Counterintuitive Scenarios in Discrete Gravity Without Quantum Effects or Causality Violations

In certain established approaches to quantum gravity, such as causal set theory and causal dynamical triangulations, discrete spacetime structure is taken to be a primary feature, not a secondary effect of “quantizing” a pre-existing classical continuum-based theory, as is done in approaches such as string theory and loop quantum gravity. For a priori discrete models, the full quantum theory is often obtained via some version of Feynman’s sum-over-histories approach, in which each “history” is a discrete object viewed as a classical spacetime. Counterintuitive physical scenarios such as Schrödinger’s cat or the grandfather paradox are typically associated with either quantum effects or causality violations, but we demonstrate that equally bizarre scenarios can arise at a purely classical level in the discrete causal context due to symmetry considerations. In particular, the graph-theoretic phenomenon of pseudosimilarity leads to situations in which alternative events occurring at physically distinguishable locations in the universe can cause different parts of the universe to “swap identities” in a fugue-like manner alien to continuum-based theories. This phenomenon is perhaps best understood as an extension of the relativity principle, which we call relativity of identity.