Symmetries at the origin of hierarchical emergence

TL;DR

This paper demonstrates that hierarchical emergence in complex systems can be fundamentally explained by underlying symmetries, linking physical, informational, and belief structures through group theory and statistical mechanics.

Contribution

It introduces a unified framework showing how symmetries induce hierarchical structures in both physical processes and Bayesian beliefs, bridging objective and epistemic levels.

Findings

Emergent macroscopic levels are organized by symmetry subgroups.

Symmetries shape Bayesian belief hierarchies and autonomous updates.

Familiar macroscopic quantities naturally arise from symmetries in models like Hopfield networks.

Abstract

Many systems of interest exhibit nested emergent layers with their own rules and regularities, and our knowledge about them seems naturally organised around these levels. This paper proposes that this type of hierarchical emergence arises as a result of underlying symmetries. By combining principles from information theory, group theory, and statistical mechanics, one finds that dynamical processes that are equivariant with respect to a symmetry group give rise to emergent macroscopic levels organised into a hierarchy determined by the subgroups of the symmetry. The same symmetries happen to also shape Bayesian beliefs, yielding hierarchies of abstract belief states that can be updated autonomously at different levels of resolution. These results are illustrated in Hopfield networks and Ehrenfest diffusion, showing that familiar macroscopic quantities emerge naturally from their symmetries. Together, these results suggest that symmetries provide a fundamental mechanism for emergence and support a structural correspondence between objective and epistemic processes, making feasible inferential problems that would otherwise be computationally intractable.