Abstract
We calculate the Casimir energy associated with abelian gauge fields in real compact hyperbolic spaces. The cosmological applications of the vacuum energies are briefly considered.
Casimir effect for differential forms in real compact hyperbolic spaces
V. S. MendesI; T. G. PradoII
IDepartamento de Física, Universidade Estadual de Londrina, Caixa Postal 6001, Londrina-Paraná, Brazil
IIDepartamento de Física, Universidade Estadual de Londrina, Rodovia Celso Garcia Cid, Caixa Postal 6001, 86051-990, Londrina, Paraná, Brazil
ABSTRACT
We calculate the Casimir energy associated with abelian gauge fields in real compact hyperbolic spaces. The cosmological applications of the vacuum energies are briefly considered.
I. INTRODUCTION
In this note we present some results concerning abelian gauge fields in locally symmetric spaces. In particular, we calculate the topological Casimir energy for abelian gauge fields (p-forms) in compact hyperbolic spaces XG = G\X = G\G/K, where G = SO1(N,1) , K = SO(N) is a maximal compact subgroup of G, and G Ì G is a discrete group of isometries. Locally symmetric spaces in general and real hyperbolic spaces in particular play important role in supergravity [8], superstring theory [9], and cosmology[1-13].
Casimir effect in spaces with non-trivial topology is an important issue in different areas of quantum field theory and cosmology (see for example [2]). Calculations involving quantum fields in X = N, N have been actively investigated, but not much has been done for hyperbolic spaces. Calculations involving scalar and spinor fields in X = N have been considered in [1-7]. Here we present some results for the case of gauge fields.
II. SPECTRAL FUNCTIONS OF HYPERBOLIC GEOMETRY
A formal expression for the Casimir energy can be written as follows: , where is the set of eigenvalues of a Laplace-Beltrami operator , acting on co-exact part of p-forms (see [15]). We can use the zeta function method and get
Calculation of the zeta function z may start from representation
The trace of the heat kernel can be calculated using the Fried formula [11]. Taking into account the physical degrees of freedom we get
The first term is the contribution from the identity element of the isometry group G, and it is given by
The second term is the contribution from the remaining elements of the isometry group,
Here c is a homomorphism G ® S1. For more details and notation see [3]. Spectral properties of Laplace-Beltrami operator is controled by Harish-Chandra-Plancherel measure (r) wich is given by
where = 22N-4G2(N/2) and a2l are the Miatello coefficients [14].
III. THE CASIMIR ENERGY
The zeta function related to the identity integral (t) is calculated to be
where km(s;d,a) dr r2m(d+r2)-ssech2(ar), and yG(z) is the logarithmic derivative of the Selberg zeta function. For odd dimensions we have
For odd N there are poles at s = -1/2, therefore Casimir energy cannot be obtained by means of zeta function regularization.
The regularized Casimir energy related to co-exact forms on real compact even-dimensional hyperbolic manifolds is
IV. CONCLUDING REMARKS
Cosmological predictions, such as the microwave background anisotropies and the current acceleration expansion of the universe [17], depend pretty much on details of theoretical model under consideration. Recent data obtained by the Wilkinson Microwave Anisotropy Probe (WMAP) [] satellite confirmed, and set new standards of accuracy to previous COBE's measurements wich are in agreement with the assumption that the topology of the universe migtht be non-trivial, with particular enphasis of a compact hyperbolic space. Combined with this observation, the WMAP satellite also indicates that around 60% of the critical energy density of the universe is contributed by a smoothly distribucted vaccum energy or dark energy, whose net effect is repulsive (leading thus to an accelerated expansion of the universe). Note also that topological component of the Casimir energy for co-exact forms on even-dimensional manifolds, associated with the trivial character is always negative.
References
[1] E. Elizalde, S. D. Odintsov, A. Romeo, A. A. Bytsenko, and S. Zerbini. Zeta Regularization Techniques with Applications. World Scientific, Singapore, 1994.
[2] A. A. Bytsenko, G. Cognola, and L. Vanzo. Phys. Rep. 266, 1 (1996).
