How to demand the angle for little group I am not here searching answers for homework, it just a theoritical problem. 
This question arise from problem 2.1 in Weinberg QFT volume 1. To solve this problem, I need to know explicitly what the angle $\theta$ of SO(3) is. I know the angle can be fixed by $\gamma $ and $p$, but I have been  thinking this for a long time, and still don't know how to solve it. I really need some help, thanks a lot if you can write things explicitly.
I have reading some papers about this, seems it is very hard to solve for it; I will post a example here: 
arXiv:math-ph/0401032v1 15 Jan 2004
 A: You don't need to go to external papers to solve this. Just apply the material in the chapter. This is definitely considered a 'homework problem', by the way.
You are given a state $\Psi_{p,\sigma}$ where $p$ is in the y-direction and $\sigma$ is some arbitrary spin state for spin 1. The way we define spin states for particles that are not at rest is just $$\Psi_{p,\sigma}\propto U(L(p))\Psi_{k,\sigma}$$
where $k$ is the rest momentum, and $L(p)$ is a standard boost. The particular convention Weinberg gives is just the obvious boost from rest to a y velocity (no extra rotation about the y-axis). I'm using a proportional symbol because I don't care about normalization factors here.
Now we boost our state into the z-direction by $\Lambda$ (again this is the obvious boost)
$$U(\Lambda)\Psi_{p,\sigma}\propto U(L(\Lambda p))\,U(L^{-1}(\Lambda p)\Lambda L(p))\,\Psi_{k,\sigma}$$
The point of writing it this way is the first factor $L(\Lambda p)$ just boosts our momentum without changing the spin (because it uses our standard boost convention), and the second is just a rotation because it takes $k$ to $k$.
So figure out what $L^{-1}(\Lambda p)\Lambda L(p)$ is as a matrix. Everything is defined in the obvious way and if you don't believe me just look up Weinberg's definition of the conventional boost. Now you have a rotation matrix, the next step is to find the corresponding matrix acting on spin one states. That is a problem from ordinary non-relativistic quantum mechanics.
