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The Casimir effect for a class of hyperbolic (D+1)-dimensional spacetimes
Open accessAbstract
The vacuum energy for a free massless scalar field defined on an ultrastatic (D+1)-dimensional spacetime R*H D / Gamma , Gamma being a torsion-free subgroup of isometries of D-dimensional Lobachevsky space H D , is explicitly computed using the Harish-Chandra-Selberg trace formula. All topologically inequivalent configurations of scalar fields are taken into account. As a result, a negative topological vacuum energy is found for a trivial line bundle related to character chi =1.
Overview
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TL;DR
The paper uses the Selberg trace formula and zeta-function regularization to compute the vacuum/Casimir energy of a free scalar field on ultrastatic spacetimes \(R \times H^D/\Gamma\) with compact hyperbolic spatial section. It finds that the topological contribution from non-identity hyperbolic conjugacy classes is negative for untwisted scalar fields, with different analytic behavior in odd and even dimensions.
Key findings
- For odd \(D=2M+1\), the analytically continued expression has a simple pole at \(s=-1/2\); the heat-kernel coefficient \(K_{D+1}=K_{2M+2}\) is non-vanishing and proportional to the residue.
- For even \(D=2M\), the analytic continuation gives a finite result at \(s=-1/2\).
- The “topological” contribution \(E_2(s)\) to the vacuum energy is negative for trivial characters \((\chi=1)\). For \(D=3\) \((M=1)\), \(E_2(s)\) has a pole, \(K_4=A_2\) is non-vanishing, and zeta-function regularization gives a finite result in the limit \(A\to 0\); this topological contribution is again negative for an untwisted scalar field.
- The paper concludes that the vacuum energy related to hyperbolic elements of the \(\Gamma\)-conjugacy class different from the unity element (the topological contribution) is negative.
Methods
- Considers a free scalar field on a \((D+1)\)-dimensional ultrastatic spacetime \(R \times H^D/\Gamma\), where \(H^D/\Gamma\) is a compact hyperbolic manifold, the quotient of \(H^D\) by a discontinuous torsion-free group of isometries; restricts to non-trivial cyclic subgroups of \(\Gamma\). The cases \(D=2\) and \(D=3\) were studied previously.
- Uses the Selberg trace formula to perform analytic continuation for \(\mathrm{Tr}\,L^{-s}\), together with zeta-function regularization and heat-kernel coefficients, for manifolds without boundary.
- Separates the odd-dimensional case \(D=2M+1\) and even-dimensional case \(D=2M\), using modified Bessel functions and the Selberg zeta function.
Limitations
- Stated: spinor fields are not treated; the analysis is restricted to scalar fields and manifolds without boundary; the arbitrary mass renormalization parameter \(\mu\) is said to summarize unknown physics.
- Stated: in general, ignorance of the spectrum prevents quantitative conclusions without symmetry; finite-temperature and Coleman-Weinberg analyses are deferred elsewhere.
- Evident from the provided text: OCR corruption and missing formulas prevent independent verification of the explicit expressions and numerical values.
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In the paper
“In this case, the E , ( s ) has a pole.”
§ The Casimir effect for a class of hyperbolic (D+1)dimensional spacetimes > Additional information
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In the paper
“The related coefficient K4 = A2 is not vanishing and the zeta function regularization gives a finite result which is given, in the limit A goes to zero, by equation (141,”
§ The Casimir effect for a class of hyperbolic (D+1)dimensional spacetimes > Additional information
plain- Asserted: passes — The authors put it forward as true.
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In the paper
“Thus we see that the topological contribution to the vacuum energy is negative for trivial characters ( x = 1).”
§ The Casimir effect for a class of hyperbolic (D+1)dimensional spacetimes > Abstract
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In the paper
“A direct computation shows that the K,,,, is non-vanishing and it is proportional to the residue at s = -6 of the expression (17).”
§ The Casimir effect for a class of hyperbolic (D+1)dimensional spacetimes > A A Bytsenko and S Zerbini
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In the paper
“Then we see that the vacuum energy is manifestly negative as soon as one deals with untwisted scalar fields defined on a Riemann surface of arbitrary genus g.”
