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- Thread starter Endervhar
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Suppose that you have some flawed theory (as QED) that gives the next energy for a collection of electrons [itex]1,2,3,...[/itex]

[tex] E_{QED} = E_0 + E_1 + E_2 + E_3 + ···[/tex]

and [itex]E_1, E_2, E_3,...[/itex] are physically acceptable but [itex]E_0[/itex] diverges (infinite). Renormalization is a procedure to eliminate the unphysical term predicted by QED and get the physical

[tex] E_{renorm-QED} = E_1 + E_2 + E_3 + ···[/tex]

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Thanks, juanrga. A glimmer of light dawns, I think.

Having started with electrons 1, 2, 3.... where does the E

Also, does it have to be the E

Sorry if these are silly questions, but you would be hard presses to underestimate my mathematical understanding.

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That's a pretty difficult task since renormalization theory is indeed a somewhat abstract business, and without the appropriate mathematics it's hard to explain. So I'll try my best at the example of the most simple (and at the same time most successful) part of the standard model of elementary particles, which is quantum electrodynamics QED.

QED is, as the name tells, the quantum version of classical electrodynamics (CED), which becomes an approximation of QED under the appropriate circumstances. The good thing with QED compared to CED is that both, the charge-carrying "particles" (here we'll only consider electrons and positrons) and the electromagnetic "field" is described by one concept, quantum fields. The problem of CED was that the electron, i.e., point-like particles doesn't really fit into the more continuum-like theory for the electric field. Already there the pioneers of what's now called "classical electron theory" faced major difficulties, when they tried to solve selfconsistently the dynamical problem of moving point particles and the electromagnetic field.

What one learns usually in the E+M lectures about CED is only an approximation (which, however, is pretty good and serves most of the practical purposes in describing everyday electromagnetic and optical phenomena and of course the demands of electrical engineering): Either one considers charges in a given motion, mostly described as the charge and current density, i.e., a continuum description of chraged matter and then calculates the electromagnetic field from it or one considers a given electromagnetic field and asks, how particles (or charged bodies) move in it.

Now the problem is that an accelerated charge radiates electromagnetic waves, i.e., besides the static Coulomb field around a charge at rest in addition an accelerated charge also generates a wave field that carries energy and momentum with it. Due to energy-momentum conservation this must have an effect on the particle since it looses the energy now carried away by the field, and that means that there is some kind of "damping force" from this radiation, which is the particle's own field. If one tries to take this radiation damping into account, i.e., the interaction of the particle with its own electromagnetic field, one runs into trouble of infinities or very unphysical solutions of self-accelerating particles ("runaway solutions").

Now, the usual approximation of a particle moving in a given field (which is due to other (!) charges and currents) but ignoring the self-interaction with its own radiation field can be seen in more modern language as a kind of perturbation theory, i.e., one ignores the interaction between the particle and its own field.

Another, closely related problem is that of the energy stored in the electric field even of the point charge at rest. If one tries to calculate the total energy of the Coulomb field of a point charge one gets a useless expression leading to an infinite amount of energy. On the other hand, this field is in a way part of the particle itself, and one should thus count the energy of the field as part of the particle's energy when it is at rest, but that's nothing else than the paritcle's mass.

Now, if you take the particle as a little ball of finite extent and calculate it's electrostatic field, you get a finite energy of this field, but then you have to take into account that equal-sign charges repell each other, and you have to assume that the charged "parts" of the particle are held together by some other force or stresses. All these stresses contain energy as well as the electromagnetic field, and at the end, taking all these energies into account, you should get the particle's rest mass (times [itex]c^2[/itex]). Then you can make the radius of you little ball smaller and smaller, but keeping its mass at the physical value. After all that's the mass that is measured when looking at the particle's motion. Thus, the infinite energy of the electrostatic Coulomb field is compensated by the stresses and perhaps some generic typo of mass, one cannot further explain by other means than just being a property of matter. At this point the physicists have done their first "renormalization" of an infinity by "renormalizing" the electron's mass by just using the physical value of the electron instead of an infinite one coming from the electrostatic field energy. Note again that this is all about classical electromagnetism of charged point particles!

All these troubles are of course inherited by QED, but there the physicists in the late 2040ies came to a much more systematic and better working solution than the physicists were able to get for CED: In QED you describe everything by quantum fields, i.e., the electromagnetic field itself becomes quantized and the electron is described by a quantized field of another type called Dirac field. Quantization is a pretty straight-forward thing, albeit a very abstract procedure. So I can only give some summary of what comes out:

One starts with non-interacting quantities, i.e., either the electromagnetic field for itself without any charges or currents around it with which it could interact or an electron without any fields around it with which is could interact. Then you make some pretty simple assumptions: (a) the energy of a particle or field should be positive, i.e., there is a state with the lowest possible energy, the ground state, and this should be the state, if neither an em. field nor electrons are present, (b) the theory should be causal, i.e., if one changes something at one point, nothing can happen which needs information exchange about this change faster than light, and (c) the theory should be local, i.e., if you perform experiments far away from each other, they do not affect each other causally (i.e., the fact that I do an experiment on earth, does not bother a physicist on alpha centauri concerning his experiments ;-)).

