I have read several articles talking about the meaning of spins and spinor spaces. They only have a mathematical, or quantum mechanical identity. Thus one hardly finds a classical (geometrical) clarification, i.e. there is no direct relationship between $SU(2)$ and geometrical rotations in Euclidean space.

I do understand the relationship between $SO(3)$ and $SU(2)$; the later being the simply connected covering group of the former. Since $so(3)$ and $su(2)$ are isomorphic and half-integer representation of $so(3)$, a vector space spanned by generators of rotations, cannot produce $SO(3)$, we are allowed to play with $SU(2)$ instead. In my thought, this is a brief compendium of the physical meaning of covering group $SU(2)$ discussed in other physics stackexchange questions (though there are a lot of additional topics such as projective Hilbert space).

However above-mentioned descriptions make me confused.

Suppose some physicists succeed in discovering the existence of spin state in an electron. Probably they can also clarify that the spin is intrinsic degree of freedom of particles and behaves as if it is an angular momentum with two possible eigenstates of $J_3$.

However since the spin angular momentum does not have any classical meaning, the physicists may cast doubt on the idea that the rotations of the spin states can be expressed by a certain representation of $SO(3)$. The situation is different from when they played with spherical harmonics. Their main problems are 1) there is no such representation and 2) the rotations of spin states cannot be related with space-time. They do not share any degree of freedom with the extrinsic (orbital) angular momentum.

Well, I don't know how to rotate the spin state of the electron in real experiments. Nevertheless, the physicists would know how to do it, know the explicit expression of $j=\frac{1}{2}$ right spinors, which are connected with $x,y,z$ up/down spins, and maybe form of Pauli matrices.

What is missing here is that they cannot know whether $SU(2)$ is a truly appropriate group of the rotations in the spinor space or not. By accident, they can find the correspondence between basis of $su(2)$ and Pauli matrices. However it does not justify the idea that a $j-$representation of $SU(2)$ represents a kind of rotation we cannot imagine explicitly, though $SU(2)$ is related with $SO(3)$. For example there is no simple reason for them to designate a name "spin-$i$" to $j=i$ representation of $SU(2)$ without any experimental detail.

Thus my questions are

1) How the physicists justified themselves that $SU(2)$ is a group of rotations of something we cannot explicitly see when they found the intrinsic angular momentum? Is this just an experimental result? Or, they have some mathematical reasoning?

2) Suppose we do not know what the spin is, but $j=0,1/2,1,3/2, \cdots$ representations of $SU(2)$. Then, can we predict the existences of the orbital and spin angular momentum? I mean, their existences are inevitable consequences of ( $SO(3)$ and $SU(2)$ ) group theory?

  • $\begingroup$ If I'm correct in thinking that this interpretation stems from the original structure of the Dirac equation, wasn't it just due to some random insight that Dirac had one night? There might not be a rigorous motivation for it; rather, somebody saw a pattern that reminded them of $SU(2)$. $\endgroup$ – probably_someone Jun 15 '17 at 10:24
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    $\begingroup$ This seems like a history question... $\endgroup$ – ZeroTheHero Jun 15 '17 at 11:28
  • $\begingroup$ While the question does refer to historical development, I think there would be pedagogical value in carefully laying out the answer (for myself as well as other similarly less-advanced students of the topic). I'm still aiming to have a go at it in the near future (earlier promised reply delayed by more urgent matters). $\endgroup$ – iSeeker Jul 1 '17 at 12:28

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