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4 component spinor isomorphic to S^7? 
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#1
May712, 11:28 AM

P: 1,368

I was told the space S^3 is isomorphic to the set of all 2 component spinors with norm 1 (see http://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! 


#2
May812, 01:20 AM

Sci Advisor
P: 1,594

"4 component spinor" is not specific enough.
2component 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 noncompact space, not a sphere. But there are other 4component 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
May912, 02:05 PM

Sci Advisor
P: 908

Sam 


#4
May1012, 08:36 AM

P: 1,368

4 component spinor isomorphic to S^7?
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. 


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