# Can a point in S^3 be uniquely labeled by a 2 component Spinor?

1. May 5, 2012

### Spinnor

Can a point in S^3 be uniquely labeled by a 2 component Spinor?

Thanks for any help!

2. May 5, 2012

### Spinnor

With a little more thought, I think my question could have been more precise. I guess what I'm really interested in is if the topology of S^3 is the same as the space of all two component spinors with magnitude (norm?) 1? Are they basically the same space? If so I wonder if a spinor times exp(iωt) could be thought of as an orbit in S^3?

Thanks for any help!

It might have been more appropriate to post in the "Topology & Geometry" group, I would move it if I could.

3. May 5, 2012

### fzero

Yes, because $SU(2)$ is isomorphic to $S^3$. We can represent a arbitrary $SU(2)$ matrix by

$$U = \begin{pmatrix} z_1 & -z^*_2 \\ z_2 & z_1^*\end{pmatrix}, ~~ |z_1|^2 +|z_2|^2=1.$$

Then the unit spinor

$$\psi = U \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} z_1 \\ z_2 \end{pmatrix}$$

also represents a point on the sphere. The most general orbits would be obtained by writing

$$U(t) = \exp [i \sigma^a \theta_a(t) ]$$

and specifying the angles of rotation.

4. May 5, 2012

### Spinnor

Thanks fzero!

S^3 I think I can picture, but a spinor is more confusing to me. I guess it is a little less so now, thanks again.

5. May 5, 2012

### Spinnor

Let ω = 1 and let t = 0, at what time t do we come back to our starting place,

a) t = ∏/2

b) t = ∏

c) t = 2∏

d) t = 4∏

Thanks for any help!

6. May 5, 2012

### Spinnor

Then can a Dirac spinor be thought of as a pair of points in a pair of three-spheres?

Or a pair of distinct points in a single three-sphere?

A general path looks like?

Thanks for any help!

7. May 5, 2012

### Spinnor

A solution of the Dirac equation?

8. May 6, 2012

### Spinnor

Let,

$$U(t) = \exp [i \sigma^a \theta_a(t)] = \exp [i \sigma^z t]$$

act on $$\begin{pmatrix} 1 \\ 0 \end{pmatrix}$$

Worked out below it looks like the answer is c.

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9. May 6, 2012

### Spinnor

S^3 is the set of points in R^4 such that,

x^2 + y^2 + z^2 + w^2 = 1

I guess we can let z_1 and z_2 above be,

z_1 = z + iw
z_2 = x + iy

then |z_1|^2 +|z_2|^2=1

Now I can plot the path in S^3, z_1 = exp(it),

z = cos(t), w = sin(t)

10. May 6, 2012

### Spinnor

Let,

$$U(t) = \exp [i \sigma^a \theta_a(t)] = \exp [i \sigma^x t]$$

act on $$\begin{pmatrix} 1 \\ 0 \end{pmatrix}$$

Worked out below.

File size:
28 KB
Views:
142