Dimensional regularization and renormalization of QED

Abstract

We give an x-space definition of dimensional regularization suited to the tree expansion method of renormalization. We apply the dimensionally regularized tree expansion to QED, obtaining sharp bounds on the size of a renormalized graph. Subtractions are made with the Lagrangian counterterms of the tree expansion, not by minimal subtraction techniques, and so do not entail a knowledge of the meromorphic structure of a graph as a function of dimension. This renormalization procedure respects the Ward identities, and the counterterms required are gauge invariant.

Euclidean Asymptotic Expansions of Green Functions of Quantum Fields (I) Expansions of Product of Singular Functions

“…It should be noted that one of the major difficulties in studies of asymptotic expansions of Feynman diagrams 4 The better known rigorous definition of the dimensional regularization is in terms of the αparametric representation [34] while our reasoning is essentially based on momentum space picture. Formal constructions of dimensional regularization in terms of position/momentum representations exist [35], [36], but the original definition of [33] is the most useful one in applications (which means, by the way, that it is this definition that should be a preferred subject of investigation; indeed, it is not difficult to adapt it for the purposes of formal proofs [37], but an in-depth discussion of this point goes beyond the scope of the present publication).…”

“…The regularization independent aspects of the original derivation have been exhaustively discussed in [20], [21] and [22]. Various ways to rigorously treat dimensional regularization without parametric representations were discussed in [35], [36], although not much practical understanding is added thereby to the heuristic treatment of [33]. Anyway, the variety of uses of dimensional regularization in practical calculations is such that there is little hope that everything that is being done will ever be "rigorously proved".…”

Finite Tsallis Gravitational Partition Function for a System of Galaxies

“…So, in this letter we will use the generalization of the dimensional regularization (GDR) [6,7,8,9,10,11] to obtain a finite or divergence free gravitational partition function. This method generalize the dimensional regularization of Bollini and Giambiagi [14,15,16,17,12,13] It may be noted that the entropy obtained from the gravitation partition function has been used to analyze the clustering of galaxies. This has been done by relating the entropy of the system of galaxies to the clustering parameter, and this can in turn be related to the observations [18,19,20,21,22].…”

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

“…It can be viewed as a particular 'analytic regularization', as introduced by Speer in the context of BPHZ-renormalization long ago [Spe71], and applied to EG renormalization by Hollands [Hol08]. A different approach had been taken by Rosen and Wright [RW90]: they implement dimensional regularization in x-space by making replacements on the level of the position space Feynmann rules. In particular, the spacetime coordinate x is replaced by X = (x, x), where x is a formal parameter corresponding to the "integration over the complex dimension".…”