Is there a geometrical interpretation of a spinor?

Asked May 16 '20 at 22:58

Active May 17 '20 at 00:30

Viewed 62 times

To give a geometrical interpretation of a vector one can associate a vector with two points in space $(A,B)$. Any vector can be thought of as an equivalence class of pairs of points.

Like wise a symmetric 2-tensor can be thought of in terms of a D-simplex in D dimensions. Which is given by D+1 points.

Is there any geometrical interpretation of a spinor in terms of geometrical objects in D dimensional space?

I'm guessing not since D dimensional space as we know it has symmetry group O(D) which can't incorporate spinors.

The only way I can sort of visualise a spinor space, is to think of an white object with black dots, and when you rotate it in your hand 360 degrees the black dots have changed white and vice versa.

Edit: Well I just realised that one could use twistor space, in which case you do have a representation but that exists in 8 dimensions (or 4 complex dimensions).

edited May 17 '20 at 00:30

asked May 16 '20 at 22:58

zooby

4,649

1

Yes: https://en.wikipedia.org/wiki/Spinor – David G. Stork May 16 '20 at 23:00
@Stork I can only see a visualisation of a spinor in 1D a mobius band. Not sure how that would work for 2D. – zooby May 16 '20 at 23:03
What would a vector pinned to a Klein bottle do? – Oscar Lanzi May 17 '20 at 00:39
Can you expand on how a 2-tensor can be thought of in terms of a D-simplex? – anon May 17 '20 at 00:52
@runway44 Yes, a DxD matrix has D rows of vectors. each vector an reprsent a coroner of the simplex with one corner at the origin. – zooby May 17 '20 at 01:46

Is there a geometrical interpretation of a spinor?

0 Answers0