Londrina, Caixa Postal 6001, Londrina-Paraná, Brazil 5 E-mail address: [email protected] 0 (Dated: September, 2004) 0 2 We consider gauge theories based on abelian p−forms on real compact hyperbolic spaces. Using thezeta-functionregularizationmethodandthetracetensorkernelformula,wedetermineexplicitly n an expression for the vacuum energy (Casimir energy) corresponding to skew-symmetric tensor a J fields. It is shown that the topological component of the Casimir energy for co-exact forms on even-dimensional spaces, associated with the trivial character, is always negative. We infer on the 3 possible cosmological consequencesof this result. 1 PACSnumbers: 04.70.Dy,11.25.Mj v 9 0 I. INTRODUCTION p forms: (δω ,ϕ )=(ω ,dϕ ). Inquantumfieldtheory 0 − p p p p 1 the Lagrangianassociatedwithωp takesthe form: dωp ∧ 0 The topological Casimir effect for scalar (or spinor) ∗dωp (gauge field), δωp ∧ ∗δωp (co−gauge field). 5 The Euler-Lagrangeequations, supplied with the gauge, 0 fieldsonspacesofformΓ X,whereΓ isadiscretegroup give: L ω = 0, δω = 0 (Lorentz gauge); L ω = \ p p p p p h/ acting on manifold X, hasebecome a very exciting and 0, dωp = 0 (co-Lorentz gauge). These Lagrangians t importantissueinareasofquantumfieldtheoryandcos- give possible representation of tensor fields or general- - e p mology[1-9]. InitialevaluationoftheCasimirenergyhas ized abelian gauge fields. e been given for X = RN, SN. In [7-15], the calculation h As an application, we evaluate the Casimir effect as- involves the case in which X is a Lobachevsky real hy- : e sociatedwithtopologicallyinequivalentconfigurationsof v perbolic space. i e Abelianco-exactformsonrealcompacthyperbolic man- X Maximally symmetric spaces, such as the hyperbolic ifolds. r spaces,playveryimportantinsupergravity[16],insuper- a stringtheory[17]anddefinitelyplaysacrucialroleincos- mology [18, 19, 20]. Besides, hyperbolic space forms are examples of a general noncompact irreducible rank one symmetric space the proper and outlook mathematical machinery could be available. In this paper we present II. THE TRACE FORMULA APPLIED TO THE a decomposition of the Hodge Laplacian and the ten- TENSOR KERNEL sor kernel trace formula for free generalized gauge fields (p forms) on real hyperbolic space forms. The main ing−redient required is a type of differential form struc- Let us consider an N dimensional compact real hy- − ture on the physical, auxiliary, or ghost variables. We perbolic space XΓ with universalcovering X and funda- evaluate spectral functions and the Casimir effect asso- mental group Γ. We can represent X as the symmetric e ciated with physical degrees of freedom of the Hodge– space G/K, where G = SO (N,1) and K = SO(N) 