4 component spinor isomorphic to S^7?

[Spinnor]

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!

 

[Ben Niehoff]

"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).

 

[samalkhaiat]

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!

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

Sam

 

[Spinnor]

Thanks to both of you, Ben and Sam, for clearing that up!

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