Let us consider the wave equation written in the form $$ \partial_{tt}\phi-\Delta\phi+\phi=0. $$ By a Fourier transform, this takes the form $$ (k_0^2-{\bf k}^2-1)\phi(k)=0. $$ This has as a solution, in the sense of distributions, $$ \phi(k)=A(k)\delta(k_0^2-{\bf k}^2-1). $$ It is easy to see that, by Fourier transforming back, $$ \phi(x)=\int\frac{d^3k}{(2\pi)^3}\left(\frac{A({\bf k},\sqrt{{\bf k}^2+1})}{2\sqrt{{\bf k}^2+1}}e^{i{\bf k}\cdot{\bf x}-i\sqrt{{\bf k}^2+1}t}-\frac{A({\bf k},-\sqrt{{\bf k}^2+1})}{2\sqrt{{\bf k}^2+1}}e^{i{\bf k}\cdot{\bf x}+i\sqrt{{\bf k}^2+1}t}\right) $$ that appears as a general solution of the equation we started from.
My questions are the following:
- Is this approach sound to solve a linear PDE?
- What does it happen to the Dirac equation $(i\gamma\cdot\partial-1)\psi=0$ using this technique?