  A non-classical, non-quantum theory, or NCQ, is any fully consistent theory
that differs fundamentally from both the corresponding classical and quantum
theories, while exhibiting certain features common to both. Such theories are
of interest for two primary reasons. Firstly, NCQs arise prominently in
semi-classical approximation schemes. Their formal study may yield improved
approximation techniques in the near-classical regime. More importantly for the
purposes of this note, it may be possible for NCQs to reproduce quantum results
over experimentally tested regimes while having a well defined classical limit,
and hence are viable alternative theories. We illustrate an NCQ by considering
an explicit class of NCQ mechanics. Here this class will be arrived at via a
natural generalization of classical mechanics formulated in terms of a
probability density functional.
