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4 component spinor isomorphic to S^7?

  1. May 7, 2012 #1
    I was told the space S^3 is isomorphic to the set of all 2 component spinors with norm 1 (see https://www.physicsforums.com/showthread.php?t=603404 ). Can I infer that the space of all 4 component spinors with norm 1 is isomorphic to S^7?

    If so is a Dirac spinor isomorphic to S^7?

    Thanks for any help!
    Last edited: May 7, 2012
  2. jcsd
  3. May 8, 2012 #2

    Ben Niehoff

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    "4 component spinor" is not specific enough.

    2-component spinors transform under Spin(3) which is isomorphic to SU(2), hence S^3. Dirac spinors transform under a reducible rep of Spin(3,1), which is going to be some non-compact space, not a sphere. But there are other 4-component spinors, such as those in Spin(4), Spin(5), or Spin(4,1). None of these are topologically S^7, though.

    S^7 is the set of unit octonions, which don't have a group structure (due to the failure of associativity).
  4. May 9, 2012 #3


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    Topologically, the manifold defined by [itex]\bar{\Psi}\Psi=1[/itex] is [itex]S^{3}\times \mathbb{R}^{4}[/itex].

  5. May 10, 2012 #4
    Thanks to both of you, Ben and Sam, for clearing that up!

    What a gem Physics Forums is, ask almost any question and get answer.
    Last edited: May 10, 2012
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