Framework for the Quantum Mechanical Sum of Possibilities and Meaning for Field Theory and Gravity

Abstract

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.