Biquaternions, Majorana spinors and time-like spin-foams

This work is developed in the context of Lorentzian spin-foams with space- and time-like boundaries. It is argued that the equations describing the semiclassical regime of the various spin-foam amplitudes admit a common biquaternionic structure. A correspondence is given between Majorana 2-spinors and time-like surfaces in Minkowski 3-space based on such complexified quaternions. A symplectic structure for Majorana spinors is constructed, with which the unitary representation theory of $\mathrm{S}\mathrm{U}(1,1)$ is re-derived. As the main result, we propose a symplectomorphism between Majorana spinor space (with an area constraint) and $T^{\ast }\mathrm{S}\mathrm{U}(1,1)$, generalizing previous studies on twisted geometries to the case of time-like 2-surfaces.