Editing 20: Sir Roger Penrose - Plotting the Twist of Einstein’s Legacy
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::The following diagram summarizes the relationship with the structures so far: | ::The following diagram summarizes the relationship with the structures so far: | ||
[[File:Spinor_construction.png|frameless| | [[File:Spinor_construction.png|frameless|]] | ||
</div> | </div> | ||
::The hook arrows denote the constructed embeddings, the double arrow gives the spin double cover of the rotations, and the downwards vertical arrows are the representations of the two groups acting on their respective vector spaces. The lack of an arrow between the two vector spaces indicates no direct relationship between vectors and spinors, only between their groups. There is however a bilinear relationship, meaning a map which is quadratic in the components of spinors to obtain components of vectors, justifying not only that spin transformations are square roots of rotations but spinors as square roots of vectors. When n is even, the spinor space will decompose into two vector spaces: <math> \Delta = S^+ \oplus S^- </math>. In a chosen spin-basis these bilinear maps will come from evaluating products of spinors from the respective subspaces under the gamma matrices as quadratic forms. | ::The hook arrows denote the constructed embeddings, the double arrow gives the spin double cover of the rotations, and the downwards vertical arrows are the representations of the two groups acting on their respective vector spaces. The lack of an arrow between the two vector spaces indicates no direct relationship between vectors and spinors, only between their groups. There is however a bilinear relationship, meaning a map which is quadratic in the components of spinors to obtain components of vectors, justifying not only that spin transformations are square roots of rotations but spinors as square roots of vectors. When n is even, the spinor space will decompose into two vector spaces: <math> \Delta = S^+ \oplus S^- </math>. In a chosen spin-basis these bilinear maps will come from evaluating products of spinors from the respective subspaces under the gamma matrices as quadratic forms. | ||