On the Axioms of Causal Set Theory

Causal set theory is a promising attempt to model fundamental spacetime structure in a discrete order-theoretic context via sets equipped with special binary relations, called causal sets. The elements of a causal set are taken to represent spacetime events, while its binary relation is taken to encode causal relations between pairs of events. Causal set theory was introduced in 1987 by Bombelli, Lee, Meyer, and Sorkin, motivated by results of Hawking and Malament relating the causal, conformal, and metric structures of relativistic spacetime, together with earlier work on discrete causal theory by Finkelstein, Myrheim, and ’t Hooft. Sorkin has coined the phrase, “order plus number equals geometry,” to summarize the causal set viewpoint regarding the roles of causal structure and discreteness in the emergence of spacetime geometry. This phrase represents a specific version of what I refer to as the causal metric hypothesis, which is the idea that the properties of the physical universe, and in particular, the metric properties of classical spacetime, arise from causal structure at the fundamental scale. Causal set theory may be expressed in terms of six axioms: the binary axiom, the measure axiom, countability, transitivity, interval finiteness, and irreflexivity. The first three axioms, which fix the physical interpretation of a causal set, and restrict its “size,” appear in the literature either implicitly, or as part of the preliminary definition of a causal set. The last three axioms, which encode the essential mathematical structure of a causal set, appear in the literature as the irreflexive formulation of causal set theory. Together, these axioms represent a straightforward adaptation of familiar notions of causality to the discrete order-theoretic context. Interval finiteness is often called local finiteness in the literature, an unfortunate misnomer. In this paper, I offer what I view as potentially critical improvements to causal set theory, including changes to the axioms, and new perspectives and technical methods. Abandoning continuum geometry introduces new types of behavior, such as irreducibility and independence of causal relations between pairs of events, inadequately modeled by conventional order theory. Transitive binary relations fail to resolve the subtleties of independent modes of influence between pairs of events. Interval finiteness permits locally infinite behavior incompatible with Sorkin’s version of the causal metric hypothesis, and imposes unjustified restrictions on the global structure of classical spacetime. Intertwined with both axioms is the use of causal intervals to study local spacetime properties, despite their failure to capture or isolate local causal structure. To address these issues, I propose to replace the axioms of transitivity, interval finiteness, and irreflexivity, with local finiteness, and, under conservative assumptions, acyclicity. I refer to a binary relation satisfying these new axioms as a locally finite causal preorder; its transitive closure is the familiar causal order. The resulting models, which I call locally finite directed sets, generalize both causal sets and Finkelstein’s causal nets. The resulting theory diverges significantly from existing approaches, particularly at the quantum level. This is due principally to a broader interpretation of directed structure, and more generally, multidirected structure, inspired by Grothendieck’s schemetheoretic approach to algebraic geometry. Entities more complex than spacetime events are viewed as elements of higher-level multidirected sets, in analogy to Isham’s topos-theoretic approach to quantum gravity, and Sorkin’s quantum measure theory. This viewpoint leads to a new backgroundindependent version of quantum causal theory. The backbone of this approach is the theory of corelative histories, an adaptation of the familiar histories approach to quantum theory. Co-relative histories serve as the “relations” of higher-level multidirected sets called kinematic schemes, via the principle of iteration of structure. Systematic use of relation space circumvents the causal set