Representation of Operators in Quantum Optics

The invariance properties of the double-integral and diagonal representations of operators in terms of the coherent states are examined. It is shown that a unique diagonal representation always exists for bounded operators, hence for every density operator, and for unbounded operators which are polynomials in the boson creation and annihilation operators. The associated weight function $P\left(\alpha \right)$ is a generalized function in the space $Z^{\prime }$. The physical significance of this result is discussed, with particular emphasis on the diagonal representation of the density operator of arbitrary radiation fields. A general formula for the weight function $P\left(\alpha \right)$ is derived and is used to calculate the particular form of the weight function for several radiation fields of interest.