On the Axioms of Causal Set Theory

TL;DR

This paper proposes improvements to causal set theory in discrete gravity, emphasizing a nontransitive causal relation and background-independent quantum histories, leading to new dynamical equations for quantum spacetime.

Contribution

It introduces a nontransitive causal relation, relation space methods, and a background-independent histories approach, advancing causal set theory and discrete quantum gravity.

Findings

Development of a nontransitive causal relation framework

Introduction of relation space methods for causal structure analysis

Derivation of causal Schrödinger-type equations for quantum spacetime

Abstract

This paper offers suggested improvements to the causal sets program in discrete gravity, which treats spacetime geometry as an emergent manifestation of causal structure at the fundamental scale. This viewpoint, which I refer to as the causal metric hypothesis, is summarized by Rafael Sorkin's phrase, "order plus number equals geometry." Proposed improvements include recognition of a generally nontransitive causal relation more fundamental than the causal order, an improved local picture of causal structure, development and use of relation space methods, and a new background-independent version of the histories approach to quantum theory. Besides causal set theory, \`a la Bombelli, Lee, Meyer, and Sorkin, this effort draws on Isham's topos-theoretic framework for physics, Sorkin's quantum measure theory, Finkelstein's causal nets, and Grothendieck's structural principles. This approach circumvents undesirable structural features in causal set theory, such as the permeability of maximal antichains, studied by Major, Rideout, and Surya, and the configuration space pathology arising from the asymptotic enumeration of Kleitman and Rothschild. The paper culminates in the theory of co-relative histories and kinematic schemes, combining the causal metric hypothesis, the histories approach to quantum theory, and Grothendieck's relative viewpoint. This leads to the derivation of causal Schr\"odinger-type equations as dynamical laws for discrete quantum spacetime.