  In this work we present the formal background used to develop the methods
used in earlier works to extend the truncated Wigner representation of quantum
and atom optics in order to address multi-time problems. The truncated Wigner
representation has proven to be of great practical use, especially in the
numerical study of the quantum dynamics of Bose condensed gases. In these
cases, it allows for the simulation of effects which are missed entirely by
other approximations, such as the Gross-Pitaevskii equation, but does not
suffer from the severe instabilities of more exact methods. The numerical
treatment of interacting many-body quantum systems is an extremely difficult
task, and the ability to extend the truncated Wigner beyond single-time
situations adds another powerful technique to the available toolbox. This
article gives the formal mathematics behind the development of our "time-Wigner
ordering" which allows for the calculation of the multi-time averages which are
required for such quantities as the Glauber correlation functions which are
applicable to bosonic fields.
