# Interpretation of field operators

In the book Field Quantization of Greiner, in section 3.2 he introduces the field operators (for bosons), that are postuleted to satisfy the commutation relations $$[\hat{\psi}(\textbf{x},t), \hat{\psi}^{\dagger}(\textbf{x}',t)] = \delta(\textbf{x} - \textbf{x}')$$ $$[\hat{\psi}(\textbf{x},t),\hat{\psi}(\textbf{x}',t)] = [\hat{\psi}^{\dagger}(\textbf{x},t), \hat{\psi}^{\dagger}(\textbf{x}',t)] = 0.$$ Defining the vacuum state $$| 0\rangle$$ such that $$\hat{\psi}(\textbf{x},t)|0 \rangle = 0$$, he argues that the state $$|\textbf{x}_{1},...,\textbf{x}_{n}; t \rangle \equiv \frac{1}{\sqrt{n!}}\ \hat{\psi}^{\dagger}(\textbf{x}_{1},t)\cdots \hat{\psi}^{\dagger}(\textbf{x}_{n},t) |0\rangle$$ can be interpreted as a state of a system of $$n$$ (identical) particles localized at positions $$\textbf{x}_{i}$$ at the instant $$t$$. Thereby, it's easy to show that $$\hat{\psi}^{\dagger}(\textbf{x},t)$$ can be interpreted as an operator that creates a particle at position $$\textbf{x}$$ in the instant $$t$$, because $$\hat{\psi}^{\dagger}(\textbf{x},t) |\textbf{x}_{1},...,\textbf{x}_{n}; t \rangle = \sqrt{n + 1} \ |\textbf{x}, \textbf{x}_{1},...,\textbf{x}_{n}; t \rangle$$

My doubt is about the interpretation of the operator $$\hat{\psi}(\textbf{x},t)$$. I know that it annihilates a particle, but I would like to interpretated its action in the state above. Applying this operator and using the commutation relations, I have $$\hat{\psi}(\textbf{x},t) |\textbf{x}_{1},...,\textbf{x}_{n}; t \rangle = \frac{1}{\sqrt{n!}} \sum_{i} \delta(\textbf{x} - \textbf{x}_{i}) |\textbf{x}_{1},..., \textbf{x}_{i-1}, \textbf{x}_{i+1}, ...,\textbf{x}_{n}; t \rangle.$$ How can I interpretated this? To show that $$\hat{\psi}(\textbf{x},t)$$ is in fact a operator that destroy a particle at position $$\textbf{x}$$? If I take $$\textbf{x} = \textbf{x}_{1}$$, for exemple, the sum will diverge.

These reasons are why, when doing quantum field theory "rigorously," one tends to work with so-called "smeared" operators. Given a function $$f$$ on spacetime, the smeared operator $$\psi_f$$ is given by
$$\psi_f=\int\mathrm{d}^dx f(x)\psi(x).$$