Bayesian Renormalization

In this note we present a fully information theoretic approach to renormalization inspired by Bayesian statistical inference, which we refer to as Bayesian Renormalization. The main insight of Bayesian Renormalization is that the Fisher metric defines a correlation length that plays the role of an emergent RG scale quantifying the distinguishability between nearby points in the space of probability distributions. This RG scale can be interpreted as a proxy for the maximum number of unique observations that can be made about a given system during a statistical inference experiment. The role of the Bayesian Renormalization scheme is subsequently to prepare an effective model for a given system up to a precision which is bounded by the aforementioned scale. In applications of Bayesian Renormalization to physical systems, the emergent information theoretic scale is naturally identified with the maximum energy that can be probed by current experimental apparatus, and thus Bayesian Renormalization coincides with ordinary renormalization. However, Bayesian Renormalization is sufficiently general to apply even in circumstances in which an immediate physical scale is absent, and thus provides an ideal approach to renormalization in data science contexts. To this end, we provide insight into how the Bayesian Renormalization scheme relates to existing methods for data compression and data generation such as the information bottleneck and the diffusion learning paradigm.Comment: 20 pages, no figures. V2: Citation format fixed, references adde

BRST Cohomology is Lie Algebroid Cohomology

In this paper we demonstrate that the exterior algebra of an Atiyah Lie algebroid generalizes the familiar notions of the physicist's BRST complex. To reach this conclusion, we develop a general picture of Lie algebroid morphisms as commutative diagrams between algebroids preserving the geometric structure encoded in their brackets. We illustrate that a necessary and sufficient condition for such a diagram to define a morphism is that the two algebroids possess gauge-equivalent connections. This observation indicates that the set of Lie algebroid morphisms should be regarded as equivalent to the set of local diffeomorphisms and gauge transformations. Moreover, a Lie algebroid morphism being a chain map in the exterior algebra sense ensures that morphic algebroids are cohomologically equivalent. The Atiyah Lie algebroids derived from principal bundles with common base manifolds and structure groups may therefore be divided into equivalence classes of morphic algebroids. Each equivalence class possesses a representative which we refer to as the trivialized Lie algebroid, and we show that the exterior algebra of the trivialized algebroid gives rise to the BRST complex. We conclude by illustrating the usefulness of Lie algebroid cohomology in computing quantum anomalies. In particular, we pay close attention to the fact that the geometric intuition afforded by the Lie algebroid (which was absent in the naive BRST complex) provides hints of a deeper picture that simultaneously geometrizes the consistent and covariant forms of the anomaly. In the algebroid construction, the difference between the consistent and covariant anomalies is simply a different choice of basis.Comment: 24 pages, 1 figure, LaTe

Abstracts from the 8th International Conference on cGMP Generators, Effectors and Therapeutic Implications

This work was supported by a restricted research grant of Bayer AG