Relation between Quantum and Semiclassical Description of Optical Coherence

The problem of relating the semiclassical and quantum treatments of statistical states of an optical field is re-examined. The case where the rule of association between functions and operators is that of antinormal ordering is studied in detail. It is shown that the distribution function for each mode corresponding to this case is a continuous bounded function, and is also a boundary value of an entire analytic function of two variables. The nature of the distribution for the normal ordering rule of association and its relation to this entire function are discussed. It is shown that this distribution can be regarded as the limit of a sequence of tempered distributions in the following sense: One can find a sequence of density operators ${\hat{\rho }}_{\left(\nu \right)}$ which converges in the norm to the density operator $\hat{\rho }$ of any given field (consisting of a single mode), such that each member of the sequence can be expressed in the form ${\hat{\rho }}_{\left(\nu \right)}=\int {\phi }_{\left(\nu \right)}\mathrm{}\left(z\right)\left|z〉〈z\right|d^{2}z$, where ${\phi }_{\left(\nu \right)}$ is a tempered distribution.