Cylinder transition amplitudes in pure AdS3 gravity

PORRATI, MASSIMO; KIM, JAYME; GARBARZ, ALAN

JOURNAL OF HIGH ENERGY PHYSICS - (Online)

Springer Berlin Heidelberg

Año: 2020 vol. 2020

1029-8479

A spacelike surface with cylinder topology can be described by various sets of canonical variables within pure AdS_{3} 3​ 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 Schrödinger equation, then by summing in a sensible way all the saddle contributions, and we discuss why both approaches are mutually consistent.