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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJan 29th 2018
   • (edited Jan 29th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have written out the rigorous formulation of _[[renormalization group flow]]_ and _[[running coupling constants]]_ (currently both redirect to the same entry). This is an exegesis of _[Brunetti-Dütsch-Fredenhagen 09, section 4.2 and 5.1](https://ncatlab.org/nlab/show/renormalization+group+flow#BrunettiDuetschFredenhagen09)_ with clues from _[Dütsch 18, section 3.5.3](https://ncatlab.org/nlab/show/renormalization+group+flow#Duetsch18)_; but I have tried to disentangle, in the writeup, the general principle from the specific case where the flow is induced by scaling transformations. (The latter is going to be written out at _[[Gell-Mann-Low renormalization cocycle]]_, but don't look at that entry just yet). Accordingly I tried to streamline presentation and notation, to make it all clear at a glance what is going on... but of course this may be in the eye of the beholder.

   I have written out the rigorous formulation of renormalization group flow and running coupling constants (currently both redirect to the same entry).

   This is an exegesis of Brunetti-Dütsch-Fredenhagen 09, section 4.2 and 5.1 with clues from Dütsch 18, section 3.5.3; but I have tried to disentangle, in the writeup, the general principle from the specific case where the flow is induced by scaling transformations. (The latter is going to be written out at Gell-Mann-Low renormalization cocycle, but don’t look at that entry just yet).

   Accordingly I tried to streamline presentation and notation, to make it all clear at a glance what is going on… but of course this may be in the eye of the beholder.

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 29th 2018
   • (edited Jan 29th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI should say: the conclusion of the proof that the "running of the couplings" is a group cocycle over RG ([this prop.](https://ncatlab.org/nlab/show/renormalization+group+flow#CocycleRunningCoupling)) needs that any S-matrix scheme $\mathcal{S} \colon LocObs \to PolyObs$ is an injective function. This is stated without comment in the reference I am following. Apparently it's meant to be trivial, but right now I don't see it. I'll think about it...

   I should say:

   the conclusion of the proof that the “running of the couplings” is a group cocycle over RG (this prop.) needs that any S-matrix scheme $\mathcal{S} \colon LocObs \to PolyObs$ is an injective function. This is stated without comment in the reference I am following. Apparently it’s meant to be trivial, but right now I don’t see it. I’ll think about it…

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeJan 30th 2018
   • PermaLink
   Author: Urs
   Format: MarkdownItexNow that I am awake again: > the conclusion of the proof that the "running of the couplings" is a group cocycle over RG ([this prop.](https://ncatlab.org/nlab/show/renormalization+group+flow#CocycleRunningCoupling)) needs that any S-matrix scheme $\mathcal{S} \colon LocObs \to PolyObs$ is an injective function. This is stated without comment in the reference I am following. Apparently it's meant to be trivial, but right now I don't see it. I'll think about it... Of course this is just the uniqueness-clause in the "[[main theorem of perturbative renormalization|main theorem]]". I have finished the proof.

   Now that I am awake again:

   the conclusion of the proof that the “running of the couplings” is a group cocycle over RG (this prop.) needs that any S-matrix scheme $\mathcal{S} \colon LocObs \to PolyObs$ is an injective function. This is stated without comment in the reference I am following. Apparently it’s meant to be trivial, but right now I don’t see it. I’ll think about it…

   Of course this is just the uniqueness-clause in the “main theorem”. I have finished the proof.

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJan 30th 2018
   • (edited Jan 30th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have polished the definition/proof that scaling transformations on Minkowski spacetime satisfy the axioms of rg-flow, and added that as an example-section [here](https://ncatlab.org/nlab/show/renormalization+group+flow#ScalingTransformationsRGFlow). Accordingly I have removed the entry where I had kept this material previously, which used to be called _[[Gell-Mann-Low renormalization cocycle]]_, and instead made that a redirect now to _[[renormalization group flow]]_. (Since it is the main and almost exclusively discussed example of RG flow, it makes little sense to keep it in a separate entry.)

   I have polished the definition/proof that scaling transformations on Minkowski spacetime satisfy the axioms of rg-flow, and added that as an example-section here.

   Accordingly I have removed the entry where I had kept this material previously, which used to be called Gell-Mann-Low renormalization cocycle, and instead made that a redirect now to renormalization group flow. (Since it is the main and almost exclusively discussed example of RG flow, it makes little sense to keep it in a separate entry.)

5. • CommentRowNumber5.
   • CommentAuthorDavidRoberts
   • CommentTimeJan 30th 2018
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   Author: DavidRoberts
   Format: MarkdownItexIs it possible to have RG be n>1 dimensional?

   Is it possible to have RG be n>1 dimensional?

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJan 30th 2018
   • (edited Jan 30th 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexSo that's exactly why I took care to disentangle the general idea from the case of scaling transformations: The definition in [this prop.](https://ncatlab.org/nlab/show/renormalization+group+flow#FlowRenormalizationGroup) of renormalization group flow makes no assumption on the nature of $RG$ apart from the defining condition that it preserves causal order of spacetime supports and its action intertwines the Wick algebra products of a correspondingly parameterized collection of free field vacua ([this equation](https://ncatlab.org/nlab/show/renormalization+group+flow#eq:IntertwiningWickProductsActionRG)). This is all one needs to conclude that, by conjugation, RG acts on the set of S-matrix renormalization schemes ([this equation](https://ncatlab.org/nlab/show/renormalization+group+flow#eq:RGConjugateSmatrix)) and hence, via the "[[main theorem of perturbative renormalization|main theorem]]", by vertex redefinitions aka "running of the coupling constants". For that alone one does not even need to assume that $RG$ is a Lie group, this enters only if one wants to express a corresponding $\beta$-function (as the derivative of the RG-action). So, no, from the math side $RG$ is not required to be a 1-dimensional Lie group. However, I don't have a big supply of examples of $RG$s beyond the case where $RG = \mathbb{R}_+$ acting by scaling transformations. I should come up with some.

   So that’s exactly why I took care to disentangle the general idea from the case of scaling transformations:

   The definition in this prop. of renormalization group flow makes no assumption on the nature of $RG$ apart from the defining condition that it preserves causal order of spacetime supports and its action intertwines the Wick algebra products of a correspondingly parameterized collection of free field vacua (this equation).

   This is all one needs to conclude that, by conjugation, RG acts on the set of S-matrix renormalization schemes (this equation) and hence, via the “main theorem”, by vertex redefinitions aka “running of the coupling constants”.

   For that alone one does not even need to assume that $RG$ is a Lie group, this enters only if one wants to express a corresponding $\beta$-function (as the derivative of the RG-action).

   So, no, from the math side $RG$ is not required to be a 1-dimensional Lie group.

   However, I don’t have a big supply of examples of $RG$s beyond the case where $RG = \mathbb{R}_+$ acting by scaling transformations. I should come up with some.