Many text books emphasize that the quantum field is not wavefunction. But because of the similarity in the format, I could not stop from wondering whether they are actually the same thing.
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A wavefunction is (typically understood to be) a complex valued function on configuration space; a wave function assigns a complex number (probability amplitude) to each point in configuration space. For a system of $N$ particles, the system's wavefunction is $3N$ dimensional.
A quantum field, on the other hand, is an operator valued function on physical space; a quantum field assigns an operator to each event in spacetime.
These are clearly fundamentally different.
One can make a connection between a single particle wavefunction and a quantum field in the following way.
$|0\rangle$ is the vacuum (no particle) state of the system, $\phi^*(x,t)$ is the operator which creates a particle with definite location $x$ at time $t$, and $\Psi(x,t)$ is a 1 particle wavefunction.
Then, a 1 particle state described described by the wavefunction $\Psi$ is given by
$$\int dx \;\Psi(x,t)\;\phi^*(x,t)|0\rangle$$
That is, the state is a superposition over all 1 particle states, $\phi^*(x,t)|0\rangle$, weighted by $\Psi(x,t)$.
For a 2 particle state, the generalization is
$$\int dx_1 dx_2 \;\Psi(x_1, x_2,t)\;\phi^*(x_1,t)\phi^*(x_2,t)|0\rangle $$
Alfred Centauri
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A quick answer is that simple wave functions can not describe the processes of annihilation and creation represented in fields. Fields support multi-particles states in a way that wave functions can not.
Whelp
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