I have a question about the quantization procedure of the Klein-Gordon field as presented in Peskin&Schroeder.
The field is expressed as a Fourier decomposition ϕ(x,t)=∫d3p(2π)3eipxϕ(p,t), with ϕ∗(p,t)=ϕ(−p,t) so that ϕ(x) is real.
To continue one introduces ladder operators: ϕ(p)=1√2ωp(ap+a†−p).
But now ϕ∗(p)=1√2ωp(ap+a†−p)∗=1√2ωp(a∗p+aT−p)=ϕT(−p)≠ϕ(−p)=1√2ωp(a−p+a†p).
So why is the difference between ϕT(−p) and ϕ(−p) not important (is there even a difference?)?
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