Renormalization Group for dummies

I actually understood most everything you were saying.. but I just want to know if you can apply this coupling constant getting bigger dependent on terms in the series too to non-QFT problems like calculating for the trajectory of a ball. This is simply what I want to know. Thanks.

One way to think about the changing constants is to realise you are just writing and "effective theory". In the Box-Jenkins philosophy: all models are wrong, but some are useful. Say you have a curve. Every point on the curve can be approximated by a straight line. Depending on which part of the curve you are approximating, the slope of the line will change. The straight line is your "effective theory" and the changing slope like your changing coupling constant. This example is not a detailed comparison, but it's the general philosophy of the renormalization group. As for detailed mathematical correspondence, apart from quantum field theory, the renormalization group has been applied in classical statistical mechanics and classical mechanics.

I read in wiki that "Geometric series are used throughout mathematics, and they have important applications in physics, engineering, biology, economics, computer science, queueing theory, and finance."

So you are saying that Renormalization Group concepts and regulator thing are also used in biology, economics, finance and not just in QFT? So in the calculations in biology. The coupling constant equivalent can become infinite in the second term but if one makes a cut-off at first term. it is finite?

Renormalization has nothing to do with infinities. QED is renormalizable and it has a cut-off - it is not a true theory valide at all energies, it is only an effective theory like gravity, valid below the Planck scale. Once you have a cut-off, there are no infinities. Sometimes you are lucky and you get a theory where you can remove the cut-off, like QCD. But in QED, as far as we know, the cut-off probably cannot be removed.

I'm asking if the techniques in Renormalizaton Group can be applied to Finance or Biology problem

I can't believe why no one is answer my simple question. I know in power series, one expand the terms. I'm asking if the techniques in Renormalizaton Group can be applied to Finance or Biology problem, not just in QFT. If one knows the answer to this. Please let me know. Thanks.

In very, very broad terms - yes - everyone uses effective theories.

Less facetiously, the philosophy of the central limit theorem is like the renormalization group. Take the heights of people. That's due to hundreds of genes, whose interactions we hardly understand. Yet when you plot a distribution of heights of populations, very often you get a Gaussian distribution characterized fully by two parameters: mean and variance. So if you are looking coarsely on a population level, you don't need the theory of all the genes with hundreds of parameters - you just need the Gaussian distribution with two parameters.

Here the Gaussian distribution is derived using the renormalization group:
http://www.math.princeton.edu/facult...gorovLec07.pdf

Its not quite like that. You write out the power series and you calculate the terms term by term using perturbation theory. The first term is finite - no problem. Second and higher terms turn out to be infinite. It took people a long time to figure out why this happened but the answer turned out to be the thing you expand the power series in, the coupling constant, was infinite and not 1/137 like they thought. Substitute infinity in any power series and its infinite or undefined. To get around this problem you impose a cutoff (this can be looked on as taking the term you are expanding about as finite and later taking its limit to infinity) redo your perturbation procedure and you find it is all OK. The reason is the coupling constant secretly depends on the cutoff - which is rather trivial the way I explained it - but it took people a long time to realize this is what is going on. The value of 1/137 they used was the value measured at a certain energy scale which in effect was measuring the value with a cutoff. But the equations they used had no cutoff so you were really using the value as the cutoff goes to infinity ie infinity. As you take the cutoff to infinity the coupling constant goes from 1/137 to infinity which is why without the cutoff terms in the power series are infinite. Now what you do is assume the coupling constant is a function of what is called the renormalised coupling constant (which is the value from experiment ie 1/137) so you know it will not blow up. You assume it is a function of the un-renormalised parameter ie the value that does blow up to infinity, expand it in a power series, substitute into the original power series, collect terms so you now have a power series in the renormalised parameter. But you have chosen it so it is the value found from experiment so does not blow up. Carry out your calculations, take the cutoff to infinity and low and behold you find the answer is finite.

The infinity minus infinity thing comes from when you analyse the behavior of the series when you use the renormalised value and take the limit - you find a term that is the original un-renormalised coupling constant and a term that is a function of the renormalised coupling constant - they in fact both blow up to infinity as you take the limit - but are subtracted from each other so the answer is finite.

If you are at the level of Calculus For Dummies its probably going to be difficult to understand the paper I linked to. I have a degree in applied math and I found it tough going. So don't feel bad you are finding it tough - I congratulate you for trying.

If you want to get your math up to the level you can understand that paper you will have to a study a more advanced textbook. The one I recommend is Boas - Mathematical Methods
http://www.amazon.com/Mathematical-M...tt_at_ep_dpi_1

Unfortunately otherwise you will have to accept the hand-wavey arguments. As I said in my original post the jig is up with this one - you need to do the math.

To give a specific answer to the questions you raised and how to relate it to renormalisation I will see what I can do. If you substitute infinity into any power series it will give either infinity or terms like infinity minus infinity that are undefined. An example of the first would be the power series e^x where each term is positive and an example of the second would be sine x which has positive and negative terms. Now one way to try and get around this is let x be finite and take the limit. Before you take the limit everything is fine - its finite and perfectly OK. Now what you do is assume the variable in the power series is a function of another variable (in this case called the re-normalized variable) that you hope does not blow up to infinity as you take the limit. You expand that out as a power series and you collect terms so you have a new power series in that variable. Now you take the limit and low and behold, for the case of what are called re-normalizable theories, everything is finite. You look deeper into why this occurred and you find changing to this new variable introduced another term in your equations that also blows up to infinity but is subtracted from the original variable that blows up to infinity - as you take the limit they cancel and you are left with finite answers.

