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 spinfoam 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 $T^{\ast}SU(1,1)$ as a symplectic quotient of 2-twistor space $\mathbb{T}^2$ 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 are 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.