A representation is introduced for the density operator of the electromagnetic field that is suitable for all density operators and that reduces to the coherent-state
P representation when the latter exists. It expresses the density operator $\ensuremath{\rho}$ as the sum of four terms, each of which is a two-dimensional weighted integral over outer products of coherent states. The first integral has the form of the
P representation, i.e., the outer products are projection operators. The absence of singularities in this term is achieved by the presence of the three supplementary integrals which vanish when the density operator possesses the
P representation. In general, for stationary density operators, only the first two terms of the regularized
P representation are necessary. A simple prescription is given for obtaining the four weight functions of this representation from the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, where $|\ensuremath{\alpha}〉$ is a coherent state and $\ensuremath{\rho}$ is the density operator. According to this prescription, the
P representation does not exist and one or more of the supplementary, regularizing terms is necessary when the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$ contains a term that decreases more rapidly than $\mathrm{exp}(\ensuremath{-}{|\ensuremath{\alpha}|}^{2})$ as $|\ensuremath{\alpha}|\ensuremath{\rightarrow}\ensuremath{\infty}$. The regularized
P representation affords nonsingular integral expressions for all density operators and for most expectation values, including, when they are finite, those of the normally ordered products of the creation and annihilation operators, ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$ and
a. The construction and use of this representation is illustrated with the aid of simple examples in which the density operator does not possess the
P representation.
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 $P(\ensuremath{\alpha})$ 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 $P(\ensuremath{\alpha})$ under which the expectation values of the normal ordered operators ${({a}^{\ifmmode\dagger\else\textdagger\fi{}})}^{n}{a}^{m}$ assume a particularly simple integral form are presented. It is shown that if $P(\ensuremath{\alpha})$ 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.
The problem of expanding a density operator $\ensuremath{\rho}$ in forms that simplify the evaluation of important classes of quantum-mechanical expectation values is studied. The weight function $P(\ensuremath{\alpha})$ of the
P representation, the Wigner distribution $W(\ensuremath{\alpha})$, and the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, where $|\ensuremath{\alpha}〉$ is a coherent state, are discussed from a unified point of view. Each of these quasiprobability distributions is examined as the expectation value of a Hermitian operator, as the weight function of an integral representation for the density operator and as the function associated with the density operator by one of the operator-function correspondences defined in the preceding paper. The weight function $P(\ensuremath{\alpha})$ of the
P representation is shown to be the expectation value of a Hermitian operator all of whose eigenvalues are infinite. The existence of the function $P(\ensuremath{\alpha})$ as an infinitely differentiable function is found to be equivalent to the existence of a well-defined antinormally ordered series expansion for the density operator in powers of the annihilation and creation operators
a and ${a}^{\ifmmode\dagger\else\textdagger\fi{}}$. The Wigner distribution $W(\ensuremath{\alpha})$ is shown to be a continuous, uniformly bounded, square-integrable weight function for an integral expansion of the density operator and to be the function associated with the symmetrically ordered power-series expansion of the density operator. The function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, which is infinitely differentiable, corresponds to the normally ordered form of the density operator. Its use as a weight function in an integral expansion of the density operator is shown to involve singularities that are closely related to those which occur in the
P representation. A parametrized integral expansion of the density operator is introduced in which the weight function $W(\ensuremath{\alpha},s)$ may be identified with the weight function $P(\ensuremath{\alpha})$ of the
P representation, with the Wigner distribution $W(\ensuremath{\alpha})$, and with the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$ when the order parameter
s assumes the values $s=+1, 0, \ensuremath{-}1$, respectively. The function $W(\ensuremath{\alpha},s)$ is shown to be the expectation value of the ordered operator analog of the $\ensuremath{\delta}$ function defined in the preceding paper. This operator is in the trace class for
Res<0, has bounded eigenvalues for
Res=0, and has infinite eigenvalues for
s=1. Marked changes in the properties of the quasiprobability distribution $W(\ensuremath{\alpha},s)$ are exhibited as the order parameter
s is varied continuously from $s=\ensuremath{-}1$, corresponding to the function $〈\ensuremath{\alpha}|\ensuremath{\rho}|\ensuremath{\alpha}〉$, to
s=+1, corresponding to the function $P(\ensuremath{\alpha})$. Methods for constructing these functions and for using them to compute expectation values are presented and illustrated with several examples. One of these examples leads to a physical characterization of the density operators for which the
P representation is appropriate.