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The Tortoise-Man

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I have a pretty naive question about quantization of real Klein-Gordon (so scalar) field ##\hat{\phi}(x,t)##.

The most conventional form (see eg in this one ; but there are myriad scripts) is given by

##\hat{\phi}(x,t)= \int d^3p \dfrac{1}{(2\pi)^3} N_p (a_p \cdot e^{i(\omega_pt - p \cdot x)}+b_p^\dagger \cdot e^{-i(\omega_pt - p \cdot x)})##

where ##N_p## is a momentum-dependent normalization factor, and ##\omega_p= \sqrt(p^2+m^2)##, ##a_p## annihilation and ##b_p^\dagger## creation operators.

My question is what is/are the reason(s) such that ##a_p## is "weighted" with positive exponential factor ##e^{i(\omega_pt - p \cdot x)} ## and ##b_p^\dagger## with negative exponential ##e^{-i(\omega_pt - p \cdot x)} ## and not another way around?

I'm seeking for two reasonings, the pure formal ("sober" calculation), but also heuristical/physical (intuitive interpretation of this factor)

The most conventional form (see eg in this one ; but there are myriad scripts) is given by

##\hat{\phi}(x,t)= \int d^3p \dfrac{1}{(2\pi)^3} N_p (a_p \cdot e^{i(\omega_pt - p \cdot x)}+b_p^\dagger \cdot e^{-i(\omega_pt - p \cdot x)})##

where ##N_p## is a momentum-dependent normalization factor, and ##\omega_p= \sqrt(p^2+m^2)##, ##a_p## annihilation and ##b_p^\dagger## creation operators.

My question is what is/are the reason(s) such that ##a_p## is "weighted" with positive exponential factor ##e^{i(\omega_pt - p \cdot x)} ## and ##b_p^\dagger## with negative exponential ##e^{-i(\omega_pt - p \cdot x)} ## and not another way around?

I'm seeking for two reasonings, the pure formal ("sober" calculation), but also heuristical/physical (intuitive interpretation of this factor)

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