Timelike twisted geometries

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 ${\mathrm{T}}^{*}\mathrm{SU}(1,1)$ 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 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.