There is nothing quantum about half-integer spin.
To quote Weinberg in his Lectures on QM: "there is one topic I was not sorry to skip: the relativistic wave eq. of Dirac. It seems to me that the way this is usually presented in books on QM is profoundly mistaken".
When the great mathematician Élie Cartan first introduced the concept of spinor in 1913, it's a purely classical concept of geometry. Quantum mechanics has not been fully developed yet back in 1913. The classical spinors have NOTHING to do with quantum mechanics. The rotation property of a spinor associated with the Poincare group is a purely classical spinor behavior, unlike the other quantum properties of a spinor when the spinor field is second-quantized in QED.
The spinor field $\Psi$ endowed with half-integer intrinsic spin is NOT a quantum wave function and does NOT belong to the Hilbert space $\cal H$. A good counter-example: The Majorana spinor is real and has half-integer spin. Apparently, the real Majorana spinor does NOT belong to the complex Hilbert space of quantum mechanics.
Let's take a look at the specific definition of spin:
$$
\vec{S} = \frac{\hbar}{2} \vec{\sigma}
$$
where $\sigma_i$ are Pauli matrices. Folks see the Planck constant $\hbar$ in the above definition and automatically associate spin with quantum mechanics.
But that is a false impression. Let's look at the spin rotation
$$
U(\vec{\theta}) = e^{i\vec{\theta} \cdot \vec{S}/\hbar}= e^{\frac{i}{2}\vec{\theta} \cdot \vec{\sigma}}
$$
As you can see, the $\hbar$ in the definition of spin is canceled out by the denominator $\hbar$ in spin rotation, $\hbar$ has just simply disappeared!
The Planck constant $\hbar$ has been manually put into the definition of spin due to historical reasons. It is redundant and unnecessary, since spin per se (without QFT second quantization) is purely a classical concept.
For more details, see a related answer here.
Added note:
The other answers/comments erroneously stated that:
- "Spin is a purely relativistic property"
- "Dirac equation, which is both relativistic and quantum mechanical"
- "Spinors do not exist outside of quantum mechanics since they belong to projective vector spaces"
All the above are FALSE statements, given that:
- Spin is NOT a purely relativistic property. One can surely define the Spin group $Spin(3) \sim SU(2)$ in 3-D non-relativistic Euclidean space.
- Dirac equation is relativistic but NOT quantum mechanical. If you are wondering, the spinor $\Psi$ per se is NOT a quantum wave function. There is an addition step to "second quantize" $\Psi$
to make it quantum in the context of QFT, where $\Psi$ is turn into an amalgam of quantum creation and annihilation operators. However, before taking this additional step of "second quantization", there is nothing quantum about $\Psi$.
- Pure quantum states are described by the elements of the projective space of a complex Hilbert space $\cal H$. However, as we stated in the previous bullet point, the spinor $\Psi$ is NOT a quantum wave function, hence $\Psi$ (including its spin characteristics) does NOT belong to the complex Hilbert space $\cal H$! Therefore, any attempt using "the projective space of a complex Hilbert space $\cal H$" to justify the quantumness of spin/$\Psi$ is misguided.
More Added note:
This is to answer @tparker's question: "So you’re saying that spin is dimensionless in a classical context"?
Yes, spin is dimensionless in a classical context. Let's take a look at the Pauli equation:
$$
-q\frac{\hbar\vec{\sigma} \cdot \vec{B}}{2m}\Psi=i \hbar\frac{\partial}{\partial t}\Psi
$$
where for simplicity sake we omitted space-dependence for the spinor field $\Psi$.
Pauli equation in the above format gives you the FALSE impression that the spin is of dimension $\hbar$. But do you notice that $\hbar$ appears on BOTH sides of the Pauli equation? We can simmplly divide both sides by $\hbar$, and arrive at
$$
-q\frac{\vec{\sigma} \cdot \vec{B}}{2m}\Psi=i \frac{\partial}{\partial t}\Psi
$$
Now we can clearly see that the above time evolution of the spinor field $\Psi$ is actually NOT dependent on the Planck constant $\hbar$.
For historical reasons, folks erroneously thought that the spinor field $\Psi$ is a quantum wave function and manually multiply both sides by $\hbar$ to make the Pauli equation resemble the Schrödinger equation. But in reality spinor field $\Psi$ is NOT a quantum wave function and the multiplication by $\hbar$ is unnecessary and misleading.
