On the Axioms of Causal Set Theory Benjamin F. Dribus Louisiana State University [email protected] November 8, 2013 Abstract 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 el- ements 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 implic- itly, 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 formula- tion 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, includ- ing 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 re- lations 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 scheme- theoretic 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 background- independent version of quantum causal theory. The backbone of this approach is the theory of co- relative 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 problem of permeability of maximal antichains, leading to the derivation of causal Schr¨odinger-type equations describing quantum spacetime dynamics in the discrete causal context. 1 Contents 1 Introduction 3 1.1 Overview of the Causal Sets Program . .3 1.2 Purposes; Viewpoint and Style; Intended Audience . .8 1.3 Underlying Natural Philosophy . .9 1.4 Notation and Conventions; Figures . 11 1.5 Summary of Contents; Outline of Sections . 12 2 Axioms and Definitions 20 2.1 The Causal Metric Hypothesis . 20 2.2 The Axioms of Causal Set Theory . 22 2.3 Acyclic Directed Sets; Directed Sets; Multidirected Sets . 25 2.4 Chains; Antichains; Irreducibility; Independence . 28 2.5 Domains of Influence; Predecessors and Successors; Boundary and Interior . 30 2.6 Order Theory; Category Theory; Influence of Grothendieck . 33 3 Transitivity, Independence, and the Causal Preorder 38 3.1 Independent Modes of Influence . 38 3.2 Six Arguments that Transitive Binary Relations are Deficient . 41 3.3 The Causal Preorder . 44 3.4 Transitive Closure; Skeleton; Degeneracy; Functoriality . 44 4 Interval Finiteness versus Local Finiteness 47 4.1 Local Conditions; Topology . 47 4.2 Interval Finiteness versus Local Finiteness . 53 4.3 Relative Multdirected Sets over a Fixed Base . 57 4.4 Eight Arguments against Interval Finiteness and Similar Conditions . 61 4.5 Six Arguments for Local Finiteness . 63 4.6 Hierarchy of Finiteness Conditions . 63 5 The Binary Axiom: Events versus Elements 68 5.1 Relation Space over a Multidirected Set . 69 5.2 Power Spaces . 80 5.3 Causal Path Spaces . 85 5.4 Path Summation over a Multidirected Set . 93 6 Quantum Causal Theory 97 6.1 Quantum Preliminaries; Iteration of Structure; Co-Relative Histories . 98 6.2 Abstract Quantum Causal Theory via Path Summation . 107 6.3 Schr¨odinger-Type Equations in Quantum Causal Theory . 113 6.4 Kinematic Schemes . 117 7 Conclusions 127 7.1 New Axioms, Perspectives, and Technical Methods . 127 7.2 Omitted Topics and Future Research Directions . 131 7.3 Acknowledgements; Personal Notes . 137 A Index of Notation 139 References 147 2 1 Introduction Causal sets are discrete order-theoretic models of classical spacetime. A causal set is a set C, assumed to be countable, whose elements correspond to spacetime events, together with a binary relation on C, satisfying the additional axioms of transitivity, irreflexivity, and interval finiteness. The binary relation defines an interval finite partial order1 on C, called the causal order, with the physical≺ interpretation that x y in C if and only if the event represented by x exerts causal influence on the event represented≺ by y. The physical interpretation of C is completed by fixing a discrete measure2 µ on C, assigning to each≺ subset of C a volume equal to its number of elements in fundamental units, up to Poisson-type fluctuations.3 The role of and µ in modeling classical spacetime is summarized by the phrase, \order plus number equals geometry," coined by Rafael Sorkin, the foremost architect and advocate of causal set theory. This phrase≺ represents a special case of what I refer to as the causal metric hypothesis (CMH), 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. Approaches to fundamental physics involving some form of the causal metric hypothesis may be collectively referred to as causal theory. Besides causal set theory, these include other discrete causal theories, such as causal dynamical triangulations and causal nets, as well as theories involving interpolative causal models of spacetime, such as domain theory. The purpose of this paper is to offer potentially critical improvements to the causal set program, and to causal theory in general. This effort is based on the conviction that causal theory is among the best-motivated existing approaches to the outstanding problems of fundamental physics, and that causal set theory is perhaps the cleanest and best-balanced existing version of causal theory. Despite important conceptual distinctions, and critical technical differences, between the existing formulation of causal set theory and the approach I offer in this paper, the two are close enough to be considered part of the same broad program. For example, they are closer to each other than they are to the other versions of causal theory mentioned above. In this introductory section, I sketch a broad conceptual context for the more specific material to follow. Section 1.1 is a brief overview of causal set theory, describing its origins, outlining its principal methods, and providing a glimpse of its current state of development. I have included references to more thorough treatments, carefully selected for reliability and quality of exposition. Section 1.2 outlines in more detail the purposes of this paper, its method and approach, and its intended audience. Section 1.3 introduces general principles of natural philosophy used throughout the paper. Section 1.4 describes the notation and conventions of the paper in general terms; appendix A is much more thorough. Section 1.5 provides a hyperlinked topical outline of the succeeding material, more detailed than that provided by the table of contents above. 1.1 Overview of the Causal Sets Program Historical Antecedents. The study of causality long predates any formal mathematical notion of order, and it is important to resist viewing relevant early developments in a teleological sense, as mere steps along a path toward present convention. For example, Zeno's dichotomy paradox, proposed around 450 B.C., is often described today in terms of the subdivision of continuum intervals, and \resolved" by the convergence of geometric series. However, this particular version of Zeno's paradox was originally stated in a physical context, in terms of sequences of events, with the physical issue at stake being whether or not fundamental causal structure is interpolative.