Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. III. A Generalized Wick Theorem and Multitime Mapping

The new $c$-number calculus for functions of noncommuting operators, developed in Paper I and employed in Paper II to formulate a general phase-space description of boson systems, deals with situations involving equal-time operators only. In the present paper extensions are presented for the treatment of problems involving boson operators at two or more instants of time. The mapping of time-ordered products onto $c$-number functions is studied in detail. The results make it possible to evaluate time-ordered products of boson operators by phase-space techniques. The usual Wick theorem for boson systems is obtained as a special case of a much more general theorem on time ordering. Our method of derivation appears to provide the first direct proof of Wick's theorem as well as a clear insight into its true meaning. A closed expression is also obtained for the time-evolution operator in terms of the solution of the $c$-number differential equation for the phase-space equivalent of this operator. The new calculus is also applied to the problem of evaluating normally ordered time-ordered, and also the antinormally ordered time-ordered, correlation functions.