Why there are no spinors for GL(n)

  1. Does anybody know a simple proof of the fact that there are no finite-dimensional extensions of the [tex]\textsl{so(n)}[/tex]-spinor representation to the group of general linear transformations. The proof seems can be based on the well-known fact that when rotated [tex]2\pi[/tex] a spinor transforms [tex]\psi\rightarrow-\psi[/tex]. But i have found no elementary proof...
     
  2. jcsd
  3. samalkhaiat

    samalkhaiat 1,101
    Science Advisor

     
  4. Thank you very much. These are well or less-known but nontrivial facts. For example, highly nontrivial is the fact that the double covering of the real general linear group is not a matrix group. I wonder if there is a simple proof with no mention of double-coverings etc
     
  5. Haelfix

    Haelfix 1,748
    Science Advisor

    Theres a cute little book called "spin geometry" by Lawson et al that proves the result (chapter 5) by noting there is no faithful finite dimensional representations possible. The double cover is called the metalinear group incidentally.
     
  6. Excuse me, I found no chapter 5 in this book. Paragraph 5 deals with representation, but non a single word had Lawson said about general linear group and its infinite-dimensional spinors.
     
  7. Haelfix

    Haelfix 1,748
    Science Advisor

    Eeep, apologies, The PDF I have is evidently a lot different than the published book. Unfortunately the book is checked out of our library so I can't find the appropriate corresponding sections. Eyeballing the google book chapter content, maybe look in the representation section on page 30, or somewhere where they talk about Dirac operators?

    Its possible that its not there though, in which case I apologize.
     
  8. samalkhaiat

    samalkhaiat 1,101
    Science Advisor

     
    Last edited: Jun 25, 2008
  9. I agree with you that modern language is best suited, and I would strongly recommand to use it as you do. However, I want to point out that spinors were first defined by Cartan, as you know since you have his book, at a time where vector bundles were not even known. So I do believe one could construct a proof without explicit use of fiber bundles (although one will not get away without topological consideration of course, such as double-covering) :smile:
     
  10. samalkhaiat

    samalkhaiat 1,101
    Science Advisor

     
  11. You are too skilled! One need not "spinor bundles" to prove that it is not possible to extend spinorial representation to a representation of general linear group without enlargring the representation space. It is all about spinors and vectors, not about sections of bundles. Thank you for trying.
     
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