 Sudarshan's paper was received at Physical Review Letters on March 1, 1963, and published on April 1, 1963, while Glauber's paper was received at Physical Review on April 29, 1963, and published on September 15, 1963.Despite many useful applications in laser theory and coherence theory, the Glauber–Sudarshan P representation has the drawback that it is not always positive, and therefore is not a true probability function.We wish to construct a function formula_1 with the property that the density matrix formula_2 is diagonal in the basis of coherent states formula_3, i.e.,

We also wish to construct the function such that the expectation value of a normally ordered operator satisfies the optical equivalence theorem.