If one follows this approach, is every state also the eigenvector of some "mathematical observable" (ie. is the condition for a mathematical observable the same as in micromass's approach, that it just has to be self-adjoint, or are there additional conditions from the dynamical group)?atyy,
Not sure whether I should get involved in this thread, but I get the feeling you're looking at all this in a cart-before-horse manner...
To construct a quantum model of a particular physical scenario, one starts with a physically-meaningful dynamical group (which maps solutions of the equations of motion into other solutions), and attempts to construct a unitary representation thereof (i.e., a representation of the group as operators on a Hilbert space, or generalization thereof). One way to do this is to find the group's Casimirs, and a maximal commuting set of generators within the dynamical algebra of group generators, determine their joint spectrum, and construct a Hilbert state space from that spectrum (verifying of course that the rest of the generators and the whole group are also satisfactorily represented on the state space so-constructed).
[Edit: In more general cases, it may be more physically-sensible to construct a POVM on the spectrum, since a POVM is closer to what real apparatus does. Cf. http://en.wikipedia.org/wiki/POVM]
So what I'm trying to say is: the dynamical group comes first, followed by the states which are (in a sense) derived from the dynamical group and the requirement of unitary representation. However, in this thread, it seems you're trying to do it the other way around, therefore encountering some difficulty with the physical interpretation.
BTW, Ballentine chapters 1-3 give a better introduction to this way of looking at things than the Preskill notes (imho).