Semi-classical Locality for the Non-relativistic Path Integral in Configuration Space

In an accompanying paper Gomes (arXiv:1504.02818, 2015), 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 {Q}) of some field over 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. Locality in full can only emerge dynamically, and is not postulated. This new understanding of locality is based solely on the properties of extremal paths in configuration space. I do not demand locality from the start, as it is usually done, but showed conditions under which certain systems exhibit it spontaneously. In this way we recover semi-classical local behavior when regions dynamically decouple from each other, a notion more appropriate for extension into quantum mechanics. The dynamics of a sub-region O within the closed manifold M is independent of its complement, (M-O), if the projection of extremal curves on (mathcal {Q}) onto the space of extremal curves intrinsic to O is a surjective map. This roughly corresponds to (e^{ihat{H}t}circ mathsf {pr}_{mathrm{O}}= mathsf {pr}_{mathrm{O}}circ e^{ihat{H}t}), where (mathsf {pr}_{mathrm{O}}:mathcal {Q}rightarrow mathcal {Q}_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 mutually independent according to the criterion proposed here, the semi-classical path integral kernel factorizes, showing cluster decomposition which is the ultimate aim of a definition of locality.