The geometric β-function in curved space-time under operator regularization

Abstract

In this paper, I compare the generators of the renormalization group flow, or the geometric β-functions, for dimensional regularization and operator regularization. I then extend the analysis to show that the geometric β-function for a scalar field theory on a closed compact Riemannian manifold is defined on the entire manifold. I then extend the analysis to find the generator of the renormalization group flow to conformally coupled scalar-field theories on the same manifolds. The geometric β-function in this case is not defined.

A mathematical perspective on the phenomenology of non-perturbative Quantum Field Theory

“…In physical theories with strong couplings such as Hadron Physics, we should deal with hadrons such as protons and neutrons as the composite particles build up from quarks and gluons (as elementary particles under strong interaction). In general, QCD, as a nonabelian gauge theory with the symmetry group SU (3), has provided a modern understanding of the complicated nature of hadrons and nuclei where we study the strong interactions of quarks and gluons under confinement and chiral symmetry breaking. The appearance of nuclear weak force enables us to describe any change quark's flavor via W bosons.…”

“…This setting has been lifted onto a uni-versal Tannakian formalism where a renormalizable Quantum Field Theory is studied via a category of geometric objects which can be recovered by a category of finite dimensional representations of the affine group scheme G Φ := Hom(H FG (Φ), −). [2,3,16,18,19,20,21,24,22,26,33,34,37,65,66,87,95,96,97,98,125,155,156,168,174] Perhaps the most fundamental result in this direction would be the discovery of a very deep interrelationship between Feynman integrals and theory of motives in Algebraic Geometry where a motivic renormalization machinery has been formulated to deal with divergencies in the language of Picard-Fuchs equations and other powerful tools.…”

“…It studies the hadronic interactions involving quarks and gluons at both long and short distances. Its symmetry group is SU (3) where it includes N f family of quarks ψ i f and gluons A i µ . Some experimental evidences inform us that at a critical temperature around T c ≈ 170 MeV QCD matter undergoes a deconfining phase transition into quark-gluon plasma.…”