# I Intrinsic spin symmetry of spacetime

1. Apr 8, 2017

### AlephClo

I try to understand where the intrinsic spin symmetry of spacetime (ISSS) is established. I read articles but still do not understand how to put together all the information to make a clear picture of where ISSS comes from.
Minkowski space - Lorentz group O(1,3) - Covering SO(1,3) with SL(2)c - Homomorphism of SL(2)c and SO(1,3), Lie Algebra so(1,3), construction of representations of sl(2)c and SO(1,3), Poincaré group and the classification of the representations, relationship to Noether theorem.

Any references and ⁄ or that would give me a explanations of why all these steps and what they each mean, and give me physical interpretations⁄reasons of all these mathematical structures would be more that wellcome :-)

Thank you very much, AlephClo

2. Apr 8, 2017

### Staff: Mentor

Please provide some specific references. We can't know what background knowledge you have if we don't know what references you've actually read.

3. Apr 8, 2017

4. Apr 8, 2017

### Staff: Mentor

Ok, let's try a different starting point: what do you mean by "intrinsic spin symmetry" of spacetime? Feel free to give a reference to a specific part of the paper you linked to; even though I can't understand the French, I can probably guess from the equations what it is referring to.

5. Apr 8, 2017

### AlephClo

The fermions spin +- 1⁄2, +- 3⁄2 etc; I understand is a the fourth symmetry of spacetime CPT, and it is derived⁄established from Poincaré and Lorentz groups. I want to unserstand the rational that supports that.

Thank youy for your time and patience. AlephClo

6. Apr 8, 2017

### Staff: Mentor

Do you mean CPT is the fourth symmetry? What do you think are the first three?

Also, you do realize that CPT symmetry doesn't just apply to fermions, right? It applies to all particles.

7. Apr 9, 2017

### strangerep

Let's try a slightly different route...

Angular momentum (in general) arises from spatial isotropy of the system under consideration. (IOW, invariance under 3D spatial rotations.)

Classically, there is a distinction between orbital angular momentum and intrinsic angular momentum. See Box 5.6 of Misner, Thorne & Wheeler for details. I gave a sketch of its contents long ago in this post.

Quantum mechanically, it turns out that angular momentum is quantized in half-integral steps. You don't need the full mechanics of spacetime, the Poincare group, etc, to derive this. You just need rotations of 3-dimensional space. See section 7.1 of Ballentine for a quick derivation. Basically, the half-integral quantization arises simply because we require that the group elements of SO(3) -- i.e., rotations of 3-dimensional space -- be represented as unitary operators on Hilbert space. Ballentine performs this derivation in just a few pages.

I suggest you put aside questions about CPT symmetry and the Poincare group temporarily -- until you understand the above in detail.

Last edited: Apr 9, 2017
8. Apr 9, 2017

### AlephClo

Thank you very much PeterDonis and Stangerep.
I do have MTW and I am in the process to go through it entirely, in conjuction with Schutz and Carroll. I looked at MTW and Ballentine and think I will be able to understand spin.

I went through Gravity & Light Winter School by F Schuller both lectures and tutorials. I strongly recommend that you give a look at it for further recommendations for self-learning people.

https://gravity-and-light.herokuapp.com/lectures
https://gravity-and-light.herokuapp.com/tutorials

Thank you again, AlephClo