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Please note, I am only interested in classical mechanics discussion on this. Please do not involve quantum mechanics.

Inspired by this question: Is Angular Momentum truly fundamental?

My question is:

Can there be a concept of angular momentum separate from "orbital" angular momentum in classical mechanics? For example, can there be a thing such as "intrinsic" angular momentum in a classical theory that could be distinguished from the limit of shrinking a spinning ball to size zero?


It seemed obvious to me that the answer should be no. -- No such distinction could be made in a classical theory. However searching on related topics and reading more brought me to this article in wikipedia:
http://en.wikipedia.org/wiki/Einstein%E2%80%93Cartan_theory
While I don't follow the math details, the overview comments seem to claim there is a classical notion of intrinsic spin, but GR cannot handle intrinsic angular momentum. And a different classical theory is presented which does, and differs from GR. For instance, some important tensors can be non-symmetric now, which can't happen in GR no matter the size we make a spinning ball.

Therefore there seems to be a real distinction between "intrinsic" angular momentum, and the limit of shrinking a spinning ball to size zero, even in classical mechanics. This blows my mind. So if the answer to the above questions are YES!, can someone help explain what this classical "intrinsic" angular momentum is?

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Ginsberg
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    I think your issue is with the distinction between a metric formulation of GR and a tetrad formulation of GR. You can couple spinors to the later, but not to the former. – Zo the Relativist Apr 01 '11 at 00:05
  • GR does have an intrinsic notion of angular momentum at least for certain spacetimes which are "asymptotically flat" where you can define the ADM angular momentum analogously to how the ADM mass is defined. Also take a look at this classic paper by Friedman and Sorkin on Half-integral spin from quantum gravity which shows the relation between spin and the so-called "large" diffeomorphisms, i.e. those that are not continuously transformable to the identity. –  Apr 03 '11 at 17:15

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I can give some indication of what is going on here, but I have not studied all the references as yet. This topic is complicated by the fact that there are several (possibly independent) ingredients to this issue. You can follow up on some or all aspects.

General Relativity and Einstein-Cartan Theory

The basic equations of GR we can write as $G^{ab}=P^{ab}$. Here the LHS is the geometric content of the theory and the RHS is the matter (stress-energy tensor) content of a given solution. Basic properties show that both Tensors (and several others) are symmetric. The geometric solutions are described by a metric which generates curvature. Cartan discovered later that this was not the most general form of such a geometry. Essentially the geometry could contain non-symmetric Tensors too, say $T_{a}^{bc}$ as the Torsion tensor. Thus a manifold could have two independent types of non-Euclideanness: Curvature and Torsion. The curvature in GR is well known, but the Torsion is set to zero.

It is possible to have a manifold which has zero curvature (so all parallel lines stay parallel), but has non-zero Torsion. So how does this manifest itself in the manifold? Well basically given say a unit vector (eg pencil with a point) moving it along a curve will result in the pencil having rotated from its original direction. Parallel lines stay parallel, but the space has a sort of built-in helix structure causing an intrinsic - or geometric - rotation.

Interesting mathematical idea, but does it relate to anything physical? Einstein became excited by the idea for a time, developing Teleparallel gravity - lots of torsion but no curvature (hence the Parallel aspect).

This theory was developed further in the 1960s by coupling (ie equating) the Torsion to the "Spin Tensor" $S_{a}^{bc}$. This added a second equation to the usual Einstein Tensor equation, generating EC (KS) Theory. The spin tensor is classically defined as a generalisation of angular momentum to tensors. So in the EC theory the angular momentum of an object also has gravitational effect (calculations show that it would only be significant in high spin neutron stars or maybe early universe.)

Spin Density

A simple-seeming generalisation of the idea of density from mass density ie $\rho(x)$ to linear momentum density to spin density $s(x)$ is associated with the name of Wessenhoff in 1947 who was trying to develop a "classical" Dirac theory. Here a spin value is associated with each point in space - at least in the mathematics. Later there is discussion of a "Wessenhoff fluid" - which is a strange quasi-classical fluid with vortices at all length scales. Since then the phrase "spin density" has become used in some parts of the literature without explicitly including quantum effects, but corresponds to a use of this mathematical idea.