Normally when you calculate the terms in a power series using perturbation theory it does not blow up to infinity. That's because it is very unusual to chose a variable to expand the power series in that is infinity. The only reason it was done is they did not understand the physics well enough then - they did not understand the measurement of the constant they thought was small at 1/137, and was a good thing to expand in a power series about since as it is raised to a power it gets smaller and smaller, was a measurement made with a cutoff basically in effect. The equations they used had no cutoff and it all went pair shaped. When this happened it left some of the greatest physicists and mathematicians in the world totally flummoxed - these are guys like Dirac with awesome mathematical talent. It was a long hard struggle over many years to sort out what was going on. The thing that fooled them was the parameter you expanded about as a power series secretly depended on the regulator or cutoff and as you took its limit to infinity it went to infinity. When you expanded about a different one that didn't blow up to infinity everything worked OK.

As I was penning this I remembered John Baez wrote an interesting article about re-normalisation that may be of help:
http://math.ucr.edu/home/baez/renormalization.html

Thanks
Bill

Thanks for the updated descriptions. You'd make a great PF science advisor someday.
Say, in the magnetic moment of the electron with measured value of 1.00115965219 and calculated value of 1.0011596522 in the fourth term in the power series. Did they use Renormalization Group there? Maybe the reason the second term to fourth term didn't produce an infinite coupling constant is because the calculation replaces it with 1/137 as you described. What then is the original coupling constant when no Renormalization Group procedure was used?

Renormalization inorer to remove infinities is just a dirty trick. The renormalization group itself says that you can look at a theory at different energy scales E and E' (or at different length scales which is the same) and renromalize masses and coupling constants g(E) and g'(E') such that the predictions of both theories are the same. Essentially they are THE SAME theory.

This can e.g. be interpreted as "integrating out" degrees of freedom and hiding their contribution in the renormalization of parameters. Have a look at http://en.wikipedia.org/wiki/Renorma...lization_group

Suppose you have a theory with certain coupling constants ga, gb, ...; suppose you study this theory at some energy scale E. Then you will observe that you can re-express the theory in terms of different coupling constants g*a, g*b,... at a different energy scale E* (and different interactions, i.e. changing E may turn on new couplings). It is interesting that these two 'theories' defined at different energy scales E and E* are essentially the same theory! The way how to find the description at scale E* given the description at E is via studying trajectories ga(E), gb(E) in the space of all coupling constants.

Different points on the same trajectory correspond to different scales E, E*, ... of the same theory; different theories are defined by points not connected by a trajectory. These trajectories define something called renormalization group flow. The renormalization group of a certain theory is defined via a set of coupled differential equation defining this flow of coupling constants.

Perturbative QFT is defined by studying the flow near the point ga = gb = ... = 0; in some case (QCD) it can be shown that this is reasonable in certain regimes; in other cases (QG) there are strong indications that G=0 may never be a valid starting point, neither in the UV (not asymptotically free), nor in the IR (coupling constant of Newtonian gravity G is not zero).

Bill Hobba kept saying this but he didn't explain the physical reason why

note the interesting comments by Science Advisor tom.stoer that the Renormalization Group is just a DIRTY TRICK.

The physical reason the coupling constant gets larger is the the shielding effect of the virtual particles around an electron. Landau showed that because of that for any finite charge at any distance the charge would measure zero. This created the famous zero charge problem. The only way around it is for the charge to actually be infinite and to grow to an infinite value the closer and closer you get to it - this is called the Landau pole. That is the reason for the infinities - the closer you get the higher the energy you must use. If I remember correctly this is not just theory at 90gv it measured 1/127.

As one other person posted the fact it blows up to infinity means the theory is sick and incorrect - but we already know that because it gets replaced by the electroweak theory which will probably also get replaced by something else someday and that something else will hopefully be free of these problems.

Thanks
Bill

In the Dirac Equation. The magnetic moment of the electron is calculated as 1. In the 4th term in the power series, it's equal to 1.0011596522. The interacting fields are the electron self magnetic field and the electron. What about the interactions of say two electrons, what would be the Dirac Equation counterpart of 1.0 in the magnetic moment of the electron calculation? Do you calculate the dirac equation of each electron by adding them or calculate both of them combined? And if the result is for example 3.0. After the fourth term in the power series, would the result only be 3.0111 or would it be 5.0 (I don't think it would just be a small 3.0111 isn't it because the electric field strength is bigger). Thanks.

Don't know the answer to that one - sorry.

Thanks
Bill

Do you mean that the full integral has a closed-form expression (involving modified Bessel functions) for any value of lambda, but the series expression (involving Gamma functions) has terms that only converge absolutely when lambda<1? So if we had lambda>1, and all we had was the series form, we might worry the integral doesn't exist, when in fact it does?

I can't believe why no one is answer my simple question. I know in power series, one expand the terms. I'm asking if the techniques in Renormalizaton Group can be applied to Finance or Biology problem, not just in QFT. If one knows the answer to this. Please let me know. Thanks.