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Sebas4

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- 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:

[tex] \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) [/tex]

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

The quantized field in Schrödinger picture is given by:

[tex] \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) [/tex]

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

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