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The spin-one quantum states belong to $\mathbb{C}^3$, with a set of orthonormal bases taken as the spin component along a direction, which would correspond to the direction of the magnetic fild gradient in a Stern-Gerlach apparatus. So we have a sample space (observable) with clear physics for each direction; but any $SU(3)$ transformation turns a sample space into another, leaving plenty of room for other observables - check the Gell-Mann matrices.

The question is: if a massive elementary particle ($W^\pm$, $Z^0$) is described intrinsically (by itself) with $\mathbb{C}^3$, what do the rest of sample spaces, the "observables" not identified with a spin projection, stand for? I consider the case of nonzero mass just to put aside the issues of the speed of light limit, but massless gauge fields should be understood as well.

I think this problem is connected to the fact that all known massive elementary particles are spin-half - with $SU(2)$ representing rotations, all sample spaces are spin projections - or zero, except the electroweak massive bosons, being these formulated as massless gauge fields.

An intrinsic quantum state space - with all sample spaces representing intrinsic properties - cannot be assigned to an elementary massive particle with spin greater than $\tfrac{1}{2}$, such kind of particles do not make sense by themselves; one has to consider at least a decay or "vertex" event, add more elements to the scenario.

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Given the standard basis for the Gell-Mann matrices, the spin 1 representation of $SU(2)$ (given explicitly e.g. in this question) is unitarily equivalent to $$J_x=\frac{1}{\sqrt{2}}(\lambda_4+\lambda_6),\qquad J_y=\frac{1}{\sqrt{2}}(\lambda_5-\lambda_7),\qquad J_z=\lambda_3.$$ You can certainly define expectation values of any of the other 5 independent $SU(3)$ generators too, and can put your spin state in eigenstate of one of them if you want, but you won't be able to find any extra observable that commutes with all of $J_x, J_y, J_z$. This follows from Schur's lemma, and will generalize to higher spin too.

Note that this is different from other $SU(2)$ subgroups of $SU(3)$. We could also define the $SU(2)$ subgroup $$S_x=\lambda_1,\qquad S_y=\lambda_2,\qquad S_z=\lambda_3.$$ This does have an observable $\lambda_8$ that commutes with it, but it is not an irreducible representation, and the $\lambda_8$ observable can be thought of as telling you which irreducible representation a vector falls in.

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  • $\begingroup$ The question is how to interpret any state in the Hilbert space; in C2 any state corresponds to a direction for the spin-half component, which does not hold for higher spin. $\endgroup$
    – Daniel
    Jun 6, 2021 at 2:15
  • $\begingroup$ @Daniel, Can you 'interpret' a state in the Hilbert space for wave functions on a real line in a way analogous to that? $\endgroup$
    – octonion
    Jun 6, 2021 at 2:25
  • $\begingroup$ @Daniel, it sounds like what is bothering you is that there are more spin states for spin $1$ than just the spin up states for a given direction. That is certainly true, but it is also true that eigenstates of say the $J_z$ operator span the Hilbert space. I really don't understand what you expect from the more philosophical part of your question. I was pointing out something about the relation between the 'extra' Gell-Mann observables and the angular momentum obervables. $\endgroup$
    – octonion
    Jun 6, 2021 at 2:40
  • $\begingroup$ Well, don't you think that the lack of a thought experiment to prepare certain theoretically assumed quantum states is a problem? $\endgroup$
    – Daniel
    Jun 6, 2021 at 3:29
  • $\begingroup$ @Daniel, Yes I see your point now. However I don't think this is any different in principle from other instances of coherent superposition in quantum mechanics. Consider a harmonic oscillator Hamiltonian. You can construct an arbitrary wave function as a superposition of eigenstates, and also associate an operator to projection on that arbitrary wavefunction. This seems a close analogy to what you don't like about the spin 1 Hilbert space. Every state can be written as a superposition of eigenstates of the 'good' operators we understand, but not every operator we can construct is 'good.' $\endgroup$
    – octonion
    Jun 6, 2021 at 4:08

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