quantum mechanics - Meaning of spin

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?

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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.

Answer

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.