A note on the spinor construction of Spin Foam amplitudes

Abstract

I discuss the use of spinors in the construction of spin-foam models, in particular the form of the closure and simplicity constraints for triangles that are space-like, i.e. with (area)$^2=\half S^{IJ}S_{IJ}>0$, regardless of whether they belong the tetrahedra with a space-like or time-like normal, emphasizing the role of the light-like 4-vector $u_t\sigma^I\bar u_t$. In the quantization of the model, with the representations of SL(2,$\mathbb{C}$) acting on spaces of functions of light-like vectors, one may use the canonical basis of SU(2) representations, or the pseudobasis limited to the discrete representations of SU(1,1); in alternative it is proposed to use instead a basis of eigenstates of $(L_3,K_3)$, which might give matrix elements and vertex functions with the same classical limit. A detailed example of a small triangulation is presented, which among other things indicates, on the basis of a classical calculation, that it would be impractical to limit oneself to tetrahedra with time-like normals.