Creation and annihilation operator

[Sebas4]

TL;DR Summary

Does the annihilation/creation operator on the complex exponent?

Hey, I have a short question.
The quantized field in Schrödinger picture is given by:
$$\hat{\phi} \left(\textbf{x}\right) =\int \frac{d^{3}p}{\left(2\pi\right)^3} \frac{1}{\sqrt{\omega_{2\textbf{p}}}}\left(\hat{a}_{\textbf{p}}e^{i\textbf{p} \cdot \textbf{x}} + \hat{a}^{\dagger}_{\textbf{p}}e^{-i\textbf{p} \cdot \textbf{x}}\right)$$

My question is, does the the annihilation $\hat{a}_{\textbf{p}}$ and creation $\hat{a}^{\dagger}_{\textbf{p}}$ operator act on $e^{i\textbf{p} \cdot \textbf{x}}$ and $e^{-i\textbf{p} \cdot \textbf{x}}$ respectively? In other words: does the annihilation/creation operator on the complex exponent?

 

[weirdoguy]

does the the annihilation a^p and creation a^p† operator act on eip⋅x and e−ip⋅x respectively?

No.

 

[Paul Colby]

Well, it does in the sense that $a$ and $a^\dagger$ commute with these factors.

 

[vanhees71]

No, they don't. The creation and annihilation operators are linear operators defined in the Fock space. The expeonential functions are numbers; $\vec{x}, \vec{p} \in \mathbb{R}^3$.