Are Spinors and the Bloch Sphere Connected through Complex Vector Homomorphism?

[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!

 

[eljose79]

Hello there! It's great that you're diving into the world of spinors. They can be a bit confusing at first, but once you understand their properties and applications, they can be incredibly powerful tools in physics and mathematics.

Firstly, your understanding of the homomorphism from C^3 to C^2 is correct. The dot product is used to map isotropic vectors in C^3 to vectors in C^2, and this mapping is indeed 1:2. This is because C^3 is three-dimensional, while C^2 is two-dimensional. This means that for every point in C^2, there are two corresponding points in C^3 that map to it.

To address your first question, the use of isotropic vectors in C^3 is due to the nature of spinors. Spinors are mathematical objects that are used to describe the spin of particles in quantum mechanics. In order to fully describe the spin of a particle, we need to use complex numbers, and thus we use isotropic vectors in C^3. These vectors are also called "spinors" because they are used to represent the spin of a particle.

Regarding your second question, the reason why two C^2 vectors can map to the same isotropic C^3 vector is because of the concept of "spinor equivalence." This means that two spinors that differ by a factor of -1 are considered equivalent. This is similar to how two points on a sphere that are antipodal (opposite sides of the sphere) are considered equivalent. In other words, the spinors may look different, but they represent the same physical quantity.

I hope this helps clarify some of your confusion. Spinors can be a complex topic, but with some practice and understanding, you'll be able to use them confidently in your research. Don't hesitate to ask more questions if you need further clarification. Happy learning!