Implications of Einstein-Weyl Causality on Quantum Mechanics

A fundamental physical principle that has consequences for the topology of space-time is the principle of Einstein-Weyl causality. This also has quantum mechanical manifestations. Borchers and Sen have rigorously investigated the mathematical implications of Einstein-Weyl causality and shown the denumerable space-time Q$^{\mathrm{2}}$ would be implied. They were left with important philosophical paradoxes regarding the nature of the physical real line E, e.g., whether E $=$ R, the real line of mathematics. In order to remove these paradoxes an investigation into a constructible foundation is suggested. We have pursued such a program and find it indeed provides a dense, denumerable space-time and, moreover, an interesting connection with quantum mechanics. We first show that this constructible theory contains polynomial functions which are locally homeomorphic with a dense, denumerable metric space R* and are inherently quantized. Eigenfunctions governing fields can then be effectively obtained by computational iteration. Postulating a Lagrangian for fields in a compactified space-time, we get a general description of which the Schrodinger equation is a special case. From these results we can then also show that this denumerable space-time is relational (in the sense that space is not infinitesimally small if and only if it contains a quantized field) and, since Q$^{\mathrm{2}}$ is imbedded in R*$^{\mathrm{2}}$, it directly fulfills the strict topological requirements for Einstein-Weyl causality. Therefore, the theory predicts that E $=$ R*.