It is often claimed that spin is a purely quantum property with no classical analogue. However (as was very recently pointed out to me), there is a classical analogue to spin whose action is given (in M. Stone, "Supersymmetry and the quantum mechanics of spin", Nucl. Phys. B 314 (1986), p. 560) by $$S = J\left[\int_\Gamma\vec{n}\cdot\vec{B}\,\mathrm{d}t + \int_\Omega\vec{n}\cdot\left(\frac{\partial\vec{n}}{\partial t}\wedge\frac{\partial\vec{n}}{\partial \tau}\,\mathrm{d}t\,\mathrm{d}\tau\right)\right],$$ where $\Omega$ is a region on the unit sphere bounded by the closed loop $\Gamma$. Stone goes on to derive the algebra satisfied by the quantities $J_i \equiv J\,n_i$ as the standard angular momentum algebra given by $$\{J_i,J_j\}_{PB} = \epsilon_{ijk}J_k,$$ where $\{\cdot,\cdot\}_{PB}$ denotes the Poisson bracket.

However, while this all makes a good deal of sense, and the connection to spin in quantum theories is evident through path integration over SU(2), I have to wonder: why isn't this action studied in most introductions to classical mechanics? Some authors claim it's because the dynamics generated by this action can only be understood through the use of symplectic forms, which is a rather advanced formulation of classical mechanics. But this answer isn't very satisfying to me. Instead, I have the following set of related questions:

1.) Are there even any classical systems which can be described by such an action? Alternatively, since the above action has degrees of freedom defined on the surface of the unit sphere $S^2$, are there classical systems whose dynamical degrees of freedom are simply on $S^2$?

2.) If so, what are they, and do they have direct quantum analogues (in the same way that, say, the harmonic oscillator exists in both classical and quantum mechanics, and the classical and quantum Hamiltonians are directly related to one another)?

3.) If not, why does spin appear as a dynamical variable in quantum mechanics but not in classical mechanics, despite the fact that both theories can accommodate it? If classical mechanics is simply a limiting case of quantum mechanics, what about this limiting process causes spin to disappear in classical systems?

  • 2
    $\begingroup$ I wonder if @mike stone has any comment... $\endgroup$ – Rococo Jul 13 '17 at 17:54
  • $\begingroup$ I think the key word to answer this question is entanglement. $\endgroup$ – Diracology Jul 17 '17 at 11:56

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.

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The classical model of spin is given by the spinning top


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.

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The title of the question is not quite correct. I offer a a partial answer and I hope it helps to some extent! I might expand it a bit later.

The way the angular momentum is first defined in classical mechanics is $\textbf{r}\times\textbf{p}$ or $\epsilon^{ijk}x_jp_k$. This should really be called orbital angular momentum. However, spin can arise even in classical physics as an intrinsic angular momentum which is independent of space and time.As an supplementary thought, it would not be completely out of place to look at what happens in a classical field theory.

Under an infinitesimal Lorentz Transformation (LT) $$x^{\prime\mu}=x^\mu+\omega^{\mu\nu}x_\nu,$$ the conserved charges of a classical field $\Phi$ (with components $\{\phi^a\}$ which mix under LT) is $$J^{\lambda\rho}=\int d^3x \mathcal{M}^{0\lambda\rho}\tag{1}$$ where $$\mathcal{M}^{\mu\lambda\rho}=\frac{\partial\mathscr{L}}{\partial(\partial_\mu\phi^a)}(\Sigma^{\lambda\rho})^a_{~b}\phi^b-(T^{\mu\nu}x^\rho-T^{\mu\rho}x^\lambda),\tag{2}$$ and $T^{\mu\nu}$ is the energy-momentum tensor, and $\Sigma^{\mu\nu}$ is the Spin matrix. The space-space components of $(1)$, $J^{jk}$, are related to the components of the classical angular momentum of a field by $$J^i=\frac{1}{2}\epsilon^{ijk}J_{kj}.\tag{3}$$ To demonstrate that $J^i$ are indeed the components of the classical angular momentum, it is sufficient to show that the Poisson Bracket (PB) $\{J_i,J_j\}_{\rm PB}=\epsilon_{ijk}J_k$. In doing so, the first term of $(1)$ throws a problem. Then one proceeds by introducing a similar definition as $(3)$: $$\Sigma^{i}=\frac{1}{2}\epsilon^{ijk}\Sigma_{jk}\tag{4}$$ and assumes a PB relation of the form: $$\{\Sigma_i,\Sigma_j\}_{\rm PB}=\epsilon_{ijk}\Sigma_k\tag{5}$$ to exist classically.

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