# Spinors and the Bloch sphere

1. Feb 25, 2010

### Talisman

So I tried learning about spinors yesterday, and got myself confused. Hopefully someone can tell me if I'm barking up the right tree...

The way they were introduced was by exhibiting a homomorphism from C^3 to C^2 by using the dot product:

(x1, y1, z1) . (x2, y2, z2) = x1*x2 + y1*y2 + z1*z2

Then a C^3 vector can be isotropic or "orthogonal to itself" (v.v = 0), e.g., v = (1, i, 0).

Now we can map isotropic C^3 vectors into C^2, but it's a 1:2 mapping. It seems this has something to do with the Bloch sphere, whose surface can be represented by vectors in C^3, or more simply in C^2.

There seems to be a natural analog in R^3, where I can either specify unit-length vectors or give a "latitude" and "longitude" (R^2). Two things escape me though:

1) In R^3, I'm using unit vectors, whereas in C^3 I'm using isotropic ones.
2) The 2:1 homomorphism should reflect the fact that two C^2 vectors map to the same isotropic C^3 vector, which probably indicates that the former specify unique points, while the latter specify lines -- but I haven't figured out why the C^3 vectors don't specify unique points, as in R^3. That is, antipodal points on the Bloch sphere should have distinct spinors but the same isotropic C^3 vector.

Any help is greatly appreciated!