I often feel think that the three-dimensional space and the one-dimensional time can be viewed as arising due to a basic spinorial structure. I am drawing an analogy with the triplet and singlet states which arise when two spin 1/2 are added. Can we say that space and time basically arise from the coupling of two spinorial structures, the triplet combination giving the 3-d space and the singlet combination giving the 1-dimensional time? Please bear with me if my observation looks very crazy.
There is a relationship between spacetime and spinors, but it isn't the one that you think (although I will ponder upon it). You can assign to every 4-vector satisfying [tex]v^2 = 1[/tex] a 2x2 special unitary matrix satisfying [tex]\det u = 1[/tex], etc. Now SU(2) matrices transform as two copies of SU(2), whereas spinors transform as one copy. So it is often said that spinors are "square roots" of vectors. There's a related construction for null vectors and this goes by the name of twistors.
the basic space time group : Lorentz group can be decomposed into SU(2)XSU(2). So you see all inertial frames are basically connected by a bispinor structure...