I am embarrassed to ask this question. But I came across the following in a physics book:
$$\int e^{ipx}e^{ipx} d^{3}x = 0$$
$d^{3}x = dydydz$, as @Semiclassical shows below.
This came up in the context of showing the orthogonality of eigensolutions of the Dirac equation for spinors. The author needs this relation to hold. Also, the author usually implies over all space in the context of integrals and uses a discrete summation over a finite space.
I would appreciate help as to how to see this. Especially since the LHS looks like the Fourier transform of $e^{ipx}$.
The text is Klauber, "Student Friendly QFT." The integral is in (4-27) where he is claiming the orthogonality of $\langle \psi^{(1)}|\psi^{(3)}\rangle$ (Assuming this is standard notation.)
Thanks very much.
Best Answer
I am the author of the text referenced. The integral over infinite volume is indeed the delta function (times (2pi)^3) and seems in this context more straightforwardly expressed as e^2ipx. (There was a pedagogic reason for expressing it as two factors in (4-27).) In the text, the integral is actually over finite volume V and (4-27) has another factor in front of it. The integral is zero except when p = 0 (with the usual boundary conditions at the edges of V.) When p = 0, the integral is V, but the other factor is zero. Hence (4-27) is zero in any case. This could have been shown more clearly, and I am posting a comment on this on the book website.