  In quantum mechanics, the measureable quantities of a given theory are
predicted by performing a weighted sum over possibilities. We show how to
arrange the possibilities into bundles such that the associated subsums can be
viewed as well-defined theories on their own right. These bundles are
submani$\textit{folds}$ of $\textit{possi}$bilities which we call possifolds.
We collect and prove some basic facts about possifolds. Especially, we show
that possifolds are ensembles of what in a certain broadly defined sense that
we explain can be regarded as soliton excitations (soliton-possifold
correspondence). We provide an outlook on some applications. Among other
things, we illustrate the use of the developed framework for the example of the
Lieb-Liniger model. It describes non-relativistic bosons with an attractive
interaction. We derive a dual theory describing the lowest-lying energy
excitation modes. While the standard Bogoliubov-approximation breaks down at
the critical point, our derived summation prescription stays regular. In the
Bogoliubov-limit we observe the summation to possess an enhanced symmetry at
this point while the summation cannot be ignored there. We finally provide a
glimpse on the restrictions black hole physics implies in this context for the
gravitational path integral.
