Below, I am reading through how to quantise a complex scalar field, beginning with an example of the Klein-Gordon field.
After arriving to the generic solution to the system corresponding to equation (3.26), it is not clear to me how we obtain equations (3.27) based on defining $b(\overrightarrow{k}) = -\tilde{b}(\overrightarrow{k})$. I, rather simplistically, would have thought such a definition would yield,
$$ \phi(t,\overrightarrow{x}) = \int\frac{d^{3}x}{(2\pi)^{3}}\frac{1}{(2w_{k})^{1/2}}[a(\overrightarrow{k})e^{i\overrightarrow{k}\cdot \overrightarrow{x} - iw_{k}t} - b^{*}(-\overrightarrow{k})e^{i\overrightarrow{k}\cdot \overrightarrow{x} + iw_{k}t}] $$
I am particularly confused over how the argument of the exponential changes sign in the first term, $i\overrightarrow{k} \cdot \overrightarrow{x}$. Please could someone help go through the maths of this?
Also, this manipulation was done with aim of creating a Lorentz covariant inner product. Why was the initial equations (3.26) not Lorentz covariant, while equations (3.27) are? I understand the key result of being Lorentz covariant means that if the expression holds in one inertial frame, then it holds in all inertial frames, however I have not actually dealt with many examples of such expressions. Therefore, I am not sure what to look out for so as to identity what quantities are, or are not, Lorentz covariant.
Any help is much appreciated :)
I am already having trouble to see that one of the authors particular solutions, namely, $\phi(\boldsymbol{x},t)=e^{i\boldsymbol{k}\cdot\boldsymbol{x}+i\omega t}$ is a solution of the KG equation. As far as I see it solves the equation $\partial^2_t\phi+\Delta\phi=i^2(|\boldsymbol{k}|^2+\omega^2)\phi\,.$ The three plus signs we used so far seem wrong. One of the standard books on QFT [1] uses the solutions $$ \phi(\boldsymbol{x},t)=e^{\pm i\boldsymbol{k}\cdot\boldsymbol{x}\mp i\omega t}\,. $$ [1] Peskin, Schroeder, An Introduction to Quantum Field Theory.