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

  1. May 5, 2012 #1
    Can a point in S^3 be uniquely labeled by a 2 component Spinor?

    Thanks for any help!
     
  2. jcsd
  3. May 5, 2012 #2
    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.
     
  4. May 5, 2012 #3

    fzero

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

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

    Then the unit spinor

    [tex] \psi = U \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} z_1 \\ z_2 \end{pmatrix}[/tex]

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

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

    and specifying the angles of rotation.
     
  5. May 5, 2012 #4
    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.
     
  6. May 5, 2012 #5
    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!
     
  7. May 5, 2012 #6
    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!
     
  8. May 5, 2012 #7
    A solution of the Dirac equation?
     
  9. May 6, 2012 #8
    Let,

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

    act on [tex] \begin{pmatrix} 1 \\ 0 \end{pmatrix} [/tex]

    Worked out below it looks like the answer is c.
     

    Attached Files:

  10. May 6, 2012 #9
    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)
     
  11. May 6, 2012 #10
    Let,

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

    act on [tex] \begin{pmatrix} 1 \\ 0 \end{pmatrix} [/tex]

    Worked out below.
     

    Attached Files:

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