A Perspective on Regularization and Curvature

Abstract

A global connection on the Connes Marcolli renormalization bundle relates $\beta$-functions of a class of regularization schemes by gauge transformations, as well as local solutions to $\beta$-functions over curved space-time.Comment: As publishe

Geometrically Relating Momentum Cut-Off and Dimensional Regularization

“…It remains to check that β σ defines a one to one correspondence between G(A ) and g(A ) for each σ. This is just an extension of the proof of the similar statement in [9,1].…”

“…Following [7], I call this connection ϕ * ω. Following [1], I show that these connections are defined by β σ , . The physical β functions for a regularization scheme appear as pullbacks along specific sections.…”

“…All calculations in this paper are done in Euclidean space, all integrals have been Wick rotated. These integrals, (1), are generally divergent as written. The process of regularization and renormalization extracts physical, finite values from these divergent integrals.…”

“…Recent literature has emerged geometrically describing this process of renormalization and regularization for a QFT in which the perturbative β function for dimensional regularization is defined by a connection on a renormalization bundle in which different regularization schemes correspond to sections [5,6,7,8,9,1]. I extend the analysis in these papers in two directions.…”

The GLOBAL Β-Functions FROM SOLUTIONS OF DYSON–SCHWINGER EQUATIONS

“…1 Start with an equation DSE with the associated Hopf algebra H DSE . The target space A for characters of H DSE which is a commutative algebra encodes the regularization prescription.…”