Derivation of the Klein-Gorden solution via Fourier Transofmations
I recently graduate with a bachelor's in physics, and I've been trying to take the next steps toward learning QFT. To this end, I have been working through <PERSON> and <PERSON>'s textbook step-by-step. 346
Currently, I am confused on a detail from chapter two, where the authors solve the Klein-Gordon equation using a Fourier transformation of the field.
In particular, I have looked through online notes and other Stack Exchange answers to understand how to work with the following integral, but I haven't found an answer that goes through the intermediate steps to evaluate it properly.
$$(\Box + m^2)\int e^{i\vec{p}\cdot\vec{x}}\phi(\vec{p},t)d^3p =0$$
I can following the process by which we exchange differentiation and integration and then use the linearity of the operator inside of the integral. As such, I can arrive at the integrand:
$$\Box\left[e^{i\vec{p}\cdot\vec{x}}\phi(\vec{p},t)\right]$$
The operator here is the sum of second-order derivatives, and so I think we can use a product rule for higher derivatives:
$$\Box\left[e^{i\vec{p}\cdot\vec{x}}\phi(\vec{p},t)\right]= \Box\left[e^{i\vec{p}\cdot\vec{x}}\right]\phi(\vec{p},t)+2\partial\cdot\left[e^{i\vec{p}\cdot\vec{x}}\right]\partial\cdot\left[\phi(\vec{p},t)\right] + e^{i\vec{p}\cdot\vec{x}}\Box\left[\phi(\vec{p},t)\right]$$
Tackling the first term, the exponential component doesn't depend on $t$ so this term should evaluate to $|\vec{p}|^2e^{-\vec p\cdot\vec x}\phi(\vec p, t)$ after taking the spacial derivatives and summing right?
Then the second term is smiliar, but we only need first order derivatives so I evaluated it as $2i(p_1+p_2+p_3)e^{i\vec p\cdot \vec x}\partial\cdot\phi(\vec t, t)$.
Then the third term would stay the same so factoring out the exponential we would in total we should get
$$\int e^{i\vec p\cdot\vec x}(\partial_t^2 - \nabla^2+|\vec p|^2 +m^2)\phi(\vec p, t)d^3p + 2i\int e^{i\vec p\cdot\vec x}(p_1+p_2+p_3)\partial\cdot\phi(\vec p, t)d^3p$$
I know that I should have gotten
$$\int e^{i\vec p\cdot\vec x}(\partial_t^2 +|\vec p|^2 +m^2)\phi(\vec p, t)d^3p$$
But I cannot figure out the steps in the middle. Am I over-complicating the derivation and heading in the wrong direction or are there terms that I can cancel somewhere? 281