# 4 component spinor isomorphic to S^7?

1. May 7, 2012

### 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!

Last edited: May 7, 2012
2. May 8, 2012

### 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).

3. May 9, 2012

### samalkhaiat

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

Sam

4. May 10, 2012

### Spinnor

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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