§ The Casimir effect for a class of hyperbolic (D+1)dimensional spacetimes > Additional information
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Citations
Cites
Zeta function regularization of path integrals in curved spacetime
Hawking, S. W. (1977) · Communications in Mathematical Physics · doi:10.1007/bf01626516 · Cited by 1,160
Properties of the vacuum. I. Mechanical and thermodynamic
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Finite temperature and boundary effects in static space-times
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The Asymptotics of The Laplacian on a Manifold with Boundary
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Zeta functions and the Casimir energy
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Harmonic analysis and propagators on homogeneous spaces
Camporesi, R. (1990) · Physics Reports · doi:10.1016/0370-1573(90)90120-q · Cited by 257
Dowker, J. S., & Banach, R. (1978) · Journal of Physics A Mathematical and General · doi:10.1088/0305-4470/11/11/012 · Cited by 117
Zeta functions of Selberg's type for compact space forms of symmetric spaces of rank one
Gangolli, R. (1977) · Illinois Journal of Mathematics · doi:10.1215/ijm/1256049498 · Cited by 100
Effective actions and conformal transformations
Brown, M. R., & Ottewill, A. C. (1985) · Physical Review D · doi:10.1103/physrevd.31.2514 · Cited by 88
Cited by
New developments in the Casimir effect
Bordag, M., Mohideen, U., & Mostepanenko, V. M. (2001) · Physics Reports · doi:10.1016/s0370-1573(01)00015-1 · Cited by 1,629
Quantum fields and extended objects in space-times with constant curvature spatial section
Bytsenko, A. A., Cognola, G., Vanzo, L., & Zerbini, S. (1996) · Physics Reports · doi:10.1016/0370-1573(95)00053-4 · Cited by 272
Self-citationSpectral functions and zeta functions in hyperbolic spaces
Camporesi, R., & Higuchi, A. (1994) · Journal of Mathematical Physics · doi:10.1063/1.530850 · Cited by 185
Vacuum energy on orbifold factors of spheres
Chang, P. C. Y., & Dowker, J. S. (1993) · Nuclear Physics B · doi:10.1016/0550-3213(93)90223-c · Cited by 64
One-loop effective potential on hyperbolic manifolds
Cognola, G., Kirsten, K., & Zerbini, S. (1993) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.48.790 · Cited by 23
Self-citationTopological Casimir energy for a general class of Clifford–Klein space–times
Williams, F. L. (1997) · Journal of Mathematical Physics · doi:10.1063/1.531871 · Cited by 17
Zeta-function regularization for Kaluza-Klein finite temperature theories with chemical potentials
Bytsenko, A. A., Vanzo, L., & Zerbini, S. (1992) · Physics Letters B · doi:10.1016/0370-2693(92)90114-j · Cited by 13
Self-citationThe large-distance limit of the gravitational effective action in hyperbolic backgrounds
Bytsenko, A. A., Odintsov, S. D., & Zerbini, S. (1995) · Classical and Quantum Gravity · doi:10.1088/0264-9381/12/1/002 · Cited by 12
Self-citationThe conformal anomaly in N-dimensional spaces having a hyperbolic spatial section
Bytsenko, A. A., Elizalde, E., & Odintsov, S. D. (1995) · Journal of Mathematical Physics · doi:10.1063/1.531215 · Cited by 11
Self-citationThe stress tensor conformal anomaly and analytic regularizations
Svaiter, B. F., & Svaiter, N. F. (1994) · Journal of Mathematical Physics · doi:10.1063/1.530574 · Cited by 11
Vacuum energy for 3+1 dimensional space-time with compact hyperbolic spatial part
Bytsenko, A. A., Cognola, G., & Vanzo, L. (1992) · Journal of Mathematical Physics · doi:10.1063/1.529530 · Cited by 10
Self-citationBose-Einstein condensation of scalar fields on hyperbolic manifolds
Cognola, G., & Vanzo, L. (1993) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.47.4575 · Cited by 8
The local zeta function for symmetric spaces of non-compact type
Godoy, T., Miatello, R. J., & Williams, F. L. (2010) · Journal of Geometry and Physics · doi:10.1016/j.geomphys.2010.08.008 · Cited by 5
Casimir effect for gauge fields in spaces with negative constant curvature
Bytsenko, A. A., Guimarães, M. E. X., & Mendes, V. S. (2004) · The European Physical Journal C · doi:10.1140/epjc/s2004-02080-3 · Cited by 4
Self-citationCasimir effect in hyperbolic polygons
Ahmedov, H. (2007) · Journal of Physics A Mathematical and Theoretical · doi:10.1088/1751-8113/40/34/016 · Cited by 1
Casimir Effect in Hyperbolic PolygonsPreprint
Ahmedov, H. (2006) · Cited by 1
Renormalization group equations in curved space-time with non-trivial topologyPreprint
Elizalde, E., & Odintsov, S. D. (n.d.) · Cited by 1
Metadata
- Authors
- Bytsenko, A. A., & Zerbini, S.
- Affiliations
- University of Trento (IT)
- ORCID
- Zerbini, S
- Venue
- Classical and Quantum Gravity
- Publisher
- IOP Publishing
- Published
- 1992-05-01
- Volume
- 9
- Issue
- 5
- Pages
- 1365-1370
- ISSN
- 0264-9381 (ISSN-L)
- 1361-6382
- OpenAlex
- W2070531991
- Semantic Scholar
- 90d76f3e1d5cf65db183823b2b86bc5e4ab658e1
- Cited by
- 16
- References
- 18
- FWCI
- 2.14
- Type
- journal article
- Language
- en
- Open access
- bronze
- doi.org
- Locations
- Classical and Quantum Gravity (publishedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Atomic and Molecular Physics, and Optics › Quantum Electrodynamics and Casimir Effect
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Keywords
- Casimir effect, vacuum energy, massless scalar field, hyperbolic spacetime, ultrastatic spacetime, hyperbolic space, character varieties, line bundles, higher-dimensional spacetime
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