I can derive $$a(k) = \int d^3 x e^{ik_{\mu} x^{\mu}} (\omega_{\vec{k}} \psi + i \pi)$$ for a free real scalar Klein-Gordon field in three ways mathematically: the usual Fourier transform way in Peskin/Srednicki, an awesome direct a = ½(2a) = ... way (exercise!), and as a by-product of a clever way of mode-expanding the Hamiltonian, but I can't tell you why $$a(k) = \int d^3 x e^{ik_{\mu} x^{\mu}} (\omega_{\vec{k}} \psi + i \pi)$$(adsbygoogle = window.adsbygoogle || []).push({}); shouldbe the answer we'll get before doing any mathematical work in a qualitative way.

How would you interpret $$a(k), \psi, \pi$$ in such a way so as to make the above expression, & similarly for the creation operator, in the real and complex case, obvious - without sweeping the problem under the rug by referring to the analogous expression for quantum harmonic oscillators which should also be explainable with such a description?

Thanks!

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# A Interpreting the K-G Annihilation Operator Expression

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