Causal Sets and Generalizations

Abstract

This chapter describes properties of directed sets and multidirected sets, which supply the basic structural content of discrete causal theory. Mathematically, these objects are usually called directed graphs and directed multigraphs, respectively, but set-theoretic terminology seems preferable for physical applications, despite the fact that the term “directed set” has a more specific conventional meaning. Directed sets and multidirected sets may be viewed as generalizations of causal sets, which have been studied as discrete models of spacetime since the 1980s. One of the main tasks of this chapter is to begin to describe some of the ways in which the version of discrete causal theory developed in this book differs from causal set theory, and to explain the reasons for these differences. Section 3.1 describes the preliminaries of causal set theory, along with some historical details. Section 3.2 explains how Sorkin’s ansatz for causal sets, “order plus number equals geometry,” may be viewed as an early version of the classical causal metric hypothesis. Section 3.3 briefly outlines quantum causal set theory, which serves as a point of comparison for the version of discrete quantum causal theory developed in Part II. Section 3.4 describes some topics in causal set dynamics and phenomenology, with a focus on Sorkin and Rideout’s theory of sequential growth dynamics. Section 3.5 presents a formal axiomatic description of causal set theory. Section 3.6 introduces the more general classes of directed sets and multidirected sets necessary to describe the version of discrete causal theory developed in this book. Directed sets serve as models of classical spacetime, while multidirected sets supply the “higher-level” structure of configuration spaces of classical histories arising in the corresponding quantum theory. Section 3.7 describes basic structural notions for multidirected sets, such as chains, antichains, and domains of influence. Section 3.8 outlines useful analogies involving order theory and category theory, with emphasis on the contributions of Grothendieck. Section 3.9 explains why transitive binary relations, such as those used in causal set theory, are likely inadequate to model fundamental causal structure in the discrete setting. A few prominent physicists, such as David Finkelstein, have previously recognized this problem. Section 3.10 introduces the causal relation, which is a generally nontransitive binary relation that generates the transitive relation usually associated with causal structure in conventional contexts.