Cylinder transition amplitudes in pure AdS 3 gravity
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Springer
Received: April 2, 2020 Accepted: May 10, 2020 Published: May 28, 2020
Cylinder transition amplitudes in pure AdS3 gravity
a
Physics Department & IFIBA-Conicet, University of Buenos Aires, Pabell´ on 1, Ciudad Universitaria, 1428 Buenos Aires, Argentina b Department of Physics and Astronomy, Montclair State University, 1 Normal Ave, Montclair NJ 07043, U.S.A. c Center for Cosmology and Particle Physics, Department of Physics, New York University, 726 Broadway, New York, NY 10003, U.S.A.
E-mail: [email protected], [email protected], [email protected] Abstract: A spacelike surface with cylinder topology can be described by various sets of canonical variables within pure AdS3 gravity. Each is made of one real coordinate and one real momentum. The Hamiltonian can be either H = 0 or it can be nonzero and we display the canonical transformations that map one into the other, in two relevant cases. In a choice of canonical coordinates, one of them is the cylinder aspect q, which evolves nontrivially in time. The time dependence of the aspect is an analytic function of time t and an “angular momentum” J. By analytic continuation in both t and J we obtain a Euclidean evolution that can be described geometrically as the motion of a cylinder inside the region of the 3D hyperbolic space bounded by two “domes” (i.e. half spheres), which is topologically a solid torus. We find that for a given J the Euclidean evolution cannot connect an initial aspect to an arbitrary final aspect; moreover, there are infinitely many Euclidean trajectories that connect any two allowed initial and final aspects. We compute the transition amplitude in two independent ways; first by solving exactly the time-dependent Schr¨odinger equation, then by summing in a sensible way all the saddle contributions, and we discuss why both approaches are mutually consistent. Keywords: Models of Quantum Gravity, Black Holes, Solitons Monopoles and Instantons ArXiv ePrint: 2003.08955
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2020)147
JHEP05(2020)147
Alan Garbarz,a Jayme Kimb and Massimo Porratic
Contents 1
2 Classical dynamics of the cylinder 2.1 Σ as annulus and as cylinder 2.2 Solving the constraints 2.3 Exact classical evolution 2.4 Canonical coordinates and transformations 2.5 Exact quantum evolution 2.6 Complex structure of a BTZ initial surface 2.7 Analysis of the Euclidean evolution
3 4 4 5 6 7 8 9
3 Semiclassical transitions between conformal structures 3.1 Euclidean saddle 3.2 Summing over the instanton moduli space
10 11 14
4 Conclusions
15
A Annulus embedding
18
1
Introduction
The study of quantum transitions between geometries in General Relativity is a central piece in the construction of a complete Quantum Gravity. It is however a prohibitively difficult matter in four dimensions but theories of gravity in lower dimensions may be less intractable. A particularly simple yet rich playground consist of GR with a negative cosmological constant in three dimensions. This AdS3 pure gra
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