Making all this quite plausible assumptions and using a quite complicated bunch of math, one arrives at some remarkable facts:

(a) The theory must contain not only electrons but also another kind of particle, called an antielectron (or positron) with exactly the same mass as an electron but with precisely the opposite charge of the electron. Since the electron carries charge -e the positron carries charge +e. The values of mass and charge must be taken from experiment. They are not predicted by the theory.

(b) Electrons, as particles with spin 1/2 (which one has to know from experiment!) must be fermions, i.e., they obey the Pauli principle which inhibits any two particles to be in exactly the same state.

(c) Photons, the quanta of the electromagnetic field (in the same way as electrons are the quanta of the Dirac field) have spin 1 (also an empirical fact, not predicted by the theory!) and thus must be bosons, which do not obey the Pauli principle. In addition from experiment one knows that photons have a mass of zero, but that doen't need to bother us here.

Then one can consider interactions between electrons, positrons, and photons, and this also gives the electromagnetic interactions between these charges as well.

One immediate consequence of quantum theory is that there are always fluctuations since due to uncertainty relations between field components, quite similar to the famous Heisenberg uncertainty relations between position and momentum components. Even "the vacuum", where one has no real particles and fields around (by definition), there are still fluctuations of the electromagnetic field and electron and positron fields. The net charge of the vacuum is of course 0, but there are fluctuations of electron-positron pairs and also electromagnetic fields around.

The first consequence of this is that even the vacuum reacts to real charges and fields to a certain extent like a medium. Thus, if for example one puts an electron somewhere at rest, the vacuum with its fluctuating electron-positron pairs becomes polarized. Far from the real electron put somewhere, the electromagnetic field of this electron looks like a Coulomb as in classical electrodynamics, but if one comes closer to the electron, one finds more charge than seen from far away. This means that the QED vacuum acts as screening like a plasma: If you put a charge in a plasma, a medium of real particles, this screening effect is already understandable in the classical theory: An electron in the plasma is surrounded by some positive charge cloud since it attracts the positive ions in the plasma and repels the negative electrons in the plasma. The net charge is of course -e, the charge of the electron, but the cloud of other charges around it, screens this charge partially so that the range of the interaction becomes smaller. The same holds true for the QED vacuum: The net effect of the charge fluctuations is a screening of the Coulomb field, and coming close to an electron gives a larger interaction than one would expect by the charge of the electron when measuring the Coulomb field from far away.

In particle physics this is usually put in terms of momenta in particle collisions: To measure the charge structure of a charged particle, one shoots with other charges on it, and QED predicts that the net charge of the electron seen by another electron becomes larger when the reaction takes place at higher exchanged momentum. Indeed, this effect has been measured at high precision at LEP (the former electron-positron collider at CERN, where now the LHC is built).

To figure out this and other quantum effects, one has to calculate higher orders in perturbation theory, and these higher orders give usually divergent results, but one can show that all infinities can be systematically subtracted by lumping them into unobservable "bare" parameters of the theory, which are the normalization of the electromagnetic and Dirac fields, the mass, and the charge of electrons and positrons (measured at some momentum scale!). Due to some beautiful mathematical feature of the theory called "gauge symmetry", one can show that you do not need to introduce a photon mass, i.e., the photon mass stays 0 at any order of perturbation theory without lumping some infinities into some non-zero bare photon mass, etc. So with a finite number of parameters, you can do perturbation theory to any order, by subtracting infinities and using the finite physical values of these quantities.

The numerical values of the parameters (couplings and masses) depend on the scale of exchange momenta in scattering reactions between electrons, positrons and photons, where you measure them, but the theory tells you precisely by which (finite!) amount these quantities change when you change the energy scale. This "running" of the constants has been shown to hold true at high precision.

Anyway, QED is a triumph of quantum field theory, including the renormalization program. It's the most accurate theory of an important part of phenomena in nature! Some predictions of the therory for quantities like the magnetic moment of the electron or the Lamb shift in the energy levels of the hydrogen atom are confirmed at a level of 12 or more significant digits!

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Thanks vanhees71. This is exactly the sort of thing I was looking for. I've printed it as I feel sure I shall have to read it several times, and do lots of looking up. Should keep me busy for a while. Thanks again.

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Having started with electrons 1, 2, 3.... where does the E_{0}come from?

Also, does it have to be the E_{0}that diverges?

Sorry if these are silly questions, but you would be hard presses to underestimate my mathematical understanding.

I selected the most simple example of divergence. In this example the infinite E

There are other infinites in quantum field theory that are related to the interactions, for instance to the electron-electron interaction, but this is much more complex to explain. At this moment you would just know that the QED potential energy is also flawed and needs to be renormalized to give physically acceptable (i.e., finite) values.

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Some useful intoductory references http://www.ate.uni-duisburg-essen.de/data/postgraduate_lecture/AJP_2011_Olness.pdf" [Broken]

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DarMM

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I wouldn't put it like this. The QED potential energy is finite, it just appears to be infinite when you expand it in terms of the bare charge, expanded in terms of the physical charge it's fine.At this moment you would just know that the QED potential energy is also flawed and needs to be renormalized to give physically acceptable (i.e., finite) values.

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Thanks again, folks. After a long spell of not posting on this forum, to come back and get this level of response so quickly is fantastic!

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