TRANSITION AMPLITUDE IN (2+1)-DIMENSIONAL CHERN-SIMONS GRAVITY ON A TORUS
Open accessAbstract
In the framework of the Chern-Simons gravity proposed by Witten, a transition amplitude of a torus universe in (2+1)-dimensional quantum gravity is computed. This amplitude has the desired properties as a probability amplitude of the quantum mechanics of a torus universe, namely, it has a peak on the “classical orbit” and it satisfies the Schrödinger equation of the (2+1)-dimensional gravity. The discussion is given that the classical orbit dominance of the amplitude is not altered by taking the modular invariance into account and that this amplitude can serve as a covariant transition amplitude in a particular sense. A set of the modular-covariant wave functions is also constructed and they are shown to be equivalent to the weight-½ Maass forms.
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Such transition amplitudes are given explicitly in [118], where they are shown to be peaked around the classical trajectory.
“such amplitudes are given explicitly in [118], where it is shown that they are peaked around the classical trajectory.”
From Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe (2005) · cites this paper as [118]
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Research works printed but not linked: 5
- [6][6] A. Anderson, preprint Imperial/TP/92-93/02, gr-qc/9210007 · arXiv
- [7][7] E. Witten, Nucl. Phys. B323 (1989) 113; Y. Fujiwara, S. Higuchi, A. Hosoya, T. Mishima and M. Siino, Phys. Rev. D44 (1991) 1756 and 1763; S. Carlip, preprint UCD- 92-16, hep-th/9206103 · arXiv
- [13][13] J. P. Serre, A. Course in Arithmetic ( Springer-Verlag, New-York, Heidelberg, Verlin ) Chap. VII.
- [15][15] G. Mess, preprint IHES/M/90/28; S. Carlip, Class. Quant. Grav. 8 (1991) 5.
- [19][19] J. Louko and D. M. Marolf, preprint SU-GP-93/7-6, gr-qc/9308018. 19 http://arxiv.org/abs/gr-qc/9210007 http://arxiv.org/abs/hep-th/9206103 http://arxiv.org/abs/gr-qc/9308018 Figure Captions Fig.1a A torus universe embedded into the Minkowski space. We take the T -axis as a baseline B and draw lines which are obtained by transforming the baseline by Λ1, Λ2 and Λ1Λ2 ( see eq.(17) ), The torus universe is expressed by the region inside these lines, with the boundary surface identified by the above transformations. The case of (a, u) = (1, 0), (b, w) = (0, 1) is shown explicitly. The torus universe is the shadowed region inside the bold lines, where the two dotted lines and the two dashed lines are respectively identified. Fig.1b An evolution of the time-slice developed into the (Y, θ/τ)-plane. The baseline B is the T -axis, i.e. Y = θ = 0. Three parallelograms (drawn by bold lines) intersecting with B at T = 0, 1 and 2 represent time-slices at the time τ = ∞, 1 and 1/2, respectively. On each parallelogram, opposite sides are regarded to be identified. The figure shows how a "wide" torus shrinks to the wire-like singularity as time elapses. From this picture, if we wish, we can deduce the trajectory of the moduli parameters, which becomes a semicircle centered on the m1-axis ( see ref.[10]). Note that a point (a, b;w, u) in the C-S phase space corresponds to a τ -evolution of the torus universe. Fig.2 The "classical orbit" Cc.o. of moduli parameters. Cc.o. is a circle in the upper-half plane. We can see that the orbit expands as τ2 gets larger (or smaller), starting from τ1. 20 · arXiv
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Quantum Gravity in 2 + 1 Dimensions: The Case of a Closed Universe
Carlip, S. (2005) · Living Reviews in Relativity · doi:10.12942/lrr-2005-1 · Cited by 172 · see citing works
- Authors
- EZAWA, K.
- Affiliations
- The University of Osaka (JP)
- Corresponding author
- EZAWA, KIYOSHI
- ORCID
- EZAWA, KIYOSHI
- Venue
- International Journal of Modern Physics A
- Publisher
- World Scientific Pub Co Pte Lt
- Published
- 1994-10-30
- Volume
- 09
- Issue
- 27
- Pages
- 4727-4745
- ISSN
- 0217-751X (ISSN-L)
- 1793-656X
- OpenAlex
- W2062082645
- Cited by
- 16
- References
- 0
- Citations by year
- 2025: 1 · 2024: 1 · 2020: 1 · 2017: 1 · 2015: 5 · 2014: 1 · 2012: 1
- FWCI
- 0.54
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Locations
- International Journal of Modern Physics A (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- arXiv (Cornell University)
- Topics
- Physical Sciences › Physics and Astronomy › Nuclear and High Energy Physics › Black Holes and Theoretical Physics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Noncommutative and Quantum Gravity Theories
- Physical Sciences › Physics and Astronomy › Astronomy and Astrophysics › Relativity and Gravitational Theory
- Keywords
- Chern-Simons gravity, three-dimensional gravity, transition amplitudes, modular invariance, Schrödinger equation
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