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Shortly after the invention of quantum electrodynamics, it was discovered that the theory had some very bad properties. It took twenty years to discover that certain naturally arising infinities could be overcome by a process called renormalization.

One might state the physical reason behind this that we are only aware of effective theories which are reliable on certain energy scales given by more or less fundamental constants. Renormalization tells us how to deal with this situation and only consider effects in some specific energy range.

The technique to perform the calculations is called the renormalization group. It is a powerful tool and it is no wonder it is under heavy development since nothing can be calculated without it in quantum field theories. A priori, this procedure is not limited to its root and one might ask the following:

How can we use the renormalization group to find effective theories for classical field theories?

I suppose, an example where this has been done very recently can be found in Renormalization Group Analysis of Turbulent Hydrodynamics.

PS.: The question is naturally linked to How to calculate the properties of Photon-Quasiparticles and in a loose line with A pedestrian explanation of conformal blocks.
Since I am no expert in the field please advice me if something is not clear or simply wrong.

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I love how astoundingly formal this question is. –  Noldorin Jan 8 '11 at 18:03

3 Answers 3

up vote 7 down vote accepted

I'm not an expert in this topic too, but I'm trying wrap my head around it.
Right now I'm trying to make an adequate hierarchy of concepts related to renormalization. Let me list them and tell how they are related:

  1. Fields, Lagrangian (Hamiltonian) and coupling constants.
  2. Perturbative calculations.
  3. Different scales.
  4. Self-similarity.
  5. Quantum fields.
  6. Ultraviolet divergences.
  7. Renormalization.
  8. Renormalization group and running couplings.
    (Let me stress that the "renormalization" and "renormalization group" are different concepts.)

Of course the concept of a field and the way to describe (1) it is a starting point.

Now, It seems to me (while I can be wrong) that every time we talk about renormalization we always deal with some perturbative approach (2). There is always something that we want to neglect. And if there is a way to make calculations without any approximations then one needn't to use techniques related to renormalization.

One of simplest examples is a hydrodynamics -- you don't want to "get down" to the level of molecules to describe a stream of water. You would like to work with some "integral" quantities, like viscosity. And the viscosity can be used to describe processes at many different scales (3): bloodstream, butterfly, submarine, internals of the star, e.t.c.

The hydrodynamics works at different scales because of the self-similarity (4): by going several orders of magnitude larger you are still able to describe your system with the same Lagrangian, but, maybe, with some parameters changed. When one does the transition from one scale to another one always neglects some peculiarities (2), that occur at smaller scale.

This is the essence of renormalization group(8) techniques. The changing parameters are also called the running couplings. I recommend you to read about Kadanoff transformation, to get more insight about it.

Note that I never mentioned divergences by far. Because this is a slightly different topic. And one can use renormalization group even if there is no infinities.

UV divergences appear due to our ignorance about smaller scales. When we talk about hydrodynamics we know that there is a "fundamental scale" -- the aforementioned molecules. But when we talk about quantum fields (6) (like electromagnetic field or some fermion field) we don't know what is the "fundamental" scale for it. We don't even know if it exists at all.

Different methods of dealing with the divergences are called the renormalization (7) methods. They are based on changes of the parameters of Larangian too, but now these changes are "infinite" because one have to "compensate the infinities" appearing from small scales. After cancelling the infinities this way one is still left with arbitrariness of choosing a finite values of the parameters. You can fix the parameters by getting them from experiment at certain scale(3) and use renormalization group (8) to go from one scale to another.

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+1, nice summing up of the main points of renormalization. –  Marek Jan 7 '11 at 19:16
thank you very much for your explanations. I even found a paper on the application of renormalization group on hydrodynamics: arxiv.org/PS_cache/arxiv/pdf/1012/1012.0461v1.pdf Greets –  Robert Filter Jan 8 '11 at 17:56

First, the full notion of renormalization group, as studied in QFT, is definitely not needed in the classical theory. This is because QFT actually doesn't make sense without a renormalization scheme and for any theory one has to always investigate the flow of couplings towards some fixed points (corresponding to conformal field theories) to check whether the given theory is renormalizable in the first place. So renormalization is an integral part of the QFT in the great contrast with classical theories.

