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## Main Question or Discussion Point

Hello! I am a bit confused about the KG equation in the context of QFT. In QM, the KG equations describes the evolution of a wavefunction, ##\phi(x,t)##, in space and time (I will assume we have no potential). This function gives the probability of finding a particle described by this wavefunction at a given position, at a given time. (Please let me know if anything I said or I will say is wrong.) Now in QFT, ##\phi(x,t)## is an operator, which acting on the vacuum state ##|0>## creates a particle at position ##x##. I am confused about the meaning of KG equations in this context? What do the space and time evolution of ##\phi(x,t)## mean now? When it acts, it doesn't create a probability distribution, but an actual particle at a point x (it is at x with 100% probability). Also mathematically speaking, if ##\phi(x,t)## is an operator, the KG equation written as ##(\partial^2+m^2)\phi(x,t)=0## doesn't make much sense to me as on the left side you have an operator ##(\partial^2+m^2)## acting on another operator ##\phi(x,t)## while on the right side you have a number, which is 0. How should I understand this equation now? Lastly, once ##\phi(x,t)## acts on the vacuum state and creates a new particle, how does this particle evolve in space and time? I assume it is still the KG equation that dictates this, but I am not sure I understand how to apply it. Any explanation would be greatly appreciated. I am really new to this and I kinda got stuck with understanding this. Thank you!