Framework for the quantum mechanical sum of possibilities and meaning for field theory and gravity

In quantum mechanics, the measurable 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 of 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 nonrelativistic 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.