I've heard that in QFT a particle is a local excitation of a quantized field, but I can't understand how can I imagine this. For example in the second quantization of the Klein-Gordon field we get an operator like this $$\Phi = \int d^3pN_p(a_p^{\dagger}e^{i(\omega_pt-px)}+a_pe^{-i(\omega_pt-px)})$$ What does this physically mean? Where do the particles come from? Is it correct to imagine a particle like a perturbation of this field, whatever it means?
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$\begingroup$ Welcome to Physics! I have made a couple of edits to your question to make the language sound more natural. Please feel free to revert these edits (or make further edits) if I have changed the meaning of your question. $\endgroup$– Michael SeifertFeb 7, 2022 at 14:17
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2$\begingroup$ Possible duplicates: physics.stackexchange.com/q/163691/50583, physics.stackexchange.com/q/127141/50583 $\endgroup$– ACuriousMind ♦Feb 7, 2022 at 14:30
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When you quantize a free Klein-Gordon field, you find the spectrum is described by a Fock space. Each Fourier mode obeys the equation for a harmonic oscillator. Therefore, each of the $a_p^\dagger$ operators acts like a creation operator, creating an excitation in mode $p$, much like in the harmonic oscillator in quantum mechanics, the creation operator creates an excitation. For each value of $p$, you can create an infinite tower of states, labeled by $n_p$, the number of excitations with momentum $p$. We interpret $n_p$ as the number of particles with momentum $p$.
This is covered in much more detail in any books or lecture notes on quantum field theory; for example, see Chapter 2 of David Tong's QFT notes https://www.damtp.cam.ac.uk/user/tong/qft.html
