Normalization of the real Klein Gordon Field in Peskin and Schroeder chapter 2 In Peskin & Schroeder's QFT, how do you get from equation 2.35 to 2.37? (In particular, how does the invariant normalization of the Klein-Gordon real field imply that 
                     $U(\Lambda)|p> = |\Lambda p>$ ?)
Also, on a more general note, could some explain why for the real Klein-Gordon field we need to make the effort to define invariant normalization? In particular, why do we care if the expression $<q|p>$ is invariant if it is $|<q|p>|^2$ which bears physical meaning?
 A: In answer to the first part of your question, let's work backward. We write
\begin{align}
U(\Lambda)|\mathbf{p}\rangle &= \sqrt{2E_{\mathbf{p}}}U(\Lambda)a^\dagger_{\mathbf{p}}U^\dagger(\Lambda)U(\Lambda)|0\rangle \\
&= \sqrt{2E_{\mathbf{p}}}[U(\Lambda)a^\dagger_{\mathbf{p}}U^\dagger(\Lambda)]|0\rangle,
\end{align}
where we used $U(\Lambda)|0\rangle=|0\rangle$ and we have
\begin{align}
a^\dagger_{\Lambda\mathbf{p}}&=\sqrt{\frac{E_{\mathbf{p}}}{E_{\Lambda\mathbf{p}}}}
U(\Lambda)a^\dagger_{\mathbf{p}}U^\dagger(\Lambda),
\end{align}
which must holds since
\begin{align}
U(\Lambda) a^\dagger_{\mathbf{p}} \sqrt{E_{\mathbf{p}}}&= a^\dagger_{\Lambda\mathbf{p}}U(\Lambda) \sqrt{E_{\Lambda\mathbf{p}}}.
\end{align}
Applying this to the vacuum state demonstrates the equation you're asking about. (Note that we've neglected spin throughout.)
Your second question needs to be clarified. You seem to be conflating Lorentz covariance with the issue of Lorentz invariance. The amplitudes are not invariant. They are covariant.
