Reference: References: Robert D. Klauber, *Student Friendly Quantum Field Theory*, (Sandtrove Press, 2013) – Chapter 3, Problem 3.2.

The plane wave solutions of the Klein-Gordon equation are

We can redefine a couple of terms by introducing

Then

The and are orthonormal functions. We have

where the integral is over the volume , and the wavelengths of the plane waves fit an integral number of times within , so that the amplitudes of the waves at the boundaries are all zero. The four-vector is defined as

If , the integrand is 1 and is integrated over , so the result is

If , consider the integral over (for the purposes of this derivation only, refers to the single dimension of the 3-vector and should not be confused with the four-vector used in 1):

where and are the limits of , where by assumption the wave amplitude is zero. Therefore

The same result follows for by just replacing by throughout the derivation, so

For mixed terms, we have

In this case, the exponent cannot be zero, so the integral always comes out to zero, so that

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