Deducing Degeneracy in Spin from Commutation Relations

[Manchot]

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.

 

[strangerep]

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?

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.

On a somewhat unrelated note, does anyone know of any good references on rigged Hilbert spaces?

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.)

 

[Manchot]

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.

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?

(If you can be more specific, I might be able to suggest something else.)

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