The properties of a certain class of functions $f({\alpha }^{\prime }, {\beta }^{\prime }, {\gamma }^{\prime }, \cdots )$ of noncommuting observables $\alpha , \beta , \gamma , \cdots$ are investigated. They have all the usual properties of distribution functions, except that they are complex. They satisfy simple Markoff-type stochastic equations and permit the calculation of the expectation values unambiguously. Conversely, quantum theory can be formulated in terms of such distribution functions having the prescribed properties.

• Received 29 July 1957

DOI:https://doi.org/10.1103/PhysRev.108.565