Quantifying Relations as a Foundation for Physics

Quantum theory possesses remarkable mathematical features whose physical origin remains undetermined. The elucidation of the origin of these features is key to better understanding the physical meaning of quantum theory, to taking us closer to the deep structure of physical reality, and to guiding us in the building up of a theory of quantum gravity. Over the last several years, the authors have developed a new methodology by which to reveal the general structures in physical theories, the key idea of which is to derive a calculus by quantifying over the elements of a logic. Applying this to quantum theory, we have identified an experimental logic and shown that, by pair-valued quantification of this logic, it is possible to derive Feynman’s rules of quantum theory. We have thereby shown that many of the key mathematical features of quantum theory (particularly its use of complex numbers) arise naturally. More recently, we have applied this methodology to derive a calculus of events in space and time by quantifying a causal set of events, thereby showing that Minkowski space-time structure is a special case of a general mathematical structure. In this project, we seek to build upon these innovative findings in three ways. First, we will extend our reconstruction to derive the finite- and infinite-dimensional von Neumann quantum formalism, and will seek a deeper understanding of the physical origin of the pair-valued representation via Hurwitz's theorem. Second, we shall investigate the implications of our reconstruction for the interpretation of quantum theory, and then attempt to develop a new interpretation on the basis of the assumption that the future is not causally determined by the present. Third, we shall investigate the degree to which the geometry of space can be derived from the quantification of causal sets, and investigate the derivation of the Dirac equation via the Feynman checkerboard model within our causal framework.