Other place where notion of renormalization group flow is important is condensed matter theory. This is because these flows have fixed points that (when they are non-trivial) correspond to critical points (this is again connected with the mentioned conformal symmetry). The renormalization group is then used to see how the flow behaves around this point and this gives valuable information about the behavior of macroscopic quantities (like specific heat) at the critical point.

But the notion by itself is not terribly important if all you care about is integrating UV degrees of freedom. I don't think you need any flow in classical theory. All you have to do is integrate some energetic interaction with an ambient field to obtain an effective mass (for a one concrete example). While renormalization group gives a useful framework for general understanding of scales of theories and their being effective, it's not really needed most of the time.

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Dear Marek, thank you for your answer! The point is that you can also define some Green's function for differential equations and thus introduce the path integral formalism in classical field theories. By this meaning, it should be possible, at least in principle, to use the Renormalization Group for further insight. I might add this to the question. Furthermore, may I ask if you could please specify your idea of flow and effective mass for classical theories, e.g. by an example? I am not sure if I can follow your idea :) –  Robert Filter Jan 7 '11 at 15:49
@Robert: yes, something like path integral definitely needs to be formulated in order to make some sense of partition sums. You can then obtain the effective theory by summing over just the states that represent microscopic behavior (in QFT you introduce an energy scale and in lattice model there is natural lattice spacing present). But I am actually not quite sure how this is done in classical theory. If you could point me to the reference for that path integral approach maybe I could help you more. –  Marek Jan 7 '11 at 17:14
@Marek: I thought there might be an easy relation I am not aware of since one could interprete the Greens function of some differential operator as propagator. But this concept seems to be erroneous, I could not find any path integral reference for classical field theories. I only found one paper, arxiv.org/PS_cache/hep-th/pdf/9903/9903136v3.pdf concerning classical mechanics but the approach used there is not what we would need imho. I will try to find some classical renormalization examples :) Sincerely –  Robert Filter Jan 8 '11 at 17:19
@Robert: well, Green functions in classical theory are used completely differently than correlators or propagators (which are often also called Green functions) in QFT and condensed matter theories. But it's certainly possible there is some connection, just that I don't know about it. –  Marek Jan 8 '11 at 17:25
@Marek: Found it. Look at en.wikipedia.org/wiki/Propagator_%28Quantum_Theory%29 in the part for relativistic propagators :) Greets –  Robert Filter Jan 8 '11 at 18:06

Marek wrote: "First, the full notion of renormalization group, as studied in QFT, is definitely not needed in the classical theory...."

Marek, the renormalization of mass first appeared in Classical Electromagnetism, didn't it? Take the "definition": $m_{physical} = m_{bare} + \delta m$. This is one constraint for two addenda so there is a one-parametric "invariance" group even in the CED. As soon as the mass renormalization (discarding $\delta m$ to originally physical $m $) is done exactly, only once, it is not really interesting, to say the least. In QED this "liberty" is extended to the charge and is done perturbatively, but the main sense remains - the renorm-group is a "liberty" in choosing the two terms to satisfy one constraint: $e_{physical} = e_{bare}(\Lambda) + \delta e(\Lambda)$ where $(\Lambda)$ is a cutoff.

If we return to the original sense of renormalizations as to discarding unnecessary perturbative corrections to originally right, physical, fundamental constant values, no group appears, no stupid relationship between "bare" and "physical" constants are derived (no Landau pole), and everything is simple: one discards infinite or finite contributions of self-action. Self-action is an erroneous concept: it does not lead to any change (no action by definition), only to wrong terms to be discarded in the end.

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I'll agree with you up to the point that problems occur also in classical theory. But to cure them (and I mean really go to the root of the problem) one needs QFT and renormalization. In pure classical physics renormalization is not needed. Well, that's my opinion anyway and I've yet to be convinced about the contrary. –  Marek Jan 17 '11 at 23:04
Marek, try to read my arguments here arxiv.org/abs/0811.4416. It is very simple and comprehensible paper, and it is relevant to the issue. –  Vladimir Kalitvianski Jan 17 '11 at 23:22
Please give explanation every time you give a negative score. –  Vladimir Kalitvianski Jan 19 '11 at 22:52
Dear downvoters, tell me where I am wrong, please. I would like to learn. –  Vladimir Kalitvianski Jan 31 '11 at 16:26

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