Hamiltonian Renormalisation: A Categorical Perspective
M. Rodriguez Zarate
TL;DR
This work formulates Hamiltonian renormalisation (HR) in a category-theoretic framework to connect functional and lattice renormalisation. It introduces two categories, Seq and Func, built from UV resolution spaces with Dirichlet and Haar embeddings, and develops functorial relations between them via the Dirichlet embedding and its adjoint, clarifying how discretisation choices influence the HR flow. The paper analyzes the convergence of the HR flow for the $U(1)^3$ model in 3+1 Euclidean gravity under different embedding combinations, showing that Dirichlet-based constructions can reach a fixed point while Haar-based approaches yield more intricate flows. Overall, the categorical perspective provides a principled, axiomatic toolkit to compare discretisation strategies, identify equivalences, and guide the selection of embeddings and derivatives in HR.
Abstract
We present a categorical formulation of the Hamiltonian renormalisation programme for quantum field theories, establishing a systematic bridge between functional and lattice renormalisation. To this end, we introduce two categories, $Seq$ and $Func$, whose objects correspond to resolution spaces at different ultraviolet scales, and whose morphisms encode embeddings, projections, coarse-graining maps, and discrete derivatives. Focusing on Dirichlet-type embeddings, we construct the corresponding subcategories $Seq_D$, $Func_D$ and prove that the embedding and its adjoint define functors between them. Furthermore we revisit and extend the analysis of the convergence rate to the fixed point for the couplings of the $U(1)^3$ model for $3+1$ Euclidean quantum gravity, analysing different combinations of Haar and Dirichlet embeddings.