# Why does spin appear in quantum systems but not classical systems?

Note: herein "relativistic" excludes Galilean relativity unless otherwise stated.

At first even spin's appearance in quantum mechanics was mysterious. Although one could easily write down a theory consistent with the observed spin of some particles, no-one knew why, for example, the electron is spin-$$\frac{1}{2}$$. The technical term for spin is "intrinsic angular momentum", $$\vec{S}=\vec{J}-\vec{L}$$. Why were there particles for which this contribution to the angular momentum that exists without externally imposed torque is non-zero?

Eventually, Paul Dirac hit upon the answer when he formulated a first-order wave equation consistent with both special relativity and quantum theory. His hope was that a "square root" of $$E^2=p^2+m^2$$ (with $$c=1$$) would avoid negative eigenenergies. The identification $$E=i\partial_t,\,\vec{p}=-i\vec{\nabla}$$ with $$c=\hbar=1$$ implies we need a result of the form $$i\gamma^\mu\partial_\mu\psi=m\psi$$ with anticommutators $$\left\{\gamma^\mu,\,\gamma^\nu\right\}=2\eta^{\mu\nu}$$. Since spacetime is $$4$$-dimensional, there are $$4$$ of these gamma matrices. It can be shown their anticommutation relations require $$\psi$$ to have at least $$4$$ components (the numbers aren't equal in general). Dirac thereby predicted antimatter and explained the electron's spin degeneracy $$2S+1=2$$ (so the electron and positron each contribute $$2$$ components to the field), so $$S=\frac{1}{2}$$. Meanwhile, the photon is spin-$$1$$ because it's associated with a massless vector field $$A^\mu$$.

The explanations I gave above for the spins of electrons and photons are very different in both length and details, but in both cases, a relativistic formulation is necessary to provide any such explanation. Thus any classical system in which a spin analogue is found will be relativistic, which precludes a discussion in most introductions to classical mechanics. (In this context "classical" means "not quantum", so special relativity is classical, but it's a bit more advanced!) You might think an alternative explanation is possible, because of Stone's paper relating classical spin to supersymmetry. But supersymmetry, if it exists, is also a relativistic effect! (You can see that from how often this article says "Poincaré".)

More precisely, the list of symmetries for nature that are consistent with both quantum and relativistic effects is provided by the Coleman-Mandula and Haag–Łopuszański–Sohnius theorems, which show supersymmetry is the only non-trivial consistent way to combine the two types of continuous symmetry in physics, spacetime and internal symmetries. The distinction between these can be made even in a non-relativistic classical theory, whereby the spacetime symmetries are Galilean while the internal symmetries are due to "rotations" of multi-component fields. As a simple example, the Lagrangian $$\frac{1}{2}\sum_{a=1}^N\left(\dot{\phi}_a^2-\omega^2\phi_a^2\right)$$ has an $$O\left( N\right)$$ internal symmetry.

The classical model of spin is given by the spinning top

https://en.wikipedia.org/wiki/Rigid_body_dynamics#Rotation_in_three_dimensions

but because it isn't yet quantized, the usual features of quantum spin are not visible; so it isn't in the center of attention in classical physics.

Moreover, one cannot see classically the associated anticommuting representation, which is central to its quantum treatment.

Both facts together fully explain its lack of covering in the literature.