Abstract
A geometric formulation of Wilson’s exact renormalisation group is presented based on a gauge invariant ultraviolet regularisation scheme without the introduction of a background field. This allows for a manifestly background independent approach to quantum gravity and gauge theories in the continuum. The regularisation is a geometric variant of Slavnov’s scheme consisting of a modified action, which suppresses high momentum modes, supplemented by Pauli–Villars determinants in the path integral measure. An exact renormalisation group flow equation for the Wilsonian effective action is derived by requiring that the path integral is invariant under a change in the cutoff scale while preserving quasi-locality. The renormalisation group flow is defined directly on the space of gauge invariant actions without the need to fix the gauge. We show that the one-loop beta function in Yang–Mills and the one-loop divergencies of General Relativity can be calculated without fixing the gauge. As a first non-perturbative application we find the form of the Yang–Mills beta function within a simple truncation of the Wilsonian effective action.
Highlights
Background independent exact renormalisationKevin Falls1,2,a AbstractA geometric formulation of Wilson’s exact renormalisation group is presented based on a gauge invariant ultraviolet regularisation scheme without the introduction of a background field
An exact renormalisation group flow equation for the Wilsonian effective action is derived by requiring that the path integral is invariant under a change in the cutoff scale while preserving quasi-locality
In this work we have presented a manifestly background independent formalism which can be applied to both quantum gravity and gauge theories to address perturbative and nonperturbative questions within these fields
Summary
We will construct a regularised path integral where we have in mind regularising a theory with a classical action I which is second order in spacetime derivatives. We will adopt the geometric approach to gauge theories [84,85,86] by considering the space of all field configurations as an infinite dimensional manifold. For gravity the fields are components of spacetime metric and the gauge transformations are diffeomorphisms whereas for Yang–Mills the fields are the gauge fields and the gauge transformations are local SU (N ) transformations. We will use the geometric approach to construct the path integral over /G in a manifestly gauge invariant manner without having to fix the gauge and we will never introduce any gauge breaking terms into the theory. We will assume that spacetime has no boundaries such that we can freely integrate by parts and drop the boundary terms
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