Abstract
According to Smolin’s taxonomy, time is either emergent in some limit a la string-theory, it’s an illusion a la Barbour/Wheeler-Dewitt or it’s fundamental a la his new Heraclitean evolving laws approach. We offer a fourth possibility regarding the nature of time, i.e., time as part of a fundamental (pregeometric) regime wherein the notions of space, time and matter are co-defined and co-determining such that we ultimately speak of RaumZeitMaterie (to borrow from Weyl). This pregeometric formalism is motivated by, and inextricably linked to, our Relational Blockworld (RBW) interpretation of quantum mechanics and paints a picture of “spacetimematter” more integrated than that of general relativity (and, a fortiori, quantum physics). In order to generate a discrete action for the transition amplitude at the basis of physics, we propose a self-consistency criterion (SCC) in the context of discrete graph theory a la Wise; the SCC is our counterpart to “quasiseparability” per Albrecht and Iglesias. To do this, we use graph theory and identify the invariant core of the discrete action which we call the “actional.” We then construct the source-free portion of the actional consistent with its counterpart for coupled harmonic oscillators on the graph. Defining the discrete source relationally via links of the graph then suggests an SCC fundamental to the discrete action of the transition amplitude, or what we call the "symmetry amplitude." As predicted by Toffoli, our basis for the action (SCC -> actional) results from a mathematical tautology, viz., the topological maxim “the boundary of a boundary is zero,” which already guarantees the consistency of fields and divergence-free sources in general relativity, classical electromagnetism and quantum electrodynamics. We then explain how RBW holds out the promise of unification that treads a middle path between the timelessness of Barbour and the fundamentality of time in Smolin.
Mark Stuckey