It is known that the thermodynamics of quantum fields in an Einstein universe for some radius is equivalent to that of an instantaneously static closed ...
Brazilian Journal of Physics, vol. 34, no. 3B, September, 2004
1193
Thermodynamics of Abelian Forms in Real Compact Hyperbolic Spaces A. A. Bytsenko, Departamento de F´ısica, Universidade Estadual de Londrina, Londrina, Brazil
V. S. Mendes, Departamento de F´ısica, Universidade Estadual de Londrina, Londrina, Brazil
and A. C. Tort Departamento de F´ısica Te´orica, Instituto de F´ısica, Universidade Federal do Rio de Janeiro, Rio de Janeiro, Brazil Received on 4 December, 2003 We analyze gauge theories based on abelian p−forms in real compact hyperbolic manifolds. The explicit thermodynamic functions associated with skew–symmetric tensor fields are obtained via zeta–function regularization and the trace tensor kernel formula. Thermodynamic quantities in the high–temperature expansions are calculated and the entropy/energy ratios are established.
1 Introduction It is known that the thermodynamics of quantum fields in an Einstein universe for some radius is equivalent to that of an instantaneously static closed Friedmann–Robertson– Walker universe. The field thermodynamics of positive curvature Einstein spaces was discussed by several authors before. In particular, the so-called entropy bounds or entropy to thermal energy ratios were calculated and compared with known bounds such as the Bekenstein bound or the CardyVerlinde bound. For example, for a massless scalar field in S3 space this was done in [1] and for a massive scalar field in [2]. Here we wish to extend the evaluation of those type of bounds to the case of skew symmetric tensor fields in real hyperbolic spaces. We shall work with a D−dimensional compact hyperbolic space X with universal covering M and fundamental group Γ. We can represent M as the symmetric space G/K, where G = SO1 (D, 1) and K = SO(D) is a maximal compact subgroup of G. Then we regard Γ as a discrete subgroup of G acting isometrically on M , and we take X to be the quotient space by that action: X = Γ\M = Γ\G/K. Let τ be an irreducible representation of K on a complex vector space Vτ , and form the induced homogeneous vector bundle G ×K Vτ (the fiber product of G with Vτ over K) → M over M . Restricting the G action to Γ we obtain the quotient bundle Eτ = Γ\(G ×K Vτ ) → X = Γ\M over X. The natural Riemannian structure on M (therefore on X) induced by the Killing form ( , ) of G gives rise to a connection Laplacian L on Eτ . IfPΩK denotes the Casimir operator of K – that is ΩK = − yj2 , for a basis {yj } of the Lie algebra k0 of K, where (yj , y` ) = −δj` , then τ (ΩK ) = λτ
for a suitable scalar λτ . Moreover for the Casimir operator Ω of G, with Ω operating on smooth sections Γ∞ Eτ of Eτ one has L = Ω − λτ 1 ; see Lemma 3.1 of [3]. For λ ≥ 0 let Γ∞ (X , Eτ )λ = {s ∈ Γ∞ Eτ |−Ls = λs }
(1)
be the space of eigensections of L corresponding to λ. Here we note that since X is compact we can order the spectrum of −L by taking 0 = λ0 < λ1 < λ2 < · · · ; limj→∞ λj = ∞. It will be convenient moreover to work with the normalized Laplacian Lp = −c(D)L where c(D) = 2(D − 1) = ∞ 2(2N − 1). Lp has spectrum {c(D)λj , mj }j=0 where the multiplicity mj of the eigenvalue c(D)λj is given by mj = dim Γ∞ (X , Eτ (p) )λj . It is easy to prove the following properties for operators and forms: dd = δδ = 0, δ = (−1)Dp+D+1 ∗ d∗, ∗ ∗ ωp = (−1)p(D−p) ωp . Let αp , βp be p−forms; then R the invariant inner product is defined by (αp , βp ) := M αp ∧ ∗βp . The operators d and δ are adjoint to each other with respect to this inner product for p−forms: (δαp , βp ) = (αp , dβp ). In quantum field theory the Lagrangian associated with ωp takes the form: L = dωp ∧ ∗dωp (gauge field); L = δωp ∧∗δωp (co–gauge field). The Euler–Lagrange equations supplied with the gauge give Lp ωp = 0 , δωp = 0 (Lorentz gauge); Lp ωp = 0 , dωp = 0 (co–Lorentz gauge). These Lagrangians give possible representation of tensor fields or generalized Abelian gauge fields. The two representations of tensor fields are not completely independent. Indeed, there is a duality property in the exterior calculus which gives a connection between star–conjugated gauge tensor fields and co–gauge fields.
