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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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0 Valid · 0 Not valid · 5 Not checked · 17 Own results and interpretations
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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
plain- Asserted: passes — The authors put it forward as true.
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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
Ambjørn, J., & Wolfram, S. (1983) · Annals of Physics · doi:10.1016/0003-4916(83)90065-9 · Cited by 377
Finite temperature and boundary effects in static space-times
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The Asymptotics of The Laplacian on a Manifold with Boundary
Branson, T., & Gilkey, P. (1990) · Communications in Partial Differential Equations · doi:10.1080/03605309908820686 · Cited by 276
Zeta functions and the Casimir energy
Blau, S. K., Visser, M., & Wipf, A. (1988) · Nuclear Physics B · doi:10.1016/0550-3213(88)90059-4 · Cited by 273
Ellis, G. (1971) · General Relativity and Gravitation · doi:10.1007/bf02450512 · Cited by 272
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
Rigorous extension of the proof of zeta-function regularization
Elizalde, E., & August Romeo, A. (1989) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.40.436 · Cited by 73
Regularization dependence of vacuum energy in arbitrarily shaped cavities
Cognola, G., Vanzo, L., & Zerbini, S. (1992) · Journal of Mathematical Physics · doi:10.1063/1.529948 · Cited by 51
Self-citationInhomogeneous multidimensional Epstein zeta functions
Kirsten, K. (1991) · Journal of Mathematical Physics · doi:10.1063/1.529045 · Cited by 46
Elizalde, E., & August Romeo, A. (1990) · International Journal of Modern Physics A · doi:10.1142/s0217751x90000751 · Cited by 24
Conformal transformation of the effective action
Dowker, J. S. (1986) · Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields · doi:10.1103/physrevd.33.3150 · Cited by 22
Topological Casimir effect for a class of hyperbolic three-dimensional Clifford-Klein spacetimes
Bytsenko, A. A., & Goncharov, Y. P. (1991) · Classical and Quantum Gravity · doi:10.1088/0264-9381/8/12/011 · Cited by 17
Self-citationTopological Casimir effect for a class of hyperbolic four-dimensional Clifford-Klein spacetimes
Goncharov, Y. P., & Bytsenko, A. A. (1991) · Classical and Quantum Gravity · doi:10.1088/0264-9381/8/10/003 · Cited by 14
Self-citationBytsenko, A. A., & Goncharov, Y. P. (1991) · Modern Physics Letters A · doi:10.1142/s0217732391000683 · Cited by 9
Self-citation
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-citation
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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