[3] A. A. Bytsenko, M. E. X. Guimarães, and V. S. Mendes. Eur. Phys. J. C 39, 249 (2005).
[4] A. A. Bytsenko, G. Cognola, E. Elizalde, V. Moretti, and S. Zerbini. Analytic Aspects of Quantum Fields. World Scientific, Singapore, 2003.
[5] A. A. Bytsenko and Yu. P. Goncharov. Class. Quantum Gravi. 8, 2269 (1991); 8, L211 (1991); Mod. Phys. Lett. A6, 669 (1991).
[6] G. Cognola, L. Vanzo, and S. Zerbini. J. Math. Phys. 32, 222 (1992).
[7] G. Cognola, K. Kirsten, and S. Zerbini. Phys. Rev. D 48, 790 (1993).
[8] M. J. Duff, B. E. Nilsson, and C. N. Pope. Phys. Rep. 130, 1 (1986).
[9] J. Maldacena. Adv. Theor. Math. Phys. 2, 231 (1998).
[10] E. Elizalde. e-Print arXiv: hep-th/0311195.
[11] D. Fried. Invent. Math. 84, 523 (1986).
[12] R. Aurich, S. Lustig, F. Steiner, and H. Then. e-Print arXiv: astro-ph/0403597.
[13] N. Kaloper, J. March-Russell, G. D. Starkman, and M. Trodden. Phys. Rev. Lett. 85, 928 (2000).
[14] R. Miatello. Trans. Am. Math. Soc. 260, 1 (1980).
[15] Yu. N. Obukhov. Phys. Lett. B 109, 195 (1982).
[16] A. A. Bytsenko. Nucl. Phys. (Proc. Suppl.) B 104, 127 (2002).
[17] S. Perlmutter et al. Ap. J. 517, 565 (1999).
[18] H. V. Peiris et al. Ap. JS 148, 213 (2003).
Received on 14 October, 2005
- [1] E. Elizalde, S. D. Odintsov, A. Romeo, A. A. Bytsenko, and S. Zerbini. Zeta Regularization Techniques with Applications. World Scientific, Singapore, 1994.
- [2] A. A. Bytsenko, G. Cognola, and L. Vanzo. Phys. Rep. 266, 1 (1996).
- [3] A. A. Bytsenko, M. E. X. Guimarăes, and V. S. Mendes. Eur. Phys. J. C 39, 249 (2005).
- [4] A. A. Bytsenko, G. Cognola, E. Elizalde, V. Moretti, and S. Zerbini. Analytic Aspects of Quantum Fields World Scientific, Singapore, 2003.
- [5] A. A. Bytsenko and Yu. P. Goncharov. Class. Quantum Gravi. 8, 2269 (1991);
- 8, L211 (1991); Mod. Phys. Lett. A6, 669 (1991).
- [6] G. Cognola, L. Vanzo, and S. Zerbini. J. Math. Phys. 32, 222 (1992).
- [7] G. Cognola, K. Kirsten, and S. Zerbini. Phys. Rev. D 48, 790 (1993).
- [8] M. J. Duff, B. E. Nilsson, and C. N. Pope. Phys. Rep. 130, 1 (1986).
- [9] J. Maldacena. Adv. Theor. Math. Phys. 2, 231 (1998).
- [11] D. Fried. Invent. Math. 84, 523 (1986).
- [13] N. Kaloper, J. March-Russell, G. D. Starkman, and M. Trodden. Phys. Rev. Lett. 85, 928 (2000).
- [14] R. Miatello. Trans. Am. Math. Soc. 260, 1 (1980).
- [15] Yu. N. Obukhov. Phys. Lett. B 109, 195 (1982).
- [16] A. A. Bytsenko. Nucl. Phys. (Proc. Suppl.) B 104, 127 (2002).
- [17] S. Perlmutter et al. Ap. J. 517, 565 (1999).
- [18] H. V. Peiris et al. Ap. JS 148, 213 (2003).
Publication Dates
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Publication in this collection
16 Jan 2006 -
Date of issue
Dec 2005
History
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Received
14 Oct 2005