Why do different quantum field theory books have different conventions regarding the normalization? Why are there several different versions of the scalar field solution, with a different coefficient in front of the exponential in the solution? Why don't all the authors use the same convention? I mean why do different quantum field theory books have different conventions regarding the normalization?
 A: Some books wish to have a simple-looking commutator for the annihilationa and creation operators:
$$
[\hat a_p,\hat a^\dagger_q]= (2\pi)^3 \delta^3(p-q)
$$
while other prefer the normalization
$$
[\hat a_p,\hat a^\dagger_q]= (2\pi)^3  2E_p\delta^3(p-q), 
$$
which is is a Lorentz covariant normalization in that states created by $a^\dagger_p$ have a higher particle number per unit volume when the energy is higher. This is  Lorentz covariant because a boost to higher energy Lorentz-contracts the volume while keeping the particle number constant.
In the latter normalization  the  on-shell momentum integrals
$$
\phi(x)=\int \frac{d^3p}{(2\pi)^3 2E_p} a_p e^{-ipx}+... 
$$
are know as Lorentz-invariant phase space (LIPS) integrals because they   arise from  doing the $p_0$ integral in the manifestly Lorentz invariant delta function
$$
\delta^4(p^2-m^2) = \frac 1{2E_p}(\delta(p_0-\sqrt{p^2+m^2})+ \delta(p_0+\sqrt{p^2+m^2})), 
\quad E=+\sqrt{p^2+m^2}.
$$
The latter normalization may look more complicated, but has the advantage that matrix elements of    Lorentz scalar fields are themselves Lorentz invariant, whereas  in the simpler-looking (no $2E$'s)  normalization  they have non-Lorentz invariant factors of $1/\sqrt{2E}$ everywhere. For example
if we define a one particle state by $|p\rangle =a^\dagger_p |0\rangle$ then the LIPS normalization has
$$
\langle0|\phi(x)| p\rangle= e^{ipx}
$$
with no extra $1/\sqrt{2E}$.
