just discovered (thanks to pointer from Igor Khavkine!) that what I thought now should exist has indeed already been developed: a genral theory of Feynman amplitudes recast from singular distributions on $M^n$ to smooth functions on Fulton-MacPherson-type compactifications of configuration spaces of points (“wonderful compactifications”). Have added the following to the References-section here:

An alternative to regarding propagators/time-ordered products/Feynman amplitudes as distributions of several variables with singularities at (in particular) coincident points, one may pullback these distributions to smooth functions on Fulton-MacPherson compactifications of configuration spaces of points and study renormalization in that perspective.

This approach was originally considered specifically for Chern-Simons theory in

- {#AxelrodSinger93} Scott Axelrod, Isadore Singer,
*Chern–Simons Perturbation Theory II*, J. Diff. Geom. 39 (1994) 173-213 (arXiv:hep-th/9304087)

which was re-amplified in

{#BottCattaneo97} Raoul Bott, Alberto Cattaneo, Remark 3.6 in

*Integral invariants of 3-manifolds*, J. Diff. Geom., 48 (1998) 91-133 (arXiv:dg-ga/9710001){#CattaneoMnev10} Alberto Cattaneo, Pavel Mnev, Remark 11 in

*Remarks on Chern-Simons invariants*, Commun.Math.Phys.293:803-836,2010 (arXiv:0811.2045)

A systematic development of perturbative quantum field theory from this perspective is discussed in

{#Berghoff14a} Marko Berghoff,

*Wonderful renormalization*, 2014 (pdf, doi:10.18452/17160){#Berghoff14b} Marko Berghoff,

*Wonderful compactifications in quantum field theory*, Communications in Number Theory and Physics Volume 9 (2015) Number 3 (arXiv:1411.5583)

For more see at *Feynman amplitudes on compactified configuration spaces of points*.

Still on that prop. establishing UV-regularization by counterterms:

Michael Dütsch kindly points out to me by private email that the afterthought of the proof, generalizing from one particular S-matrix to all of them, follows immediately by invoking again the main theorem. I have edited accordingly: paragraphs starting here

]]>I have considerably further expanded the proof of that prop..

In particular I made explicit (here) where the Hörmander topology in the definition of UV-cutoff (this def.) really enters.

]]>I have now spelled out the proof of UV-regularization via counterterms, this prop..

Hints for the idea this proof were offered DFKR 14, theorem A.1. I have tried to expand that out a little.

]]>I have been compiling more material (from separate entries that I have been writing, with some glue added) on the rigorous formulation of renormalization in causal perturbation theory; in the section:

Not done yet, but it is taking shape now.

]]>I am now beginning to spell out a comprehensive account at *renormalization*.

To begin with, I added statement and proof that renormalization of time-ordered products is inductively in the arity given by extension of the corresponding distributions to the diagonal: here

]]>I have expanded the Idea-sections at *renormalization* and *perturbative quantum field theory* and added more references.

I started writing a summary of the key steps in Kevin Costello’s discussion of renormalization. At *renormalization* in the section *Of theories in BV-CS form*:

The text still needs polishing and glue. You shouldn’t look at this right now if you don’t want to see an unfinished writeup. But I have to take a break now and get some dinner.

]]>at *renormalization* I made the Definition-section have three different subsections, a new one of which is now *Definition – Of theories in BV-CS form* on Kevin Costello’s apprach. On the other hand, so far this contains mostly just a pointer to his article.