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!

What a gem Physics Forums is, ask almost any question and get answer.

 

[blue_raver22]

I would like to clarify that the statement about the 4 component spinor being isomorphic to S^7 is not entirely accurate. While it is true that the space of all 2 component spinors with norm 1 is isomorphic to S^3, the 4 component spinor is not isomorphic to S^7.

The reason for this is that S^7 is a seven-dimensional sphere, while a 4 component spinor is a mathematical object with four complex components. These two structures are not equivalent and therefore cannot be isomorphic.

Furthermore, a Dirac spinor is a specific type of 4 component spinor that satisfies certain mathematical equations. It is not isomorphic to S^7 either, as it is a distinct mathematical object with its own properties and structure.

In summary, while there may be some similarities between the space of 2 component spinors with norm 1 and S^3, and between Dirac spinors and S^7, they are not isomorphic and should not be confused as such. It is important to understand and accurately represent the mathematical concepts and structures we are discussing.