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

Question

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?

Answer 1

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.