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I am only focusing on non-relativistic quantum mechanics in this post. My current understanding is that a particle of spin $\ell$ can be described as an element of the tensor product $L^2(\mathbb{R^3}) \otimes V$ where $V$ is a vector space of dimension $2\ell + 1$ carrying a representation of the Lie group $SO(3)$.

I am trying to better understand this from a physical point of view. For example, the Stern-Gerlach experiment tells us that an electron has two internal degrees of freedom (up and down). My question is how can one start from the observations of this experiment (that an electron has two internal degrees of freedom, spin up and spin down) and relate them to SO(3) and angular momentum? What lead physicists to believe that the number of internal degrees of freedom a particle has is related to the dimension of $V$, and hence the dimension of the $SO(3)$ representation it carries, and also angular momentum?

Also as an aside question, since only when $\ell$ is an integer can the Lie algbera $\mathfrak{so}(3)$ can be exponentiated to give representations of $SO(3)$, would it be more accurate to say $V$ carries a representation of $\mathfrak{so}(3)$? Thus when exponentiated $V$ now carries a representation of either $SO(3)$ or $SU(2)$ depending on whether $\ell$ is an integer or a half integer?

Qmechanic
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CBBAM
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    As you start to address in your final paragraph, it is precise to say that $V$ carries a representation of $\mathfrak{su}(2) \cong \mathfrak{so}(3)$. This Lie algebra representation induces a Lie group representation of $SU(2)$ over $V$. Hence, $V$ carries a representation of both $\mathfrak{su}(2) \cong \mathfrak{so}(3)$ and $SU(2)$. – Silly Goose Feb 14 '24 at 05:56
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    Moreover, I would like to emphasize that we consider projective representations of $SO(3)$ over $V$, which by a chain of theorems, is equivalent to looking for representations of $SU(2)$ over $V$. This is where the irreducible representations associated with half-integer spin come from in the first place. – Silly Goose Feb 14 '24 at 05:59
  • @SillyGoose Think about expanding this into an answer. – Tobias Fünke Feb 14 '24 at 08:08
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    Related: This and this. – Tobias Fünke Feb 14 '24 at 08:10
  • @SillyGoose Thank you, that helped clear up some of the theory. On the physical side, what motivated physicists to think of angular momentum when observing that the electron has two degrees of freedom in the Stern-Gerlach experiment. – CBBAM Feb 14 '24 at 16:22
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    @CBBAM because it couples to a magnetic field, which you'd expect of a magnetic dipole ~ angular momentum. And the two states were like clockwise vs. counterclockwise rotating currents. But physicists were actually skeptical that intrinsic spin was a real thing until Einstein (I think) showed it followed from representations of the Lorentz group SO(3,1)$_+$, the symmetries of special relativity. This also predicts antimatter. – just a phase Mar 29 '24 at 19:20
  • @justaphase Thank you! So because the electron coupled to a magnetic field physicists thought of angular momentum and that naturally led to looking at representations of $SO(3)$ with two degrees of freedom? – CBBAM Mar 29 '24 at 19:28
  • I'm not sure about the precise historical timeline, but I know they figured out that the generators of SU(2) and SO(3) are the same (i.e., the Lie algebras are the same). The Lie group SU(2) is a projective representation of the Lie group SO(3), in that for $u,v \in$SU(2), there are corresponding elements $\Omega(u),\Omega(v)\in$SO(3) such that $\Omega(u) \Omega(v) = e^{i \theta (u,v)} \Omega(uv)$, i.e., it reproduces the structure of SO(3) up to a complex phase. Really though, we have SO(3,1)$_+ \sim$ SU(2)$\times$SU(2), where one SU(2) is spin and the other is matter/antimatter. – just a phase Mar 29 '24 at 19:46
  • There's also the spin-statistics theorem that tells us that fermions have to have half-odd integer spin and bosons have integer spin representations. The integer spin representations are also (regular) representations of SO(3), but the half-odd integer ones are only projective representations of SO(3). But fundamentally, they are legitimately SU(2). But intuitively, that's why intrinsic spin behaves very much like angular momentum in experiments. – just a phase Mar 29 '24 at 19:48
  • @justaphase I see, so if I have understood correctly they both behave differently because they're both related to SU(2)? – CBBAM Mar 29 '24 at 20:01
  • SO(3) and SU(2) are closely related to each other, so they have similar physics. The Lorentz group SO(3,1)$_+$ includes rotations of 3d space, and intrinsic spin with group SU(2) comes from that. So both SU(2) spins and SO(3) angular momentum come from rotations. Their Lie algebras are the same, and the Lie groups are almost the same. They have very similar physics, which is why the interpretation of Stern Gerlach as involving angular momentum makes sense. But they are slightly different mathematical groups. Does that answer your question? – just a phase Mar 29 '24 at 20:35
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    @justaphase Yes it does, thank you very much for all your help! If you would like to post it all as an answer I will accept it. – CBBAM Mar 29 '24 at 20:57

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Historically, the Stern-Gerlach experiment showed the quantization of some internal degree of freedom, as you say. In particular, we expect a Hamiltonian term of the form $\vec{\mu} \cdot \vec{B}$, where $\vec{B}$ is the applied magnetic field and $\vec{\mu}$ is the magnetic dipole moment of the particle in the field. The experiment revealed that the component of $\vec{\mu}$ parallel to $\vec{B}$ is quantized with values $\pm \mu$.

