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In Mark Srednicki's QFT book, section $5$, he mentions following things:

$a^{\dagger}({\bf k})$ creates a particle with momentum $k$ and is given by \begin{equation} a^{\dagger}(k)=-i\int d^3x [e^{ikx}\partial_{0}\phi(x)-\phi(x)\partial_0(e^{ikx})].\tag{5.2} \end{equation} In the next, he defines another operator $a_1^{\dagger}$ (see equation 5.6) near momentum $k_1$ by \begin{equation} a_1^{\dagger}\equiv\int d^3k f_1({\bf k})a^{\dagger}({\bf k}),\tag{5.6} \end{equation} where \begin{equation} f_1({\bf k})\propto \exp{[-({\bf k}-{\bf k}_1)^2/4\sigma]}\tag{5.7} \end{equation} is an appropriate wave packet. My confusion is: what is the physical meaning of $a_1^{\dagger}$? And what does the "wave packet" mean here? I guess $a_1^{\dagger}$ is some operator that creates one-particle state of momentum "near" the given $k_1$, but why is the integral defined in whole momentum space?

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$a^{\dagger}_k$ creates a particle with a definite momentum $k$; definite in the sense of Dirac delta function.

While $a^{\dagger}_1$ creates a state in which the momentum is not definite but almost smeared over $3\sigma$ range about $k_1$. Wave packet is proper term to refer such kind of state since this is how we create wave packet in QM.

Also such wave packet are used while approximating initial and final state in LSZ formula since in a scattering setup we can't produce particle having only single momentum value because of experimental limitations.

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