Some authors seem to map the idea directly to quantum spin without considering such quasi-classical models - so the ideas can do double-duty.

Geometric Algebra

David Hestenes has expressed all the above concepts in Geometric Algebra (known as STA) which also allows discussion of spin density in this sense.

A paper on Spinning (Wessenhoff-like) fluids in Cosmology (under EC theory): http://deepblue.lib.umich.edu/bitstream/2027.42/49195/2/cq940917.pdf

A paper by David Hestenes: http://geocalc.clas.asu.edu/pdf-preAdobe8/Decouple.pdf

Intrinsic Angular Momentum

Also one should make a more elementary comment about the possible use of this phrase in Classical Mechanics. A given system of particles will have an angular momentum M' in a given frame K' in which it is otherwise at rest. From the perspective of a different frame K there will be a translation formula:

M = M' + R x P

where R is the radius to centre of mass in K and P is the momentum in K. Thus from the perspective of K the angular momentum consists of the "intrinsic angular momentum" M' plus the angular momentum of the whole system. [Landau/Lifshitz Vol1 (9.6) actually use the expression "intrinsic angular momentum".]

Roy Simpson
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The classical Dirac and EM fields have intrinsic angular momentum. For the Dirac field, for example (see $\S$2-1-3 in Itzykson&Zuber, Quantum Field Theory, McGraw-Hill), there is a component $-i\sigma_{\mu\nu}\omega^{\mu\nu}/4$ of the infinitesimal generators of Lorentz transformations as well as the orbital component $x_\mu\partial_\nu\omega^{\mu\nu}$. That is to say, the Dirac equation is invariant under infinitesimal transformations $$\psi'(x)=\left(I-\frac{i}{4}\sigma_{\mu\nu}\omega^{\mu\nu}+x_\mu\partial_\nu\omega^{\mu\nu}\right)\psi(x).$$

Peter Morgan
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  • Yes, but how does one alleviate the discomfort we feel at the thought of "classical" Dirac fields? E&M fields you can think of as classical waves. How do you do so for spinors? –  Apr 04 '11 at 14:45
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    @Deepak Properly a Question! If we want to do such a thing, we can map the spinors to quadratic objects that satisfy a constrained system of DEs. The 2-1 ambiguity of the spinor at each point can be fixed by continuity up to a global 2-1 ambiguity that is not observable. eg: the Dirac spinor may be mapped to an orthogonal tetrad ($\bar\psi\gamma^\mu\psi$, $\Re[\bar\psi\gamma^\mu\psi^c]$, $\Im[\bar\psi\gamma^\mu\psi^c]$, $\bar\psi\gamma^5\gamma^\mu\psi$). This tetrad together with the complex phase generated by $i\gamma^5$ satisfy a system of constrained DEs. Too rushed? – Peter Morgan Apr 04 '11 at 15:18
  • So "classical" Dirac spinors == classical gravity (in connection formalism)? –  Apr 04 '11 at 16:41
  • @Deepak, I went with spinors mostly because I found a clear citation for the detailed algebraic construction of the Lorentz group generators that leave the Dirac equation invariant easily, whereas I didn't instantly find a clear citation for the same construction for the electromagnetic field. I don't want to get into GR here, except perhaps to note the obvious, that the Dirac spinor has only 8 degrees of freedom, far fewer than the 24 degrees of freedom of the torsion or contorsion tensor. – Peter Morgan Apr 04 '11 at 17:08
  • Ok cool. These are issues that people like to discreetly avoid. I do have something precise in mind when I say "classical gravity". More on that if you are interested. It is not so much whether we can construct a classical model but whether there is some nice visual representation - like the Dirac "belt trick". Something you can "see"! –  Apr 04 '11 at 17:14