Why is half-integer spin not observed classically? [duplicate]

Question

It is usually stated that half-integer phenomena is purely quantum. The way in which "half-integerness" manifests itself seems very counterintuitive to me, or I simply do not understand it. For the sake of concreteness, I state two questions (i) and (ii) at the end of the post.

I assume:

1. The observed symmetry group of nature under rotations is $SO(3)$ and not $SU(2)$, since we know no experiment in which rotating the apparatus by 360° flips the sign of the result.

2. In quantum mechanics states are defined up to a phase, so in turn projective representations are allowed . We are then allowed to use a Hilbert space which furnishes any representation of $SU(2)$, the universal cover of $SO(3)$, giving all ordinary as well as intrinsically projective representations of the original symmetry group.

I do not see though, why and how $SU(2)$ is forbidden from naturally translating into the classical description of physics.

• Sure a purely quantum feature, the Hilbert space, had to be invoked in order to bring up the universal cover $SU(2)$. This seems however more like a technicality involved in the mathematical formalism. The original question of "how things rotate", whose answer lies in the representation theory of $SO(3)$, is in principle unrelated to quantum features.

• Spinor fields giving half-integer representations of $SU(2)$ can be perfectly constructed and in fact they are, for example, fundamental in the description of quantum electrodynamics. Moreover, prior to imposing (anti-)commutation relations or writing a path integral partition function, there usually exists a Lagrangian containing such fields which should in principle make perfect sense as a classical theory (and even need not be taken as Grassmanns).

So, only to avoid being too vague or abstract, I will formulate two concrete questions. Suppose we wanted to force a classical interpretation of a -say 1/2- spin field.

(i) We never directly measure such thing. Why, if our description of nature is plagued with them? Of course the Dirac field is non-hermitian, but what about a Majorana field?

(ii) For such an object, we should be able to make some measurement for which under a 360° rotation the result shifts $\psi\to-\psi$. Crearly this is against my assumption 1. Why can't we, and how does nature keeps this from happening, even though its description is full of these objects -at a quantum level at least-?

Yet, half-integer fields are essential to describe behaviour which manifests clearly in experiments such as Stern-Gerlach, or neutron interference (and relatedly if you will, exclusion principle). How is it that half-integer spin manages to manifest itself in such an elusive way and hiding away from a classical analogue, disguised in $SO(3)$?

Answer 1

The other answers are great, and it's very much true that you can totally write down a classical field theory with half-integer-spin fields. That notion still feels kind of odd, though, because all of the fields in the classical field theories that seem to arise in the world have integer spin. It's worth asking why this is the case.

In Peierls' Surprises in Theoretical Physics §1.3, he observes that in some sense, there can be no classical field theory of fermions. With occupation number per mode $N$ and phase $\gamma$, the "classical limit" is when $ \Delta N / N$ and $\Delta \gamma / \gamma$ are both small. Because of uncertainty, $ \Delta N \Delta \gamma \gtrsim 1$, but the mode occupation number can only be 1 or 0, so we can never have $\Delta N$ large enough to allow a small $\Delta\gamma$. The spin-statistics theorem tells us that in any relativistic QFT, half-integer spin means fermions and integer spin means bosons. Because the universe is relativistic (and greater than two-dimensional), this means that all of the fields deep down fall into one of these two boxes. If we try to zoom out and consider them on the classical level, we can only get enough particles in one place for that to work if the particles are bosons, and thus integer-spin.

Part of the issue is the idea that the way that we get quantum field theories is by taking classical field theories and performing a procedure that we call "quantization." This is a good enough way to introduce things and it can often give important insights or examples, but at a fundamental level, the universe is a quantum thing, and going quantum -> classical is the only direction in which that limit can be unique and well-defined. I'll add the disclaimer that this ideology is perhaps more common in condensed matter research than elsewhere (lots of people still think a lot about quantization), and I got it from McGreevy's "Where do QFTs Come From" lecture notes from this class.

Answer 2

Why is half-integer spin not observed classically?

Why do you single out half-integer spin? Are you implying that integer spin can be observed classically? If neither integer spin nor half-integer spin can be observed classically, what makes you think that half-integer spin is more "quantum" than integer spin?

Our world is fundamentally quantum. The integer spin or half-integer spin we overserve "non-classically" are quantum excitations of classical fields. For example, a photon, which is endowed with integer spin, is the quantum excitation of electromagnetic gauge field $A_\mu$. Whereas, an electron, which is endowed with half-integer spin, is the quantum excitation of spinor field $\psi$.

However, without field quantization (aka second quantization), both electromagnetic gauge field $A_\mu$ and spinor field $\psi$ are classical fields. The classical electromagnetic gauge field $A_\mu$ transforms under Lorentz group $\mathrm{SO}(1,3)$(its 3D Euclidean version is $\mathrm{SO}(3)$). On the other hand, the classical spinor field $\psi$ transforms under spin group $\mathrm{Spin}(1,3)$ (its 3D Euclidean version is $\mathrm{Spin}(3) \sim \mathrm{SU}(2)$). Note that the classical spinor field $\psi$ is related to the "generalized classical limit" in which one has classical anticommuting Grassmann variables - the same kind as one has in fermionic path integrals(see @Arnold Neumaier's explanations here).

There is NOTHING making $\mathrm{Spin}(1,3)$/$\mathrm{SU}(2)$ more "quantum" than $\mathrm{SO}(1,3)$/$\mathrm{SO}(3)$, since the symmetries of the classical spinor field $\psi$ and the classical electromagnetic gauge field $A_\mu$ are both purely-geometric non-quantum properties.

For more discussions, please see here, here, here, and here.

Answer 3

Ferromagnets are a classical manifestation of electron spin. MRI is a classical manifestation of nuclear, mostly proton, spin. Both are spin half. What you are looking for is a -1 factor upon 360$^\text{o}$ rotation.

Answer 4

Electron spin has been interpreted classically in the context of the theory of stochastic electrodynamics (classical electromagnetism + zero-point field - the field causing the Casimir force):

Cetto, Ana María & de la Peña, Luis & Valdés-Hernández, A.. (2017).

Proposed physical explanation for the electron spin and related antisymmetry.

Quantum Studies: Mathematics and Foundations. 6. 10.1007/s40509-017-0152-8.

https://arxiv.org/pdf/1707.08674

The paper reads (page 10):

"The calculations presented here confirm the physical picture of the spin of the electron as a helicoidal motion (a zitterbewegung) around the local mean trajectory, adding an effective structure to the originally pointlike particle."