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Representations of SO(3)

  1. Jan 28, 2008 #1
    Hi all, I asked this on the Quantum Physics board but didn't get a response.

    I'm reading Cahn's book on semi-simple lie algebras and their representations.

    http://www-physics.lbl.gov/~rncahn/book.html

    In chapter 1, he attempts to build a (2j+1)-dimensional representation [itex]T[/itex] of the Lie algebra of SO(3) starting with the abstract commutation relations

    [itex][T_z,T_+] = T_+, \quad [T_z,T_-] = - T_-,\quad [T_+,T_-] = 2T_z[/itex] Eq (I.14).

    He begins by defining the action of [itex]T_z,T_+[/itex] on the vector [itex]v_j[/itex] by

    [itex]T_z v_j = j v_j, \quad T_+ v_j = 0[/itex]

    but he does not explain what the [itex]v_j[/itex]'s are. How does one even know that such vectors exist?

    Any help would be greatly appreciated.
     
  2. jcsd
  3. Jan 28, 2008 #2

    George Jones

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    [EDIT]I've decided to answer on the quantum physics forum[/EDIT]
     
    Last edited: Jan 28, 2008
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