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