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 $P(\ensuremath{\alpha})$ is a generalized function in the space ${Z}^{\ensuremath{'}}$. 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 $P(\ensuremath{\alpha})$ is derived and is used to calculate the particular form of the weight function for several radiation fields of interest.
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 optical equivalence theorem is examined for the particular class of the electromagnetic fields. It is shown that the distribution function can be taken as a non-negative function, if the density operator describing the states of the electromagnetic field is diagonal in the occupation number representation. Hence the equivalence is not only formal but also essential for this class of the electromagnetic fields, so long as the description of the optical coherence theory is concerned. As a typical example the “diagonal” representation of the density operator for blackbody radiation is studied.
A systematic structure analysis of the correlation functions of statistical quantum optics is carried out. From a suitably defined auxiliary two-point function we are able to identify the excited modes in the wave field. The relative simplicity of the higher order correlation functions emerge as a byproduct and the conditions under which these are made pure are derived. These results depend in a crucial manner on the notion of coherence indices and of unimodular coherence indices. A new class of approximate expressions for the density operator of a statistical wave field is worked out based on discrete characteristic sets. These are even more economical than the diagonal coherent state representations. An appreciation of the subtleties of quantum theory obtains. Certain implications for the physics of light beams are cited.
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.