Below is a paraphrase from the preface of my upcoming Springer book, which gives an idea of what this research project is about. The book itself focuses specifically on discrete causal theory.
Three outstanding problems of modern theoretical physics are the following:
1. Description of fundamental spacetime structure.
2. Quantum-theoretic description of gravitation.
3. Unification of physical law.
These problems are deeply intertwined. Since Einstein’s discovery of general relativity a century ago, gravity has been understood to be structural in nature, rather than a “force,” in the usual sense of the word, and it is therefore difficult to imagine the development of a successful approach to quantum gravity without simultaneous acquisition of a deeper structural understanding of spacetime. Quantum theory, meanwhile, though expressed and interpreted in a variety of different ways, is believed by most serious theorists to represent a fundamental aspect of nature; hence, any successful unification of physical law is expected to be quantum-theoretic. It seems, then, that two common ingredients necessary for the solution of these three problems are suitable structural notions, and a suitable approach to quantum theory. Discrete causal theory attempts to help identify and develop these two ingredients. The most important aspects of the theory may be summarized in a few short phrases. Cause-and-effect relationships between pairs of events are taken to be the fundamental building blocks of structure; these events are assumed to form a discrete family rather than a continuum; and quantum theory is formulated as an abstraction and adaptation of Feynman’s path summation approach.
At a formal level, discrete causal theory models spacetime via structured sets, called directed sets, whose elements represent events, equipped with special binary relations, whose elements represent causal influences between pairs of events. The term “directed set” does not have the same meaning here as in category theory. Ambitious versions of the theory attempt to treat “particles and fields” as aspects of the same directed structure, leading to a simple unified picture of nature at the fundamental scale. The theory may also be generalized to admit the possibility of causal relationships between structural components more complicated than individual events. Discrete causal theory has deep historical roots reaching back to the ancient Greeks, and was foreseen to some degree by Riemann, and even by Leibniz. However, Einstein’s relativity, the digital information theory of Shannon and others, the path summation approach to quantum theory pioneered by Feynman, and an assortment of modern mathematical tools from algebra, order theory, and graph theory, are all more-or-less necessary to create a natural context for the theory.