Hamiltonian Renormalisation: Categorical Formulation Establishes Systematic Bridge Between Functional and Lattice Renormalisation

Hamiltonian renormalisation offers a powerful approach to understanding field theories, yet connecting its functional and lattice formulations remains a significant challenge. Now, M. Rodriguez Zarate from the Institute for Theoretical Physics III at FAU Erlangen-Nürnberg, and colleagues, present a novel categorical framework that systematically bridges these two perspectives. Their work introduces mathematical categories where objects represent different scales of physical phenomena and connections between them encode crucial processes like coarse-graining and differentiation. This innovative approach not only clarifies the relationship between functional and lattice renormalisation, but also provides new tools for analysing the convergence of models towards fixed points, ultimately advancing our understanding of fundamental physical interactions.

Categorical Renormalisation and Ultraviolet Resolution Spaces

Scientists have developed a novel categorical framework to systematically investigate Hamiltonian renormalisation, a programme designed to address ambiguities in quantum field theory and quantum gravity. This study pioneers a method for bridging functional and lattice renormalisation techniques by introducing two distinct categories, each representing resolution spaces at different ultraviolet scales. Morphisms within these categories encode essential transformations, including embeddings, projections, coarse-graining maps, and discrete derivatives, providing a precise mathematical language for analysing the process of removing high-energy degrees of freedom. Researchers focused on Dirichlet-type embeddings to construct subcategories and demonstrate that both the embedding and its adjoint define functors connecting these categories, establishing a rigorous relationship between different resolution scales.

The team revisited and extended previous analyses of convergence rates towards fixed points for model couplings in Euclidean space, meticulously comparing combinations of Haar and Dirichlet embeddings. To explore the impact of discretisation choices, scientists compared functional and discrete renormalisation schemes, projecting fields either onto Hilbert subspaces or discretising them on a lattice. The study further investigated the interplay between locality and quasi-locality, employing both Dirichlet and Haar kernels for projection and discretisation. Specifically, the team compared the performance of these kernels within the U(1)3 model for 3+1 quantum gravity, revisiting and extending prior analyses of convergence rates. Scientists constructed ultraviolet resolution spaces and their associated maps, defining embeddings, projections, inclusions, and discrete derivatives to specify data at varying resolutions. This categorical approach enables the identification and classification of key structural properties of discretisation and coarse-graining procedures, providing a conceptual toolkit to systematically understand how different choices affect the corresponding renormalisation flow.

Hamiltonian Renormalisation via Category Theory

Scientists have developed a categorical framework for Hamiltonian renormalisation, establishing a systematic connection between functional and lattice renormalisation techniques. This work introduces two categories, where objects represent resolution spaces at different ultraviolet scales and morphisms encode embeddings, projections, coarse-graining maps, and discrete derivatives. The team constructed subcategories and demonstrated that embeddings and their adjoints define functors between them, providing a rigorous mathematical structure for analysing field theories. Experiments involved defining resolution spaces based on partially ordered sets, which dictate the relationship between ultraviolet resolutions.

A key aspect of this work is the definition of sets that restrict real and momentum spaces to subspaces of finite resolution. The team showed that for any given resolution, a subspace can be defined within the Hilbert space, effectively projecting fields onto a finite resolution scale. Measurements confirm that these subspaces consist of functions with cutoff frequencies, providing a discrete representation of the field. Further analysis involved exploring alternative bases for these subspaces, specifically Haar and Dirichlet kernels, which exhibit improved locality properties compared to plane waves.

The team demonstrated that Dirichlet bases are spatially concentrated and smooth, crucial for handling derivatives in quantum field theory. Measurements reveal that both Haar and Dirichlet kernels are real-valued, simplifying calculations and improving the efficiency of the renormalisation process. This categorical formulation provides a conceptual toolkit for understanding how different discretisation and projection choices affect the renormalisation flow, offering a powerful new approach to analysing complex quantum field theories.

Categorical Renormalisation and Resolution Space Embeddings

This work establishes a categorical framework for Hamiltonian renormalisation, creating a systematic connection between functional and lattice renormalisation techniques. By introducing and analysing categories representing resolution spaces at different scales, the researchers demonstrate how embeddings, projections, and coarse-graining maps can be understood as functors between these categories. Specifically, they show that Dirichlet embeddings and their adjoints preserve the categorical structure within a defined sector, allowing for a rigorous classification of the linear maps essential to the renormalisation process. The team further investigated the convergence rate of the renormalisation flow for a specific model, examining the effects of combining Dirichlet and Haar embeddings. Their results highlight the benefits of this categorical perspective, offering a means to clarify equivalences and ambiguities inherent in different discretisation and projection choices. Future research directions include rigorously proving the adjoint nature of identified functors and incorporating the concept of inductive limits to fully address convergence to a fixed point, potentially offering a powerful tool for comparing and categorising various projection and embedding choices.