In all textbooks on quantum optics I can reach (Scully, Leonhardt, Walls, etc), the Glauber-Sudardhan $P$-representation $P(\alpha)$ is introduced in the following two ways:
- Fourier transform of $\mbox{Tr}(\hat\rho \mbox{e}^{\alpha \hat a^\dagger}\mbox{e}^{\alpha ^*\hat a})$.
- "Diognal element" of density operator in terms of coherent states, i.e., $\hat\rho=\int \mbox{d}^2\alpha P(\alpha)|{\alpha}\rangle\langle{\alpha}|$.
In the first defination, there're no restrictions on $\hat \rho$. However the second defination requires $\hat \rho$ to be diagonalized. In Scully' book (page 76), he says,
The name coherent state representation is due to the following representation of density operator $\hat\rho$ by means of a diagonal representation in terms of the coherent states: $\hat\rho=\int \mbox{d}^2\alpha P(\alpha)|{\alpha}\rangle\langle{\alpha}|$.
But he didn't prove it. So, can every $\hat\rho$ be expanded into a diagonal form like above? And how to prove it?
