Timelike twisted geometries

Abstract

Within the twistorial parametrization of loop quantum gravity, we investigate the consequences of choosing a spacelike normal vector in the linear simplicity constraints. The amplitudes for the SU(2) boundary states of loop quantum gravity, given by most of the current spin foam models, are constructed in such a way that even in the bulk only spacelike building blocks occur. Using a spacelike normal vector in the linear simplicity constraints allows us to distinguish spacelike from timelike 2-surfaces. We propose in this paper a quantum theory that includes both spatial and temporal building blocks and hence a more complete picture of quantum spacetime. At the classical level, we show how we can describe T * SU(1, 1) as a symplectic quotient of 2-twistor space T 2 by area matching and simplicity constraints. This provides us with the underlying classical phase space for SU(1, 1) spin networks describing timelike boundaries and their extension into the bulk. Applying a Dirac quantization, we show that the reduced Hilbert space is spanned by SU(1, 1) spin networks and hence is able to give a quantum description of both spacelike and timelike faces. We discuss in particular the spectrum of the area operator and argue that for spacelike and timelike 2-surfaces it is discrete.

Asymptotic analysis of the EPRL model with timelike tetrahedra

“…This case is more complicated. Firstly, no EPRL like construction is known (see [48] for recent developments). Secondly, the coherent state proposition [32] in the line of Freidel-Krasnov model differs from representation theoretic construction in the style of EPRL.…”

“…The model with timelike tetrahedra is considerably more difficult than standard EPRL thus it was not considered seriously so far. However, timelike tetrahedra are natural candidates for some boundary conditions, for example in spin foam cosmology [47,48], or in spin foam black hole calculations (if such will be done in the future). We hope that providing the asymptotic analysis in the present work will support and boost research in this direction.…”

Asymptotic analysis of spin foam amplitude with timelike triangles

“…However, in the extended spin foam model by Conrady and Hnybida, some tetrahedron normal vectors are chosen to be spacelike u = (0, 0, 0, 1). As a result, the model contains timelike tetrahedra and triangles which live in 3D Minkowski subspaces [7][8][9].…”

SU(1,1) Barbero-like variables derived from Holst action

“…For instance, it would be particularly interesting to see how the results for the eigenvalues of the area operator (defined on both timelike and spacelike surfaces) compare to those of Ref. [18] as well as those obtained in the context of twisted geometries [26], when using our SU (1, 1) Barbero-like variables to construct the holonomies.…”