A correspondence $a_j\leftrightarrow {\alpha }_j$ between operators $\mathrm{a}=[a_1, a_2, \cdots a_f]$ and $c$ numbers $\alpha =[{\alpha }_1, {\alpha }_2, \cdots {\alpha }_f]$ together with an arbitrary ordering rule $\mathcal{C}$ (e.g., in sequence from 1 to $f$) permit an association $M\left(\mathrm{a}\right)=\mathcal{C}{M}^{\mathrm{}\left(c\right)\mathrm{}}\left(\alpha \right)$ between a general operator $M\left(\mathrm{a}\right)$ and an associated $c$ number function $M^{\mathrm{}\left(c\right)\mathrm{}}\left(\alpha \right)$. A quasiprobability $P(\alpha , t)$ is then defined so that a general ensemble average can be written as an ordinary integration: $〈M\left(\mathrm{a}\mathrm{}\right(t\left)\right)〉=\int M^{\mathrm{}\left(c\right)\mathrm{}}\left(\alpha \right)d\alpha P(\alpha , t)$. The equation for $\frac{\partial P(\alpha , t)}{\partial t}$ suggests that the $\alpha$ obeys a classical Markoff process. If this classical Markoff process is taken literally, multitime classical averages can be computed. Do these correspond to appropriate quantum averages? For the case of field operators such that $[b, b^{†}]=1\mathrm{}$, important in discussing laser statistics, we show that with $a_1=b^{†}$ and $a_f=b$, the classical multitime average is equivalent to the average of the corresponding quantum operators written in time-ordered, normal-ordered sequence. For the atomic operators in a laser problem, we obtain the desired correspondence, but find that the more complicated commutation rules necessarily lead to derivative correction terms when multitime averages are taken. Our derivation of multitime averages is based on the quantum regression theorem. We show that this theorem is equivalent to assuming the quantum system to be Markoffian, by showing that it leads to an appropriate factorization of a multitime density matrix and to a Chapman-Kolmogoroff-like condition on the conditional density matrix.

• Received 15 January 1968

DOI:https://doi.org/10.1103/PhysRev.172.350