Hamiltonian Renormalisation: A Categorical Perspective

TL;DR

This paper develops a categorical framework for Hamiltonian renormalisation in quantum field theories, linking functional and lattice approaches, and analyzes convergence in a quantum gravity model.

Contribution

It introduces categorical structures to formalize Hamiltonian renormalisation and extends convergence analysis for a quantum gravity model.

Findings

Established categories for resolution spaces and embeddings

Constructed functors between subcategories for Dirichlet embeddings

Analyzed convergence rates for the $U(1)^3$ model in quantum gravity

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.