1 e de Rham operators on p forms. Let ω , ϕ be exterior is a maximal compact subgroup of G. Then we regard p p − differential p forms; then, the invariant inner product Γ as a discrete subgroup of G acting isometrically on − is defined by (ωp,ϕp) d=ef X˜ωp ∧ ∗ϕp. Under the ac- X, and we take XΓ to be the quotient space by that tfoiornfoorfmtshehoHldo:dgωe ∗=op(era1Rt)opr(n−thpe)ωfo,llaowndingddpr=opeδrδtie=s taehcetioinnt:egXraΓl o=f tΓh\eXco=nstΓan\tGf/uKnc.tioLnetIVoonl(ΓΓ\GG)wditehnortee- 0, δ = ( 1)np+n+1p d .−The operatpors d and δ are ad- specttotheG invaeriantmeasureonΓ Gind\ucedbydx, joint to−eachother w∗ith∗respectto this inner product for Vol(Γ G)= − Idx. For 0 p N \1 the Friedtrace \ Γ\G ≤ ≤ − R 2 formulaappliedtotheheatkernel t =e−tLp holds[21]: III. THE SPECTRAL FUNCTIONS OF K EXTERIOR FORMS AND THE VACUUM Tr e−tLp = I(p)( )+I(p−1)( ) ENERGY Γ Kt Γ Kt (cid:0) (cid:1) + HΓ(p)(Kt)+HΓ(p−1)(Kt), (1) If Lp is a self-adjoint Laplacian on p−forms then the followingresultshold. Thereexistsε,δ >0suchthatfor whereI(p)( ),H(p)( )arethe identityandhyperbolic 0 < t < δ the heat kernel expansion for Laplace opera- orbital iΓnteKgrtals reΓspeKcttively: tors on a compact manifold XΓ is given by Tr e−tLp = a (L )t−ℓ+ (tε).Thezetafunctiono(cid:0)fL is(cid:1)the 0≤ℓ≤ℓ0 ℓ p O p IΓ(p)(Kt) d=ef χ(1)V4oπl(Γ\G) MPTheilslinfuntrcatniosnforemquaζl(ss|TLrp)L=−s(Γf(osr))s−>1R(R1+/2T)rdei−mtL(pΓts−G1)d.t. p \ × ZRµσp(r)e−t(r2+p+ρ20)dr, (2) repTrheseentrtaednsvbeyrstehpeacrto-oe(cid:0)xfatchte(cid:1)pskefwor-msymωm(CeEtr)ic=tenδsωor is, p p+1 − (CE) which trivially satisfies δω = 0, and we denote by p L(CE) the restriction of the Laplacian on the co–exact HΓ(p)(Kt) d=ef √41πtγ∈CXΓ−{1}χj((γγ))tγC(γ)χσp(mγ) opn−pftohremm. Tanhiefogldoawlnhoicwhidsetsocreixbtersactthethpehcyos–iceaxlacdtegpr−efeosrmof e−t(ρ20+p)−t42γt, (3) forfefeodrommso[2f6t,h2e7s,y2s8t]e.m, and presents by alternating sum × C Γ is a complete set of representations in Γ of its Γ ⊂ conjugacyclasses,andC(γ) is a welldefined function on A. The identity component of the isometry group Γ 1 ,ρ =(N 1)/2,andχ (m)=trace(σ(m))isthe 0 σ −{ } − character σ for m SO(N). The trace formula involves ∈ The zeta function related to the identity integral the Harish-Chandra-Plancherel measure µ (r) which is given by σp IΓ(p)(Kt) in (2) has the form χ(1)Vol(Γ G) ∞ N 1 CGπP (r)rtanh(πr), ζI(sLp) = \ ts−1dt µσp(r)=(cid:18) p− (cid:19)× C πP (r), ffoorr NN ==22nn+1 | 4πΓ(s) Z0  G (4) × ZRµσp(r)e−ty(r2;m2p)dr whereCG = 22N−4Γ(N/2)2 ,andP (r)isapolynomial, 1 which presen(cid:0)ts the following(cid:1)form = χ(1)Vol(Γ G) 4 \ jn=−02 r2+((2j+1)/2)2 = ℓn=−01a2ℓr2ℓ, × ZRµσp(r)[y(r2;m2p)]−sdr, (6) P (r)=Q (cid:2) (cid:3) fPorN =2n  jn=−01 r2+j2 = nℓ=0a2ℓr2ℓ, forN =2n+1 wherey(r2;m2p)≡r2+m2p, m2p ≡b(p)+(ρ0−p)2. Because Q (cid:2) (cid:3) P (5) ofEqs. (4)and(5)it isconvenientto considereven-and a2ℓ are the Miatello coefficients [22, 23]. For p 1 there odd-dimensional cases separately. ≥ isameasureµ (r)correspondingtoageneralirreducible σ representation σ. Let σ be the standard representation p of SO(N 1) on ΛpC(N−1). If N = 2n is even then 1. Even-dimensional manifold − σ (0 p n 1) is always irreducible; if N = 2n+1 p ≤ ≤ − theneveryσ isirreducibleexceptforp=(N 1)/2=n, p − Using Eqs. (4) and (5) in (6) for N =2n we get inwhich caseσ is the directsum oftwospin–(1/2)rep- n resentations σ± : σ = σ+ σ−. For p = n the repre- sτenn=tatτin+on⊕τnτn−ofiKs th=enSdOir(e2cnt)s⊕uomnΛonfCtw2noisspnino–t(i1r/re2d)urceipbrlee-: ζI(2n)(s|Lp) = 41χ(1)Vol(Γ\G)CGZR P(r[y)r(rt2a;nmh(2pπ)]rs)dr sentations. 1 = χ(1)Vol(Γ G)C In the case of the trivial representation(p=0, i.e. for 4 \ G smooth functions or smooth vector bundle sections) the n−1 r2j+1tanh(πr)dr measureµ(r) µ (r)correspondstothetrivialrepresen- a . (7) tation. There≡fore0, we take I(−1)( ) = H(−1)( ) = 0. × Xj=0 2jZR [y(r2;m2p)]s Γ Kt Γ Kt Since σ isthe trivialrepresentation,onehas χ (m )= 0 σ0 γ For a,δ > 0, z C, define the entire functions 1. In this case, formula (1) reduces exactly to the trace ∈ formula for p=0 [7, 8, 24, 25]. K (s;δ,a) d=ef r2m(δ + r2)−ssech2(ar)dr. Then for m R R 3 Res>j+1, j 0, one gets [25] Using the K Bessel function K (s), s C, defined by ν ≥ − ∈ K (2s)d=ef (1/2)sν ∞t−ν−1exp t s2/t dt,wehave ν × 0 {− − } r2j+1tanh(ar)dr R ZR (δ+r2)s = ζH(s|Lp) = √πΓ1(s) χ(γ)j−1(γ)C(γ) X aj! j K (s ℓ 1;δ,a) γ∈CΓ−{1} = 2 (j ℓ)!(sj−11)(s− 2−)...(s (ℓ+1)),(8) ts+21K−s+12(tγmp) Xℓ=0 − − − − × γ [2mp]s−12 ∞ χ(γ)t j−1(γ)C(γ) γ The following result follows: = Z Γ(s)Γ(1 s) 0 γ∈CXΓ−{1} − n−1 e−(t+mp)tγdt ζI(2n)(s|Lp)= π8χ(1)Vol(Γ\G)CΓ(cid:18)2np−1 (cid:19) a2jj! × (2tmp+t2)s. (13) Xj=0 j K s ℓ 1;b(p)+(ρ p)2,π j−1(cid:16) − − 0− (cid:17) . (9) 1. Logarithmic derivative of the Selberg zeta function × (j ℓ)!(s 1)(s 2)...(s (ℓ+1)) Xℓ=0 − − − − The function ψ (s;χ) defined in [29] Γ ψ (z;χ)d=ef χ(γ)t j−1(γ)C(γ)e−(z−ρ0)tγ, Γ γ X γ∈CΓ−{1} 2. Odd-dimensional manifold (14) for Res > 2ρ , is a holomorphic function in the half- 0 plane Res > 2ρ and admits a meromorphic continua- In odd-dimensional case, N =2n+1, we get 0 tion to the full complex plane. It has been shown that there is a meromorphicfunction Z (s;χ) onC such that Γ ζ(2n+1)(sL ) = 1χ(1)Vol(Γ G)C 2n (d/dz)logZΓ(z;χ) = ψΓ(z;χ). ZΓ(z;χ) suitable normal- I | p 4 \ Γ(cid:18) p (cid:19) izedistheSelbergzetafunctionattachedto(G,K,Γ,χ). Therefore, n × Xj=0a2jZRr2j[y(r2;m2p)]−sdr. (10) ζH(s|Lp)= Γ(s)Γ1(1 s)Z ∞ ψΓ((ρ20tm+t++tm2)ps;χ)dt. − 0 p (15) Using the formula r2j+1(δ2 + r2)−sdr = Canonical quantization of Abelian p forms yields a for- δ2j−2sΓ(j)Γ(s−j)/(Γ(s))R,Rwe find: mthaelCexapsirmesisrioenne(r1g/y2,)wζ(hser=e−λ1/∞2|Lpi)s−=th(e1/se2t)Pofjeλig1e/n2vfaolr- { }j≥0 ues (with multiplicity) of the Laplacian L on smooth ζI(2n+1)(s|Lp)= χ(1)4VΓo(ls()Γ\G)(cid:18)2pn(cid:19) sections of a vector bundle over XΓ =Γ\HNp induced by a finite-dimensional unitary representation χ of Γ. The n a Γ j+ 1 Γ s j 1 regularizedCasimirenergyrelatedtoco-exactforms(the 2j 2 − − 2 . (11) × Xj=0 (cid:0) m2ps(cid:1)−2j(cid:0)−1 (cid:1) caoltmerpnaacttinegvesnu-mdiomfeznesrioo-naanldhpy−pefrobrmoliccommapnoinfoelndtss)isongirveeanl as follows: 1 π E(m ) = ζ( 1/2L(CE))= χ(1)Vol(Γ G)C p N=2n 2 − | p 16 \ G B. The hyperbolic component of the isometry n−1 j a j! group 2j × Xj=0Xℓ=0 (j−ℓ)! ℓq=0(−12 −(q+1)) Q intTeghrealzeHta(p)f(unc)titoankeassstohceiafoterdmwith hyperbolic orbital × (cid:20)(cid:18)2np−1(cid:19)Kj−1(−ℓ− 23;m2p,π) Γ Kt 3 + K ( ℓ ;m2,π) 1 j−1 − − 2 0 (cid:21) ζH(s|Lp) ×= Z√4∞πtΓs(−s23)eγ−∈tCmXΓ2p−+{t142γt}dχt.(γ)j−1(γ)tγC(γ(1)2) −+ 2ψ1πΓ(Zρ00∞+ntψ+Γ(mρ00;+χ)t[2+tmm0p;+χt)2[2]t21mpdt+. t(21]621) 0 o 4 In the case of scalar field (p =0 for a trivial representa- In particular, the recent data obtained by the Wilkin- tion) the Casimir energy becomes son Microwave Anisotropy Probe (WMAP) [31] satellite confirmed,andsetnewstandardsofaccuracy,tothepre- π E(m ) = χ(1)Vol(Γ G)C viousCOBE’smeasurementofalowquadrupolemoment 0 N=2n G 16 \ in the angular power spectrum of the CMB, which is in n−1 j a j!K ( ℓ 3;m2,π) accordancewiththeassumptionthatthe topologyofthe 2j j−1 − − 2 0 ×Xj=0Xℓ=0 (j−ℓ)! ℓq=0(−12 −(q+1)) uonnitvheersceamseigohftabceomnponac-ttrhivyipale,rbwoiltihc upnairvteicrusela.rCeommpbhianseids −21π Z ∞ψΓ(ρ0+Qt+m0;χ)[2tm0+t2]12dt. (17) wthiatht thi6s0o%bsoefrvthateiocnri,ttichaelWenMerAgyPdseantseiltlyiteofaltshoeiunndiivcaerteses 0 ∼ is contributed by a smoothly distributed vacuum energy Formula(16)withpositiveparametermpgivesthereg- (Casimir energy) or dark energy, whose net effect is re- ularized vacuum energy E(mp) which is finite. From pulsive(leading,thus,toanacceleratedexpansionofthe (11) it follows that in the case of odd N the identity universe). component of E(mp) has poles at s = 1/2 and there- Inthispaper,wehaveshownthatthetopologicalcom- − fore E(mp) cannot be obtained by the method available ponentoftheCasimirenergyforco-exactformsoneven- for even-dimensionalmanifolds, whichagreeswith result dimensional manifolds, associated with the trivial char- obtained in [13]. For the trivial representation χ = 1 acter, is always negative. 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