Physics Asked on April 29, 2021
I’m working in Peskin and Schroeders book on QFT
These are the Fourier transforms of the field solutions to the Klein-Gordon equation:
I don’t understand how to get from (2.25) to (2.27)
The logical thing seems to me to try the substitution: $p$ becomes $-p$, for each of the components of the momentum in the second term in the integrand of (2.25), this should however introduce another minus sign in front of the term due to the integration measure. (2.27) lacks this minus sign.
The minus sign that would be produced by the measure can be removed by flipping the integration boundaries: $$ int_{-a}^a f(x) text{d} x stackrel{xrightarrow -x}{longrightarrow} -int_a^{-a} f(-x) text{d} x = int_{-a}^a f(-x) text{d} x $$
Correct answer by flippiefanus on April 29, 2021
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