From: Laurenz Widhalm Subject: re: Is there a better way than renormalization? Date: Thu, 25 Feb 1999 10:48:58 +0100 Newsgroups: sci.physics,sci.math To: graham_fyffe@hotmail.com Keywords: Avoiding singularities in integrals of mathematical physics On Thu, 25 Feb 1999 graham_fyffe@hotmail.com wrote: > In article <36D42666.32A5AF1@dcwi.com>, > Ralph Frost wrote: > > [...] > > > [Please don't get the idea I actually know what "renormalization" is. It > > sounds like a flattening or twisting of results back into the shape one > > wants, or a kludge -- but a necessary one, given all precursors.] > > > Say, does anyone have an example equation where renormalization was used? > I've only heard bits and pieces, but it sounds like they're trying to take a > limit as some variable approaches infinity. You'd think there'd be plenty of > conventional ways to handle that. What is it actually? It's not an easy task to explain renormalization, and it is very frequently skipped in lectures even at university. That's mostly because of the mathematic techniques used, which are quite complicated. But I think you should get the basic idea without any math: An electron has negative charge. Negative charge repells negative charge, so why doesn't the electron just blow up? This might sound like a silly question, but view it like this: every electron causes an electric field. Many electrons cause a field which is the sum of all single fields, the field can spread in space and is thought to be independent of its source. Each electron responds to the sum-field, which also includes its own contribution. The electron just can't distinguish. The fact that an electron interacts with its own field is called self-interaction and is important to understand some known physical effects. If you now calculate the effects of self-interaction straightforwardly, you come to the conclusion that the electron should blow up. But obviously it doesn't, so you can either give up your theory or try to readjust the parameters of your theory in order to give that result that agress with what you see in nature. This readjustment is no big deal, you always have to measure some fundamental constants to calibrate a theory. The problem in this case is, that you have to readjust the parameters such that they become infinite. But after all, it turns out that this is just a technical problem. The techniques are regularization and renormalization. Regularization means, that you write your terms in such a way that you get your infinite result as a limit of a new parameter, which may go to 0 or oo, depending on your regularization scheme (there are more than one possibilities of how to do that, of course each must have the same result in the end - that's also a test of what you're doing). It is important, that for any finite value of this parameter, you get a well-definied, finite result. Renormalization now means that you redefine your parameters of the theory such that the parts which get infinite cancel with the parts introduced with your readjustment of the parameters. As long as your regularization-parameter is finite, this readjustment is also finite. Only in the limit, it gets infinite. But also in the limit, it cancels out, and you end up with some finite rest,the so called "radiative corrections", which you may compare with what you measure in nature. Since what you get is very exactly what you measure, physicists nowadays trust very much in renormalization, although handling with infinities can be very subtle. That's renormalization. Everything else is purely technical and shouldn't confuse anybody. You may ask: When the parameters of my theory are infinite, why do I measure finite values for them? For instance, the charge of an electron is obviously NOT infinite. But don't forget: you don't measure the "naked" charge of an electron, but always the net effect of it's "real" charge (which is infinite) and all effects of selfinteraction (which are infinite as well). If you adjusted your parameters correcty, the both infinities have opposite sign and you get oo - oo, which may give any number, as you know from math. Hope I could answer your question without giving an explicit mathematical example, because there is the danger that you get lost in mathematical details and doesn't see the big idea. But if you are interested in this technical stuff, I would recommend to go to your next library and look for a good textbook. bye, Laurenz [Frigschneck] ============================================================================== From: Matthew Nobes Subject: re: Is there a better way than renormalization? Date: Thu, 25 Feb 1999 18:10:24 -0800 Newsgroups: sci.physics,sci.math On Thu, 25 Feb 1999 graham_fyffe@hotmail.com wrote: > In article <36D42666.32A5AF1@dcwi.com>, > Ralph Frost wrote: > > [...] > > > [Please don't get the idea I actually know what "renormalization" is. It > > sounds like a flattening or twisting of results back into the shape one > > wants, or a kludge -- but a necessary one, given all precursors.] > > > Say, does anyone have an example equation where renormalization was used? > I've only heard bits and pieces, but it sounds like they're trying to take a > limit as some variable approaches infinity. You'd think there'd be plenty of > conventional ways to handle that. What is it actually? The other response to this explained how you renormailze something, but was a bit confusing on why you need to (beyond the simple, things are inifinite that shouldn't be) so let me take a crack at this, using (what I understand to be) a more modern idea of renormalization. Let's consider the electrons self energy Ee in classical electrodynamics. It was well known early on that Maxwell's theory predicts that an electron (or any other point charge) in motion will have a self energy cause by it's interaction with the electric and magnetic fields that it creates. The problem is that when you go to calculate Ee it turns out that you get 1 Ee proportional to ------------------ radius of particle No for an electron (zero radius) this is infinite. What does this imply? Well since E=m (c=1) Ee should affect the electrons