In first quantization, a state of system is represented by wavefunction (w.f.) $\phi(x)$ (a representation of a state $|\phi\rangle$ in Hilbert space). The way I understand it is that $|\phi(x)|^2$ gives probability of finding a particle at position $x$. So, $|\phi\rangle$ is a column matrix (written in some basis). Understandable to me!
In second quantization, the many-body state of system is represented by field operators. According to Wikipedia, field operators are given in terms of creation and annihilation operators $$\Psi = \sum_\nu \psi_\nu \hat{a}_\nu \quad ; \quad \Psi^\dagger = \sum_\nu \psi_\nu^* \hat{a}_\nu^\dagger$$ where $\psi$ is ordinary first quantization w.f. and $\hat{a} (\hat{a}^\dagger$) is annihilation (creation) operator.
I don't understand that how does the field operators represent a state? How can I intuitively think about it? How to relate a field operator representation with physical system? What is physical meaning for a field operator?