A. A. Bytsenko, V. S. Mendes, and A. C. Tort
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2 The trace formula applied to the tensor Kernel We can apply the version of the trace formula developed by Fried in [4]. First we define additional notation. For σp the natural representation of SO(2N − 1) on Λp C2N −1 , one has the corresponding Harish–Chandra–Plancherel density, given for a suitable normalization of Haar measure dx on G by µ ¶ π 2N − 1 Pσp (r)r tanh(πr) , µσp (r) = 4k−4 p 2 [Γ(N )]2 for 0 ≤ p ≤ N − 1, where " ¶2 # µ p+1 Y 3 2 Pσp (r) = r + N −`+ 2 `=2 " µ ¶2 # N Y 1 × r2 + N − ` + 2
For the identity component we have Z ∞ Z 2 2 VΓ (D) s−1 dtt dr µσj e−t(r +αj ) , (4) ζI (s, j) = Γ(s) 0 R where VΓ = χ(1)Vol (Γ\G) /4π, and we define αj2 = b(j) + (ρ0 − j)2 , ρ0 = (D − 1)/2 and b(j) are constants. Replacing the Harish–Chandra–Plancherel measure, we ob(D) tain two representations for ζI (s, j) which hold for odd and even dimension: (2N )
ζI
(s, j) =
=
`=p+2
is an even polynomial of degree 2N − 2. One has that Pσp (r) = Pσ2N −1−p (r) and µσp (r) = µσ2N −1−p (r) for N ≤ p ≤ 2N − 1. Now define the Miatello coeffi(p) cients (see the ref. [5]) a2` for G = SO1 (2N + 1, 1) PN −1 (p) 2` , 0 ≤ p ≤ 2N − 1 . Let by Pσp (r) = `=0 a2` r Vol(Γ\G) denote the integral of the constant function 1 on Γ\G with respect to the G – invariant measure on Γ\G, induced by dx. For 0 ≤ p ≤ D − 1, the Fried trace formula [4] applied to the tensor kernel associated to the Laplace operator on co-exact forms LCE is [6, 7]: p ³ ´ (CE) Tr e−tLp = p X
(−1)
h
j
(p−j)
IΓ
(p−1−j)
(Kt ) + IΓ
(Kt )
j=1 (p−j)
+ HΓ
(p−1−j)
(Kt ) + HΓ
i (Kt ) − bp−j
(2)
where bp are the Betti numbers. In the above formula (p) (p) IΓ (Kt ) and HΓ (Kt ) are the identity and hyperbolic orbital integrals, respectively.
3
The spectral functions of exterior forms
The spectral zeta function related to the Laplace operator Lj can be represented by the inverse Mellin transform of the heat kernel Kt = Tr exp (−tLj ). Using the Fried formula, we can write the zeta function as a sum of two contributions: Z ∞ ³ 1 (j) (j−1) ζ(s|Lj ) = dtts−1 IΓ (Kt ) + IΓ (Kt ) Γ(s) 0 ´ (j) (j−1) + HΓ (Kt ) + HΓ (Kt ) ≡
(D)
ζI
(D)
(s, j) + ζH (s, j).
(3)
(2N +1) ζI (s, j)
Z ∞ (j) N −1 VΓ C2N X (j) a2`,2N dt ts−1 Γ(s) 0 `=0 Z 2 2 × drr2`+1 tanh(πr) e−t(r +αj ) R " (j) N −1 VΓ C2N X (j) Γ(` + 1)Γ(s − ` − 1) a2`,2N Γ(s) αj2s−2`−2 `=0 # ∞ X Γ(s + n) + (5) ξn` 2s+2n , αj n=0
=
=
Z ∞ (j) N VΓ C2N +1 X (j) a2`,2N +1 dt ts−1 Γ(s) 0 `=0 Z 2` −t(r 2 +α2j ) × drr e R (j) VΓ C2N +1
µ ¶ N X 1 Γ `+ Γ(s) 2 `=0 µ ¶ (j) a2`,2N +1 1 ×Γ s − ` − , 2 αj−2s+2`+1
(6)
where Bn is the n-th Bernoulli number. Moreover, we define ¢ `+1 ¡ (−1) 1 − 2−2`−2n−1 ξn` := B2`+2n+2 . (7) n! (2` + 2n + 2) (D)
In fact, we do not need the hyperbolic component ζH (s, j) since in the high temperature limit (see next section), only (D) the function ζI (s, j) will present contribution.
4 The high temperature limit Using the Mellin representation for the zeta function one can obtain useful formulae for the non–trivial temperature dependent part of the identity and hyperbolic orbital components of the free energy (for details see the ref. [8, 9, 10]) Z 1 (D) dz ζR (z) FI,H (β, j) = − 2πi
