Weinberg Chapter 2 Little Group Q

[Heffernana]

I'm trying to understand induced representation / little group stuff in Weinberg QFT vol. 1 chapter 2 (around page 64, 65, 66). So is this the correct way of thinking about it:

We have the Poincaré group of symmetries; we wish to know how to represent operators (derived from these symmetries) that act on physical states, $\Psi$, in the Hilbert space.

Since 4-momenta - p - commute as shown in the Lie Algebra of the group, classify the state-vectors according to p and someother quantities $\sigma$. Then use a Lorentz transformation to re-write a general momentum state $\Psi_{p,\sigma}$ in terms of a finite number of distinct "standard momenta" $k_{\mu}$, as in equation (2.5.5). This standard momentum $k_{\mu}$ is invariant under a certain group symmetry $W^{\mu}_{\nu}$ by construction.

The set of W that satisfy this $Wk=k$ are called the little group.

Correct so far? I guess my question is then what to make of this? To take an example off of Table 2.1 page 66 if my state vector is dependent on k-momentum (already in standard form), $(0,0,0,M)$ (case (a)) then are the only transformations that leave it invariant (i.e. can produce an eigenvalue-eigenvector) those of SO(3), the rotation group?

As you can see I have lots of the pieces but just not quite sure what the whole point is, but really want to get it properly. Thanks

 

[azntoon]

!Yes, that is the correct way of thinking about it. The point of the little group is to classify the different representations of the Poincaré group. In the example you gave, the little group is SO(3), so the only transformations that will leave the state vector invariant are rotations.