I'm not totally sure about the precise timeline of what I've written below, but the ideas are correct.

The connection to angular momentum is that we associate these two quantized values with a current that rotates clockwise or counterclockwise about the $z$ axis. Basically, a nonzero magnetic dipole moment is associated with nonzero angular momentum of some charged particle(s). At this point, one might suspect that the electrons and nuclei of silver atoms had some net angular momentum that, when multiplied by charge, was equal to some nonzero magnetic dipole moment, which the Stern-Gerlach experiment revealed to be quantized.

But there was still some confusion. We would later realize that electrons have magnetic dipole moments even though they are pointlike --- without a notion of "size" there isn't really any rotating current, per se. Moreover, when we worked out the solution to the hydrogen atom, we found that angular momentum --- the conserved quantity corresponding to rotations of 3d space --- is quantized to have an odd number of allowed states, rather than the two possibilities observed in Stern Gerlach.

In general, symmetries correspond to mathematical groups, and continuous symmetries correspond to Lie groups. The elements of any group can be represented as a matrix acting on some vector space (e.g., a Hilbert space), which gives some intuition for why quantum states are vectors and observables are matrices (or operators). Additionally, the elements of a Lie group $G$ are generated via exponentiation of the generators of the corresponding Lie algebra $\mathfrak{g}$. The elements of the Lie group $\mathsf{SO}(3)$ corresponding to rotations of 3d space can be written as $\exp \left( i \vec{\alpha} \cdot \vec{J} \right)$, where the objects $J_k$ generate the corresponding Lie algebra $\mathfrak{so}(3)$, and satisfy the relation $[J_i,J_j] = \epsilon_{ijk} J_k$. Other Lie algebras have different relations.

There is also the Lie group $\mathsf{SU}(2)$ of $2 \times 2$ complex unitary matrices with determinant one. Interestingly, its Lie algebra $\mathfrak{su}(2)$ is equivalent to the Lie algebra $\mathfrak{so}(3)$! In other words, the generators satisfy the same rules. However, the Lie groups are not equivalent. Instead, $\mathsf{SU}(2)$ forms a projective representation of $\mathsf{SO}(3)$. A standard representation of a group $G$ is a map $\Omega$ on the elements of $G$ such that, for any $u$ and $v$ in $G$, $$\Omega(u) \cdot \Omega(v) = \Omega (u \cdot v),$$ meaning that $\Omega$ preserves the structure of the group $G$. Usually $\Omega(u)$ is a matrix. On the other hand, a projective representation satisfies $$\Omega(u) \cdot \Omega(v) = e^{i \theta (u,v)} \, \Omega (u \cdot v),$$ meaning that it reproduces the group action up to complex phases $e^{i \theta} \in \mathsf{U}(1)$.

Representations of the Lie group $\mathsf{SU}(2)$ are generically projective representations of $\mathsf{SO}(3)$. We label representations of both groups according to their "spin," which is a math term that doesn't mean anything actually spins, per se. Anyway, these representations of different spin can be understood in terms of the Lie algebras, which for these two groups are equivalent. While $\mathsf{SU}(2)$ admits integer and half-integer spin representations, $\mathsf{SO}(3)$ only admits integer spin representations. So the half-odd-integer-spin representations of $\mathsf{SU}(2)$ are projective representations of $\mathsf{SO}(3)$, while integer-spin representation are equivalent for both Lie groups.

So intrinsic spin --- e.g., as related to the magnetic moment in the Stern-Gerlach experiment --- has some intimate mathematical connection to angular momentum, as it comes from (possibly projective) representations of the rotation group for which angular momentum is a generator.

However, physicists were initially skeptical of intrinsic spin, and were not sure of its existence. This changed when Einstein (I'm pretty sure) showed that representations of the Lie group $\mathsf{SO}(3,1)_+$ were equivalent to representations of $\mathsf{SU}(2) \times \mathsf{SU}(2)$. The group $\mathsf{SO}(3,1)_+$ is known as the "orthochronus Lorentz group," which you should think of as the symmetries of special relativity. The group corresponds to rotations of 3+1d spacetime that preserve the direction of time; it includes rotations of 3d space about the three axes and Lorentz boosts of 4d spacetime along the three axes. One of the two $\mathsf{SU}(2)$s above corresponds to the rotational part of $\mathsf{SO}(3,1)_+$, and the other corresponds to the relativistic-boost part of $\mathsf{SO}(3,1)_+$. We associate the former with intrinsic spin and the latter with matter versus antimatter! This is also where the Dirac equation comes from with its four-component spinors.

So intrinsic $\mathsf{SU}(2)$ spin really does come from rotational invariance / angular momentum --- it's just that it comes from the relativistic version thereof that accounts for boosts too. Importantly, the spin-statistics theorem also requires that fermions have half-odd-integer spin, corresponding to projective representations of $\mathsf{SO}(3)$, while bosons have integer spin, corresponding to ordinary representations of $\mathsf{SO}(3)$.

just a phase
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