Quantum Spin: Exploring the Mysterious Angular Momentum

[MadRocketSci2]

Question about quantum spin:

So for the Schrodinger equation, the angular momentum of a wave-packet, whether it is free or confined to a potential well, appears to be calculated in exactly the manner you would expect it to be: L(x) = m*rxv[Psi(x)]. Aside from quantization (decomposing the wavefunction into weighted sums of some discreet orthoganal basis of other wavefunctions), there doesn't appear to be anything in orbital magnetic moment, orbital angular momentum, that is mysterious.

But when it comes to the angular momentum of particles, we are told that this is necessarily non-classical, and cannot be described in terms of the actual rotation (or motion in some more exotic double-covering space) of a distributed object. I don't really understand this.

There is an objection raised in Griffiths, that if you use a model of the electron where the electrical self-energy corresponded to the rest mass (and got a radius from that) that the rotation rate of the object's equator would have to exceed the speed of light to explain the angular momentum. This objection seems fishy - wouldn't the angular momentum of an object approaching a rotation rate of c at the equator be going to infinity due to relativistic mass-increase? Wouldn't the current density at the equator also be going to infinity from lorenz contraction of the charge density? (Might have to use general relativity, as the frame would be highly non-inertial).

In any case, it seems counter to a general trend in physics to require that particle angular momentum and magnetic moment needs to be some irreducible thing that is different from all other angular momenta and magnetic moments. Sure, you can produce a physical theory where a point particle has these things and proceed to derive the conservation law as "angular momentum (as we know it) + this other thing that is 'fundamental' ", but we never did that for energy, nor did we do that (except as expedient or due to a lack of knowledge) for any other aspect of matter. Gas kinetic theory revealed that the "internal energy" of the gas pushing a piston and yielding mechanical energy was ... mechanical energy of the gas particles. No one decided that the charge or mass or angular momentum of something like the nucleus was 'fundamental' and that the particles being emitted from it were unrelated to any internal structure. Even the rest mass of particles isn't required to be fundamental according to thing I have heard about the Higgs boson explaining rest mass.

So why is something that looks like the moments of a distributed object in every particular (except for a factor of 2 on the gyromagnetic ratio of spin-1/2 particles) only dealt with in terms of obnoxiously abstract group-theoretic arguments?

(I'll certainly be trying to decrypt these arguments as I try to figure this out, but what is the motivation? Why these arbitrarily chosen (except for group theory properties) 2x2 complex matrices instead of something like a 3 or 4-space spinor that maps to something semiclassical?)

(And if the electron is *not* a distributed object, why would you try to explain the rest mass in terms of electrostatic self-energy?)

(Why in general is there a "no internal structure beyond this point" attitude also about the fundamental particles? Why are they not regarded as provisionally fundamental-until we can break these too? It would seem the trend in physics would have lead someone to expect hidden (if not readily accessible) internal structure all the way down!)

 

[WannabeNewton]

(I'll certainly be trying to decrypt these arguments as I try to figure this out, but what is the motivation? Why these arbitrarily chosen (except for group theory properties) 2x2 complex matrices instead of something like a 3 or 4-space spinor that maps to something semiclassical?)

These are all nebulous "why" questions as far as physics is concerned. Nevertheless, there is no obnoxiousness involved and certainly nothing is arbitrarily chosen from a physical standpoint. The $j = \frac{1}{2}$ representation of $SU(2)$ naturally yields spinors and Pauli matrices. Indeed the Pauli matrices (which is what I assume you're referring to when you say "2x2 complex matrices") are simply the matrix representations of the angular momentum operators in the $j = \frac{1}{2}$ representation that act on spinors, which are themselves just objects living in the representation space $\mathbb{C}^2$. True the spin angular momentum operator has no classical counterpart but why does there necessarily need to be a classical counterpart? There is no requirement in QM that says every observable has to have a classical counterpart, right? Moreover, there are many things in special and general relativity involving rotation that have no Newtonian counterparts. Why should we expect any differently from QM? Keep in mind the $j = \frac{1}{2}$ representation is just as natural as the $j = 0$ or $j = 1$ representations-all of them come from the eigenvalue spectra of the angular momentum operators.

If you want a deeper reason as to why the spinorial representation exists at all then you have to look to relativistic QM and Lorentz covariance because the left-handed and right-handed Weyl spinors are the most fundamental objects that transform under representations of the Lorentz group. All other objects transforming under representations of the Lorentz group can be built up from spinors. See here: http://www.physics.ox.ac.uk/users/iontrap/ams/teaching/rel_C_spinors.pdf

 

[MadRocketSci2]

What do you mean by this?:

relativistic QM and Lorentz covariance because the left-handed and right-handed Weyl spinors are the most fundamental objects that transform under representations of the Lorentz group

I thought you could come up with any arbitrary geometric objects you wanted to, and make them "invariant" under a transform (any transform) if you scale the parts proportional to geometry when you change the geometry.

Perhaps instead of spinor, I should say - why not something like a 3D or 4D antisymmetric two-tensor (something representing a rotation or angular rate)? Why are you (and Griffiths) only describing the angular momentum state two dimensionally?

 

[WannabeNewton]

I thought you could come up with any arbitrary geometric objects you wanted to, and make them "invariant" under a transform (any transform) if you scale the parts proportional to geometry when you change the geometry.

Can you give an example of a "geometric object" that transforms under an appropriate (finite dimensional) representation of the Lorentz group such that the "geometric object" cannot be built up from Weyl spinors?

Perhaps instead of spinor, I should say - why not something like a 3D or 4D antisymmetric two-tensor (something representing a rotation or angular rate)? Why are you (and Griffiths) only describing the angular momentum state two dimensionally?

Only the $j = \frac{1}{2}$ states span a 2D (complex) vector space. The $j = 0$ state spans a 1D (real) space, the $j = 1$ states span a 3D (real) space and so on. There are countably infinitely many of these in positive half-integer steps from the eigenvalue spectra of $J_z$ and $J^2$ and the associated complete set of simultaneous eigenvectors of these operators. The $j = \frac{1}{2}$ states span $\mathbb{C}^2$ simply because the $j = \frac{1}{2}$ matrix representations of the angular momentum operators are 2x2 complex matrices (the Pauli matrices)-that's just a natural consequence of the mathematical formalism of QM and the Lie algebra $[J_i,J_j] = i\epsilon_{ijk}J_k$ of the angular momentum generators. Furthermore the spinors in $\mathbb{C}^2$ are different from the angular momentum operators whose matrix representations act on the spinors. We're not replacing one with the other, they are both necessary objects.

With regards to the notion of the orbital and spin angular momentum generators being codified by an antisymmetric object with two Lorentz indices, at least in QFT, see p.3 of the following: http://www.physics.indiana.edu/~dermisek/QFT_08/qft-I-14-4p.pdf