Space-time-hap: a coordinate system for the multiverse and its application to show that free choice is observer-dependent

Abstract

This note proposes a paradigm and coordinate system that extends flat, four-dimensional Minkowski spacetime to a broader framework that identifies an event not only in space and in time, but also in terms of possible world, with a third category of coordinates called ``hap'' modelling contingency and counterfactuals. Semantically, hap is based on a Bohmian-like configuration space, in which initial conditions at a specific moment in time uniquely identify a trajectory. This framework can be used to support reasonings that rigorously distinguish between causal dependencies (through time), statistical dependencies (through space) and counterfactual dependencies (through hap). As an example, we use this framework to show that the assumption of free choice is not absolute, but rather depends on the chosen frame of reference: while Alice may see a choice made freely, which is formally a unilateral, single-coordinate translation in hap, Bob sitting in another reference frame might see this same choice not made freely and observe instead a translation in hap jointly across multiple coordinates, which indicates a counterfactual dependency. The framework is agnostic regarding which flavour of the De-Broglie-Bohm theory is considered, as it only makes general assumptions on the set of trajectories that are commonly accepted in the Bohmian mechanics community. It is also general enough to support reasoning on theories for which the separation between possible worlds might also depend on the frame of reference, i.e., two observers might disagree on whether two spacetimehap events happen in the same world or not.