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I'm pretty astounded that I did not hear about this sooner, but in my course on QFT our professor told us that the concept of spin can be used to mean three things:

  1. Mechanical spin (apparently a relativistic effect giving rise to classical spin-orbit coupling)

  2. Magnetic spin (purely quantum mechanical)

  3. Classification of representations of the Lorentz group (the manner in which the particles transforms under Lorentz-transformations)

I take it that these meanings don't generally coincide since they don't seem to do so in the case of the photon: we describe this as a spin-1 particle (3rd meaning), though it has no intrinsic magnetic moment (2nd meaning).

However, despite being swiftly explained in a few words in class, I cannot remember what exactly the first meaning is about. Furthermore I'd like to find out the exact relations between these three meanings. As the case of the photon showed the last two at least don't generally seem to coincide. Can anyone clarify?


Related questions are:

  1. What is spin as it relates to subatomic particles?

  2. What are some useful ways to imagine the concept of spin as it relates to subatomic particles?

  3. Is Angular Momentum truly fundamental?

  4. What does spin 0 mean exactly?

  5. Spin of a particle and spin quantum number

  6. Why do many people say vector fields describe spin-1 particle but omit the spin-0 part?

but the answers there don't make the light go on in my head.

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2 Answers

up vote 4 down vote accepted

The three meanings of spin are in the Quantum world equivalent. What the professor means (I guess) is the following:

  1. The mechanical spin is the proper angular momentum as you are used to from the courses of classical mechanics. If one tries to interpret the spin of the electron in this way, they usually interpret the electon as a spinning particle which gives rise to a magnetic dipole, what brings us to the second meaning of spin.
  2. The magnetic spin is purely quantum mechanical and is more or less "postulated" in non-relativistic quantum mechanics based on the way the particle interacts with an external magnetic field. For example the spin-1/2 atoms in the Stern-Gerlach experiment wil split up in two possible states due to the spin-effects, they can either have spin up or down which gives rise to a splitting. This is also demonstrated in the first chapter of "Modern Quantum Mechanics" by Sakurai and Napolitano.
  3. Spin as a classification of representations of the Lorentz group is the only one true meaning of spin for as far as I know. To obtain this one should go to field theory (not even the quantum mechanical version !) and apply Noether's theorem to a general Lorentz transformation, this is done for example in chapter 2 of "Field Quantization" by Greiner and Reinhardt. Upon applying Noethers theorem to the Lorentz-transformation we get a conserved 2-tensor $M_{\mu\nu}$. If we constrict ourselves to the spatial components, it seems that this tensor splits up in two different contributions $M_{nl} = L_{nl}+S_{nl}$. The tensor $L_{nl}$ has the form of a cross-product of the postion and momentum, this is the angular momentum that we know from classical mechanics! The tensor $S_{nl}$ depends on the internal properties of the particles and is called the spin of the particle. As we can see, spin is purely a consequence of Lorentz invariance.

These three interpretations may seem different but are all three equivalent I believe. The mechanical spin is a way of giving a classical interpretation. While the magnetic and representational spin are both the same (so it seems from the quantum-field theories). Beware, altough the three representations are equivalent the three shouldn't necessarily coexist! This is shown by the spin of the photon.

For the case of the photon for example:

The photon has two polarisations, if we take the two linear polarisations and combine them we can get two different circular polarisations. This circular polarisation gives the mechanical spin.

The magnetic polarisation for a photon doesn't exist. This is because of the fact that photons don't interact and hence don't couple to the electromagnetic field.

Upon interactions the photon is able to interchange its momentum, upon interacting with other particles this way it has a spin-1 interpretation.

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After reviewing my own notes and the course material, I came to a similar understanding. Some differences: it seems the two terms contributing to $M_{nl}$ are relatable to an orbital angular momentum ($L_{nl}$) and to an intrinsic angular momentum-like property, spin ($S_{nl}$). This spin property seems to have been dubbed a "mechanical spin" by the professor because it follows from a purely classical (relativistic) calculation. (will continue in next comment) –  Wouter Jun 22 '13 at 17:05
    
(continued) As I interpret it, the magnetic moment of the particle is then a simple consequence of its intrinsic angular momentum (the mechanical spin from before) through $\vec{\mu_s} = \frac{\mu_s}{s}\vec{s}$. From this I conclude that the magnetic spin is indeed the same, so 1) and 2) coincide. Finally, the third meaning (classification of representations of the Lorentz group) connects to the "mechanical spin" through Noether's theorem as you've written. And as both you and Trimok already stated, this is the more fundamental notion of spin, since it also connects to Dirac's spinors. –  Wouter Jun 22 '13 at 17:21
    
So 1), 2) and 3) are indeed all equivalent. Applied to the case of the photon, I conclude that the mechanical spin does coincide with the classification spin. The magnetic spin (or rather the magnetic moment) is then zero because the photon is not charged. Do I make sense in all the above? –  Wouter Jun 22 '13 at 17:27
    
I believe that the magnetic moment of the photon is 1 (has spin 1) where sz = -1 or +1 depending on the chirality. The rest of your statement seems allright. –  Nick Jun 22 '13 at 18:00
    
How do you figure? Surely photons do not split into two beams in a Stern-Gerlach experiment. I agree the spin is 1, but I'd say the magnetic moment is zero because (as far as I know) it has a vanishing gyromagnetic ratio $\gamma_{\text{photon}}=0$. –  Wouter Jun 22 '13 at 18:28
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Representations of the Lorentz group give you the correct definition for spin.

Spin magnetic moment is only meaningful for charged particles.

Spin-Orbit interaction coupling is due to "electromagnetic interaction between the electron's spin and the magnetic field generated by the electron's orbit around the nucleus".

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