Kreimer-Connes & Wilson-Polchinski

[atyy]

How are the Kreimer-Connes and Wilson-Polchinski views of renormalization related? Especially with regards to the idea that QED has an infrared fixed point and is only an effective theory, versus QCD which has a UV fixed point is well-defined to arbitrarily high energies?

 

[genneth]

Since I'm not a mathematician, the Kreimer-Connes work is hard for me to follow; below is only a statement of my understanding, not necessary the truth.

Disclaimer aside, the work of Kreimer-Connes has very little relevance for your average working physicist, at least at the moment. What they have done is to put the usual methods often used in HEP (i.e. tracking Feynman diagrams and coming up with suitable counterterms to cancel infinities) on a rigorous mathematical basis. Prior to that, it was not clear whether the procedures being used actually constituted a self-consistent set of rules. They show that it is, and in fact it is a sort of beefed up construction of a type that mathematicians have already known for a while. However, that "beefed up" part still means that they don't have any new ways (unknown to the physics community) for actually doing calculations. No doubt someday someone will exploit this fact and do something amazing, but right now that's not happening. In fact, I'd laid bets going the other way --- some heuristic, ad hoc procedure dreamt up by a physicist will get turned into a valuable theorem on the pure maths side first.

As such, the relation to the Wilson-Polchinski view is as usual for the HEP vs condensed matter points of view. At the end of the day, we have the fact that we need to measure some parameters. These are the "physical" ones. Our theory, on the other hand, deals with "bare" quantities, and it is up to us where we choose our cut-off. Changing the cut-off changes the precise relationship between bare and physical parameters. For some calculations it's easier to put the cut-offs very low, so that bare and physical parameters are almost the same. For others, it's better to put it high, and apply some combinatorial cleverness to cope with the diagrams.

The usual feeling is that the condensed matter view is clearer on what the various relationships are, but that the HEP methods are strictly more powerful. Specifically, it's hard to do momentum-shell renormalisation if you need Lorentz invariance. Gauge covariance only makes things worse. Whereas dimensional regularisation has the wonderful benefit of being an entirely orthogonal choice to any physical symmetry.

 

[atyy]

The usual feeling is that the condensed matter view is clearer on what the various relationships are, but that the HEP methods are strictly more powerful. Specifically, it's hard to do momentum-shell renormalisation if you need Lorentz invariance. Gauge covariance only makes things worse. Whereas dimensional regularisation has the wonderful benefit of being an entirely orthogonal choice to any physical symmetry.

Wow, you must be a condensed matter theorist! But really, where do you get to say the condensed matter view is clearer when Wilson was a HEP guy who "saved" the condensed matter field (ok, ok, Kadanoff should get a lot of credit too )

 

[humanino]

I have been willing to answer Genneth in the other thread about this, but it seems we pretty much agree. If you want a precise statement, they have rigorously established the BPHZ fomula (preliminary version on scholaropedia) which is most general and often used to prove the equivalence between several renormalization scheme. The reference is
Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem

 

[atyy]

I have been willing to answer Genneth in the other thread about this, but it seems we pretty much agree. If you want a precise statement, they have rigorously established the BPHZ fomula (preliminary version on scholaropedia) which is most general and often used to prove the equivalence between several renormalization scheme. The reference is
Renormalization in quantum field theory and the Riemann-Hilbert problem I: the Hopf algebra structure of graphs and the main theorem

Thanks! Yeah, it would not have been apparent that you and Genneth agree - good to know - I understand Genneth's viewpoint - the Connes-Kreimer stuff is Sumerian to me. I guess I have no choice but to try to read it carefully.

 

[genneth]

Wow, you must be a condensed matter theorist! But really, where do you get to say the condensed matter view is clearer when Wilson was a HEP guy who "saved" the condensed matter field (ok, ok, Kadanoff should get a lot of credit too )

However, it's rather telling that HEP no longer uses the Wilsonian methods of momentum shell renormalisation --- it's hard to find a suitable shell if you have a Lorentzian metric. We squidgy stuff people are always working with a Wick rotated system, so we can. I don't consider the combinatorial counting of counterterms at each order to be particularly physical --- it's a very neat piece of mathematics which implements at the end the same thing, but is more abstracted from the system. Clearly I'm just not clever enough to cope with the extra step...

Btw, the idea of renormalisation in condensed matter is a little broader than this strict parameters-moving-with-cut-off. It's biggest use to us is to justify, however weakly, huge leaps of logic through its consequence of universality. Specifically, when we dream up some effective theory, very rarely do we actually proceed from a microscopic theory and then integrate out degrees of freedom we don't care about. Usually, we jump straight to some theory, with some unfixed parameters, which do not necessarily have any simple relationship to the microscopic ones, i.e. mass of Landau quasi-particles in Fermi liquid theory vs the bare mass. Our view is that we *will* throw away information, but hopefully in a maximally helpful manner, whilst retaining the physical phenomenon we are currently interested in, and that this is an unavoidable part of dealing with complexity.

 

[atyy]

I don't consider the combinatorial counting of counterterms at each order to be particularly physical --- it's a very neat piece of mathematics which implements at the end the same thing, but is more abstracted from the system.

Yes, I wonder whether there is any conceptual clarification beyond Wilson, or if it mainly organizes the calculation more efficiently.

A paper suggesting the former is:
http://arxiv.org/abs/gr-qc/0203036
Coarse graining in spin foam models
Fotini Markopoulou

Looking through the cites, this seems interesting:
http://arxiv.org/abs/0806.4309
Wilsonian renormalization, differential equations and Hopf algebras
Thomas Krajewski, Pierre Martinetti

 

[George Jones]

How are the Kreimer-Connes and Wilson-Polchinski views of renormalization related? Especially with regards to the idea that QED has an infrared fixed point and is only an effective theory, versus QCD which has a UV fixed point is well-defined to arbitrarily high energies?

I don't of a relationship between QED and QCD fixed points, but there is a relationship between Kreimer-Connes and Wilson-Poilchinski. From Quantum Field Theory A Bridge Between Mathematicians and Physicists: I Basics in Mathematics and Physics "The renormalization groups used by physicists in specific situations are representations of one-dimensional representations of the motivic Galois groups."

See pages 859-850 section 15.4.6 The Importance of Hopf Algebras,

http://books.google.com/books?id=XY...r&dq=zeidler+quantum&cd=1#v=onepage&q&f=false

 

[Finbar]

Am I right in thinking that Kreimer-Connes renormalisation is a purely perturbabtive approach or can it be used for non-perturbabtive physics like in infrared QCD?