mass, and in this case it should be inifinite, clearly falsified by experiments. Why does this happen? Well it turns out that in order to actually derive the above result you have to sume up all of the effects of the field on the electron at all distances. This is no problem so long as you stay far enough away from the electron, but when you get close enough to it, you know that classical theory is no longer appropriate. What is actually remarkable (to me at least) is that the classical theory 'knows' about this in the sense that it doesn't give you a wrong answer when you try to overextend it it gives you absolute nonsense. Now let's move on to QED. In this case you can go ahead and calculate Ee and you will find (in slightly simplier notation) Ee ~ log(1/r) This still goes to oo as r goes to 0 but it does so much more slowly. When this first happened, people thought that it was a huge disaster, and renormalization was viewed as a stop gap procedure to get numbers out of the theory. The modern view is different. The reason the theory gives you oo is because you are trying to push it all the way to zero distance (which corresponds to oo energy), this is wrong. In the case of QED it's obviously wrong because at some small distance (like 1 fermi or less) the electromagnetic force unifies with the weak force, so you have to use some different theory. What sort of picture does this present us with then. Lets look at another problem, from this point of view, the charge of the electron. In the right units the electron charge as measured from far away is e=-1/sqrt(137). What QED tells use is that the electron interacts with itself 'surrounding' itself with a cloud of virtual stuff (e+ e- pairs mostly) which screens the bare charge inside. What is the bare charge? well to find that out we would have to penetrate this cloud to zero distance, somthing we cannot do. So let's just assume we are ignorant of that. Pictorially we have \ \ cloud of stuff \ Cloud of | \ Described by |stuff described| Classiscal inacessable | GWS electroweak |by QED | Region region to QFT | theory | | x------------------------------------------------------->distance scale ^ Proper theory | in here charge is |in here charge | | is unknown |about less then |is less then |Charge is | / about -1/sqrt(129) |-1/sqrt(137) |-1/sqrt(137) bare charge=? / | / With this picture in mind renormalization should be clear. you start far out with the classical value for e. So long as you work at low enough energies (below about 50 GeV) QED works just fine. note that this is a very different point of view that the sweeping the inifnites away one (however, the methods are the same). Matthew Nobes ============================================================================== From: kevin@cco.caltech.edu (Kevin A. Scaldeferri) Subject: Re: Is there a better way than renormalization? Date: 3 Mar 1999 01:13:54 GMT Newsgroups: sci.physics,sci.math In article <7b2tj5$6jl$1@nnrp1.dejanews.com>, wrote: >In article <36D42666.32A5AF1@dcwi.com>, > Ralph Frost wrote: > >[...] > >> [Please don't get the idea I actually know what "renormalization" is. It >> sounds like a flattening or twisting of results back into the shape one >> wants, or a kludge -- but a necessary one, given all precursors.] > > >Say, does anyone have an example equation where renormalization was used? >I've only heard bits and pieces, but it sounds like they're trying to take a >limit as some variable approaches infinity. You'd think there'd be plenty of >conventional ways to handle that. What is it actually? It's really not so bizarre as people frequently make it out to be. Say I start with some classical field theory which I describe via some Lagrangian. For concreteness, let's look at the Klein-Gordon Lagragian (plus interactions): L = (1/2) ((d_a phi)^2 - m^2 phi^2) + (interaction terms) I can use the Euler-Lagrange equation to get the equation of motion for the field phi and, lo and behold, that thing I called m turns out to be the mass. Now let's turn this into a quantum theory by either introducing canonical commutation relations between phi and its conjugate momentum or by shoving that Lagrangian into a path integral. When you do this, you get quantum corrections to anything you might have calculated in the classical theory. So, for example, the mass has quantum corrections. Now, the physical mass is the thing you get _after_ you take into account all the quantum corrections, so that thing you denoted m in the Lagrangian isn't actually the physical mass after all. Let's call it the bare mass, since it's the mass before you dress it with quantum corrections, and write it m_0. So, now we have some equation m = m_0 + delta(m) where m is the physical mass, m_0 is the bare mass - the thing in the Lagrangian, and delta(m) is the quantum correction to the mass. That's it. That's renormalization. Stop and reread the above. You now understand all the physics in the renormalization procedure. Okay, so now you're saying, but what about infinities and sweeping them under the rug? What about mathematical funny business? What happened to all that? So, I have to confess to having left out one teensy little thing. While you can cook up some theory where the above is exactly what happens and nothing at all wierd happens, in this theory I wrote down there is the peculiarity that, while m is a nice normal number, both the bare mass and the quantum corrections are infinite. But, who cares? Neither one of those is an observable quantity. You need some technique (a regularization scheme) to keep track of the infinities to make sure they do cancel out, but that's about it. Moreover, thanks to Ken Wilson, we now have a very physical picture of where the infinities come from - they merely parametrize our ignorance about high energy physics. Any statements you see from famous physicists about the unsatisfactory nature of renormalization were invariably made prior to Wilson's work. -- ====================================================================== Kevin Scaldeferri Calif. Institute of Technology The INTJ's Prayer: Lord keep me open to others' ideas, WRONG though they may be.