Dimensional regularization in position space and a Forest Formula for Epstein-Glaser renormalization

Abstract

We reformulate dimensional regularization as a regularization method in position space and show that it can be used to give a closed expression for the renormalized time-ordered products as solutions to the induction scheme of Epstein-Glaser. This closed expression, which we call the Epstein-Glaser Forest Formula, is analogous to Zimmermann's Forest Formula for BPH renormalization. For scalar fields the resulting renormalization method is always applicable, we compute several examples. We also analyze the Hopf algebraic aspects of the combinatorics. Our starting point is the Main Theorem of Renormalization of Stora and Popineau and the arising renormalization group as originally defined by Stückelberg and Petermann.

An analytic regularisation scheme on curved space–times with applications to cosmological space–times

“…in [DFKR14] which efficiently encodes the full combinatorics of Feynman diagrams in a compact form. Namely, the time-ordered product of n local functionals V 1 , .…”

“…A regularisation of the Feynman propagator similar to the one above has recently been discussed in [Da15]. In this work, we shall combine the analytic regularisation of the Feynman propagator with the minimal subtraction scheme encoded in a forest formula of the kind discussed in [Ho10,Ke10,DFKR14] in order to obtain a time-ordered product which satisfies the causal factorisation property, i.e. a product which is indeed "time-ordered".…”

“…The renormalisation scheme we propose is inspired by the works [Ke10,DFKR14] which deal with perturbative QFT in Minkowski spacetime. In these works, the authors introduce an analytic regularisation of the position-space Feynman propagator in Minkowski spacetime which is similar to the one discussed in [BG72].…”

“…In order to extend the scheme proposed in [DFKR14] to curved spacetimes, and motivated by [BG72] and by the form of Feynman propagators on curved spacetimes, we introduce an analytic regularisation ∆ where u, v and w are the so-called Hadamard coefficients and σ is 1/2 times the squared geodesic distance. This analytic regularisation, namely the construction of certain distributions by means of powers of the geodesic distance, is reminiscent of the use of Riesz distributions to define advanced and retarded Greens functions on Minkowski spacetime.…”

Massive vector bosons: Is the geometrical interpretation as a spontaneously broken gauge theory possible at all scales?

“…In order that the limit ζ → 0 exists, we subtract from the Laurent series t ζ its principle part. According to [DFKR14,Corollary 4.4] the term ∼ ζ 0 ("minimal subtraction") is an admissible extension t M of t • :…”