The internal description of a causal set: What the universe looks like from the inside

TL;DR

This paper introduces an algebraic framework for representing the causal structure of discrete spacetime using evolving sets, enabling new insights into the universe's internal causal description and potential quantum extensions.

Contribution

It develops a novel algebraic method to encode causal information via evolving sets and explores their quantum generalization, offering a new perspective on causal set theory.

Findings

Evolving sets obey a Heyting algebra with a non-standard complement

The method allows extracting causal structure from algebraic calculations

Potential for formulating a causal quantum theory based on evolving sets

Abstract

We describe an algebraic way to code the causal information of a discrete spacetime. The causal set C is transformed to a description in terms of the causal pasts of the events in C. This is done by an evolving set, a functor which to each event of C assigns its causal past. Evolving sets obey a Heyting algebra which is characterised by a non-standard notion of complement. Conclusions about the causal structure of the causal set can be drawn by calculating the complement of the evolving set. A causal quantum theory can be based on the quantum version of evolving sets, which we briefly discuss.