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Basic question about Z2 graded algebras

  1. Nov 17, 2007 #1
    Sorry, I really hate reading, and I do better by just asking stupid questions.

    The lorentz group SO(3,1) is not simply connected so its unitary representation is in a projective space. It's fundamental group is [tex]\mathbb{Z}_2[/tex]
    so picking your standard path a certain way you can get

    [tex]U(\bar \Lambda )U(\Lambda ) = \pm U(\bar \Lambda \Lambda )[/tex]

    Now if you use the universal cover of SO(3,1), [tex]SL(2,\mathbb{C})[/tex] you can get

    [tex]U(\bar \Lambda )U(\Lambda ) = U(\bar \Lambda \Lambda )[/tex]

    Now my question is by introducing the gradation are you basically just redefining your unitary transformations so that [tex]U(\bar \Lambda )U(\Lambda ) = U(\bar \Lambda \Lambda )[/tex] for SO(3,1) too?
  2. jcsd
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