I have some problems about basic concepts of quantum field theory.
First let's look at Klein-Gordon field.
Klein-Gordon equation has two branches of solutions, one of which is positive frequency and the other is negative frequency. The solution can be written as $$\psi(x)=\int \frac{d^{3}p}{(2\pi)^{3}}\frac{1}{\sqrt{2E_{p}}}(a_{p}e^{-ipx+i\omega t}+b_{p}e^{-ipx-i\omega t}).$$
Then we need to quantize above solution. In the exercise part of both books, Peskin and Srednicki, the complex scalar field operator is written as
$$\hat{\psi}(x)=\int \frac{d^{3}p}{(2\pi)^{3}}\frac{1}{\sqrt{2E_{p}}}(\hat{a}_{p}e^{-ipx}+\hat{b}_{p}^{+}e^{ipx})\tag{1}$$
The question is:
Why equ(1) is not written as $$\hat{\psi}(x)=\int \frac{d^{3}p}{(2\pi)^{3}}\frac{1}{\sqrt{2E_{p}}}(\hat{a}_{p}e^{-ipx}+\hat{b}_{p}e^{ipx})~?$$ Is this because when quantizing negative frequency, annihilation operator must be replaced by creation operator and $-p$ by $p$ in exponent?