Equivalence of Semiclassical and Quantum Mechanical Descriptions of Statistical Light Beams. E. C. G. Sudarshan in Phys. Rev. Lett. 10:277 (1963). What the paper says!?
In this work, Sudarshan introduces the $P$-representation (which he calls here $\phi$). This is the paper with which he challenges the attribution of the Nobel Prize to Glauber instead of to him. Glauber "previously" introduced a close-enough version of this diagonal representation (see Eq. (3) of Ref. [1]), using non-diagonal states, but Sudarshan does not comment on this.
This is how he describes the work of Glauber[1] and his own input:
statistical states of a quantized (electromagnetic) field have been considered reently,3 and a quantum mechanical definition of coherence functions of arbitrary order presented. It is the aim of this note to elaborate on this definition and to demonstrate its complete equivalence to the classical description as long as no nonlinear effects are considered.
It is not a particularly well-written paper as it is nowadays difficult to read. In his Eq. (2) (which is the RPCS), he uses the notation $\ud{\psi(n)}$ but previously $\psi(n)$ was the state itself ($\ket{n}$ in modern notations). From context, it thus appears that $\ud{\psi(n)}\equiv\bra{n}$.
He claims (more than shows, but this can be checked I guess) that $\rho$ can thus be written in diagonal form:
From this, he jumps to the well-known result, that is more properly stated later in Eq. (6) but delayed apparently for the sake of generality (many modes). For the single mode, his claim is that:
$$\phi(z)\equiv\sum_{n=0}^\infty\sum_{n'=0}^\infty{\bra{n'}\rho\ket{n'}\sqrt{n!n'!}\over(n+n')!2\pi r}\exp(r^2+i(n'-n)\theta)\left\{\left(-{\partial\over\partial r}\right)^{n+n'}\delta(r)\right\}$$
where $z\equiv re^{i\theta}$, in which case
With this result, he indeed first introduces the (diagonal) $P$ representation.
Then comes his interpretation of such a representation:
Here he seems to state the equivalency of the statistical theory of the quantum field with a classical description altogether. The fact that $\phi$ is not necessarily "positive definite" (i.e., it could be negative) doesn't seem to cause him much issues in this regard. Nowadays, we would understand the Fock state $\ket{n}$ as non-classical precisely because it has no probabilistic interpretation.
He then generalizes to many modes and comes to same conclusion again:
Consequently the description of statistical states of a quantum mechanical system with an arbitrary (countably infinite) number of degrees of freedom is completely equivalent to the description in terms of classical probability distributions in the same (countably infinite) number of complex variables. In particular, the statistical states of the quantized electromagnetic field may be described uniquely by classical complex linear functions on the classical electromagnetic field. This functional will be "real" reflecting the Hermiticity of the density matrix; and leads in either version to real expectation values for Hermitian (real) dynamical variables.
Here the concern for positive-definiteness seems to have disappeared.
He makes two further comments. One related to the necessity to work with positive-frequency parts of the classical field as a result of using normal ordering.
The other is also a bit cryptic, related to the fact that thermal states are not the only possible states and that one could consider off-diagonal states toon and:
this implies, in accordance with Eq. (6), that not all phase-angle sequences {θ} have equal weight. In such a case the expectation values of operators with unequal number of creation and destruction operators need not all vanish.
He also provides the inverted formula to get $\rho$ from $P$ which he attributes to a Purcell-Mandel formula.
Overall, it seems he missed the non-classical, genuinely quantum features of the field and merely, indeed, provided the mathematical decomposition.