The "diagonal" $P$ representation of the photon density operator in terms of the coherent states of the radiation field is studied. It is shown that the class of weight functions $P\left(\alpha \right)$ required to represent all density operators by means of the $P$ representation is much larger than the class of all probability distributions and that therefore the analogy linking the use of the $P$ representation with certain classical stochastic techniques is useful only for a limited class of density operators. The class of density operators for which the $P$ representation is appropriate is shown to be limited in the sense that there are density operators that require weight functions so singular as to lie outside the class of all tempered distributions. Conditions on the weight function $P\left(\alpha \right)$ under which the expectation values of the normal ordered operators ${\left(a^{†}\right)}^n{a}^m$ assume a particularly simple integral form are presented. It is shown that if $P\left(\alpha \right)$ is a tempered distribution, then these expectation values are given by a simple limiting process. A limited correspondence with classical optics is suggested by a formula for these expectation values. Conditions that a density operator must satisfy if its weight function is to be non-negative are presented and are shown to exclude all pure states except the coherent states themselves. The problem of representing an arbitrary operator in terms of the coherent states is studied. New measures of the completeness of the coherent states are established.

• Received 18 August 1964

DOI:https://doi.org/10.1103/PhysRev.138.B1566