Creation and annihilation operator

  • #1
Sebas4
13
2
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?
 
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  • #2
Sebas4 said:
does the the annihilation a^p and creation a^p† operator act on eip⋅x and e−ip⋅x respectively?

No.
 
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Likes Paul Colby
  • #3
Well, it does in the sense that ##a## and ##a^\dagger## commute with these factors.
 
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Likes topsquark
  • #4
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##.
 

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