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

[Spinnor]

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

Thanks for any help!

 

[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.

 

[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.

 

[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.

 

[Spinnor]

...

If so I wonder if a spinor times exp(iωt) could be thought of as an orbit in S^3?...

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!

 

[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!

 

[Spinnor]

...
A general path looks like?

Thanks for any help!

A solution of the Dirac equation?

 

[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!

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.

 

[Spinnor]

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.

...

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)

 

[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.