Prove Spin of a massless particle $S_z=\pm1$

Quote from Introduction to High Energy Physics Edition 4 by Donald H. Perkins chapter 3.3.1 "It can be proved as a consequence of relativistic invariance that for any massless particle of spin $s$, there are only two possible spin substates, $s_1=\pm 1$, where $z$ is the direction of motion."

My questions were that:

1. How to prove the above statement?(with mathematics)

2. What happened to fermion.

3. Graviton was also massless, but had spin $2$. Does that mean gravition was not follow relativistic invariance?

• @John Rennie : I am not sure this is a complete duplicate of the other question, as the other question is only about photons, and this one is about an arbitrary spin $s$. I agree though that some answers to the other question cover the case of arbitrary $s$ and thus subquestion 1., but they do not answer subquestions 2. and 3., which are based on an incorrect quote (please see my answer). – akhmeteli Jul 24 '18 at 5:30
• @akhmeteli Diracology and my answers specifically mention gravitons – John Rennie Jul 24 '18 at 5:35
• @JohnRennie : I agree, and I wrote that "some answers to the other question cover the case of arbitrary $s$ and thus subquestion 1." However, the answers there cannot explain why the projections of spin for graviton are $\pm 1$ (as follows from the quote in this question) for the simple reason that this is not correct (the projections of spin for graviton are $\pm 2$). – akhmeteli Jul 24 '18 at 5:45
• Well, OK, I'll reopen it but I still think it's a duplicate – John Rennie Jul 24 '18 at 6:09
• Possible duplicate of Why is the $S_z=0$ state forbidden for photons? – John Rennie Jul 24 '18 at 6:10

1 Answer

It looks like the actual quote (https://books.google.com/books?id=e63cNigcmOUC&printsec=frontcover#v=onepage&q&f=false) is different from that in your question: "It can be proved as a consequence of relativistic invariance that for any massless particle of spin $s$, there are only two possible spin substates, $s_z=±s$, where $z$ is the direction of motion." The quote in the book seems reasonable, unlike the quote in your question, and your subquestions 2. and 3. are not applicable.

• Thank you! (It was a mistake in printed version, and was corrected in electronic version.) – J C Jul 24 '18 at 6:28