Field transformation under Lorentz group

[eoghan]

Hi!
In Weinberg's book "The quantum theory of fields", chapter2, it states that the transformation
of a massive particle is
$$U(\Lambda)\Psi_{p,\sigma}= N\sum\mathcal{D}^{(j)}_{\sigma',\sigma}(W)\Psi_{\Lambda p,\sigma'}$$
where W is an element in the little-group SO(3). But than it states that
$$W=L^{-1}(\Lambda p)\Lambda L(p)$$
where L is a 4x4 boost.
But then W is not in SO(3), it's a 4x4 matrix! How can this be possible?

 

[dextercioby]

W is a 4x4 matrix, an element of the group of 4x4 matrices isomorphic to SO(3).