In Weinberg's Quantum Field Theory (Vol. I, pages 64-67) it is stated that a unitary representation of little group induces a unitary representation of the Poincare group. But I don't understand how it works.
What is the way to show that the representation induced is unitary？ And is the unitarity dependent on the normalization factor N(p)？
Here I am to make my question more detailed.
On page 64, Weinberg defines $\Psi_{p,\sigma}=N(p)U(L(p))\Psi_{k,\sigma}$ where $k$ is the standard four-momentum.
Then on page 65, he states that $U(\Lambda)\Psi_{p,\sigma}=\left(\frac{N(p)}{N(\Lambda p)}\right)\sum\limits_{\sigma^\prime}D_{\sigma^\prime\sigma}(W(\Lambda,p))\Psi_{\Lambda p,\sigma^\prime}$ where $W$ is the so-called Wigner rotation.
My question is, if the representation of little group $D(W)$ is unitary, what's the way to show that $U(\Lambda)$ is unitary? Note that $U(\Lambda)$ now is represented by a "matrix" with continuous indices $p$ (of course, as well as discrete indices $\sigma$). If it's unitary, is the unitarity dependent on the normalization factor $N(p)$?
