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Spin degeneracy

  1. Nov 2, 2007 #1
    In reviewing the derivation of the quantization of angular momentum-like operators from their commutation relations, I noticed that there is nothing a priori from which you can deduce the degeneracy of the eigenstates. While this is not a problem for angular momentum, in which other constraints come into play, it seems to me that it might be a problem for spin. Is it possible to deduce the dimensionality from the spin commutation relations alone? Or must one postulate it?

    For example, I know from the commutation relations that for an electron, any non-trivial eigenstates of Sz have eigenvalues +1/2 or -1/2. But how do I know that there are only two degrees of freedom? That is, how do I know that there's no degeneracy in the m eigenvalues?

    On a somewhat unrelated note, does anyone know of any good references on rigged Hilbert spaces? Thanks.
  2. jcsd
  3. Nov 3, 2007 #2


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    This falls out directly from the Wigner theory of unitary irreducible representations of the
    Poincare group. Weinberg vol-I covers this sort of thing is considerable detail.

    Depends on exactly what you want. Arno Bohm and colleagues have written heaps of
    stuff on many aspects of that. Do a google search to find their website - I vaguely
    remember it's in the University of Texas at Austin. Also google for "Gamow vectors"
    which are related to rigged Hilbert spaces.

    (If you can be more specific, I might be able to suggest something else.)
  4. Nov 3, 2007 #3
    I'm at the lower graduate level, and haven't taken any field theory yet. So, am I to understand that finding the dimensionality is impossible from mechanics alone? From a pedagogical perspective, would I just be better off taking it as an axiom?

    I'm looking mostly for introductory material, but I'll start with what you gave me. Thanks.
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