Semi-classical locality for the non-relativistic path integral in configuration space

Abstract

In a previous paper, we have put forward an interpretation of quantum mechanics based on a non-relativistic, Lagrangian 3+1 formalism of a closed Universe $M$, existing on timeless configuration space $\mathcal{M}$. However, not much was said there about the role of locality, which was not assumed. This paper is an attempt to fill that gap. To deal with the challenges gauge symmetries may pose to a good definition of locality, I start by demanding symmetries to have an action on $\mathcal{M}$ so that the quotient wrt the symmetries respects certain factorizations of $\mathcal{M}$. These factorizations are algebraic splits of $\mathcal{M}$ into sub-spaces $\mathcal{M}=\bigoplus_i \mathcal{M}_{O_i}$-- each factor corresponding to a physical sub-region $O_i$. This deals with kinematic locality, but locality in full can only emerge dynamically, and is not postulated. I describe conditions under which it can be said to have emerged. The dynamics of $O$ is independent of its complement, $M-O$, if the projection of extremal curves on $M$ onto the space of extremal curves intrinsic to $O$ is a surjective map. This roughly corresponds to $e^{i\hat{H}t}\circ \mathsf{pr}_{\mbox{O}}= \mathsf{pr}_{\mbox{ O}}\circ e^{i\hat{H}t}$, where $\mathsf{pr}_{\mbox{ O}}:\mathcal{M}\rightarrow \mathcal{M}_O^{\partial O}$ is a linear projection. This criterion for locality can be made approximate -- an impossible feat had it been already postulated -- and it can be applied for theories which do not have hyperbolic equations of motion, and/or no fixed causal structure. When two regions are \emph{mutually independent} according to the criterion proposed here, the semi-classical path integral kernel factorizes, showing cluster decomposition which is what one would like to obtain from any definition of locality.