(ALPHA) What IS renormalization?

This is an alpha thread, which means that we will talk about renormalization as it pertains to gauge theories as formulated by Feynman, Schwinger, Wilson and others in the latter half of the twentieth century. If you do not believe in renormalization, you should be prepared to give me a better way to calculate things in a quantum theory, or you should start your own thread. Thank you.

I will try to clear up some confusions about what renormalization is and what it is not. I can't write anything detailed at the moment, but if you have a question, post it, and I will try to construct my responses accordingly.

This is bound to fail miserably, but nevertheless I will try to make these things as clear as possible. Hopefully somewhere in my ramblings there will be something useful.

 

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In the 1920’s, quantum mechanics worked well with molecules and atoms… this was because they had a finite number of degrees of freedom. However, whenever we worked with the electromagnetic force, all that was result was erroneous calculations, because the electromagnetic field was infinite within its own degrees of freedom; in other words, two degrees of freedom in any point in spacetime. We envision these points as being oscillators, each with its own position and momentum.
It turned out that the oscillators could never be at rest because that would defy the Uncertainty Principle… Instead, the oscillators where assigned with zero-point fluctuations and even a non-zero energy! Whenever the little electron [wasn’t] being observed, the energies of all the infinite degrees of freedom would make the little bugger’s mass and charge to become infinite…
This was settled eventually in the late 1940’s with a simple procedure called ‘Renormalization.’ It in simple extracted large numbers by subtracting them with other large numbers; infinities minus infinities. But the main goal was to leave a finite sum – a non-zero remainder.

 

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Obviously weare now attempting to renormalize the strong force without any finite remainders.

 

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How so?

 

Well, it is to my understanding, we can unify the weak with the electromagnetic and magnetism to fold. With power of 100GeV, or maybe even just below it, like 90GeV, we can be able to unify the electro-magnetoweak force together... but to unify the strong with renormalizational processess, we find still a few left over remainders. Supergravity, as i believe, could be a possible candidate for helping with this hypothesis. It is itself intrinsic to spin1/2 fields, creating for the superpartner sublevels with an amazing gallery of particles, some up to 70 a time! But they, unfortunately, are no all accounted for on the standard model.
I suggest a new way to all of this. Perhaps if we have our calculations wrong on the strength of the forces, is it possible to tune the universe into configuration that has electromagnetism as the most powerful force of all? I know... you're gonna call me a whack-job, a pure crank, but i'm serious. Whilst everything should be found to extrapolate back to the gravitational field, Einstein never said in his equations that this ''Quantum Gravity Superfield'' wasn't itself a diverse combi-of electromagnetism as well... but he neglected strong force out of his calculations, because of some strange reason. I think it was because he was trying to develop his hallucination called the Grand Unificational of Gravitational Potential Fields at the time... What a waste of intelligence.We might today have known the unification of everything, if he could fix the discrepency found between electromagnetism, and the gravitational, caused by two very powerfully-symmetrical functions of weak and strong?

If the electromagnetic is the strongest, the here comes my even bigger pseudotheory... What if both weak and strong forces are electromagnetic in primordial origin? - ''A theory attempting to eliminate them as individual forces, but two sides of electromagnetic fluctuations and fields... since there are Four known electromagnetic fields... 9But consistute as one according to the dogma of physics.
In a nuclear reactor, we squeeze matter into a strong forced state, fusing matter and energy into forceful, mass-warping states... beyond really any superfluidity created in other controlled labs. What if it is something even stranger...?

We know that even quarks and gluons and neutrons, and pentaquarks, and tachyons, muons, and taons, neutrino's, and axions, and photons all have antipartners, and they will produce a strength of 511KeV converted to the second power from their interaction; two photon gamma rays.
Everything can be traced back to the electromagnetic field, as much as it can with the gravitational. Singularities, gravitationally-stressed regions of spacetime don't need to exist, according to Dr. Hawking. He can actually remove them, and replace them entirely with wormholes. So why gravity?//////???//////?
When we renormalize the vacuum (E=Mc^2+E=-Mc^2=0) we SHOULD reach a total zero compound, but we do not. Instead, we analyse a non-zero totality. This non-zero total is a problem, because it is the indication that dark matter and energy exist. But what scientists have missed out, is that Hawkings recently created Quantum Cosmolgy. A mathematically-riddled composition describing the universe at t<1, and describing it as a subatomic particle. So, just like a particle, (since electrons have a finite remainder after renormalization), so should the universe no?

 

This is not quite true. The strong theory, as is, is renormalizable. Renormalizability is something different from unification.

Supergravity is a generic term for a supersymmetric gauge theory in which the supersymmetric transformations are local, as opposed to global.

No. The electromagnetic force is well studied and its' strength is very well understood. I will make detailed posts on this subject later. Perhaps you would be well-served to have a detailed understanding of renormalization first.

Let me describe renormalization as it is currently understood by working physicsts. In the future, once we have thoroughly understood that, let us then examine alternatives.

 

When i said renormalize, i didn;et mean with the Strong. With this, the heavy three bosons allow it to be renormalized.

But, its not satisfactory - according to Hawking.

 

This is one of my favorite questions. I love thinking about this one.

I don't know why though.

 

Thinking about what Klip?

 

I'm having trouble understanding exactly what you mean. The weak force has three (massive) bosons, the strong force has eight (massless) bosons. Again, these things don't have anything to do with renormalization, other than to do calculations beyond first order, on has to renormalize them. But this is true for ANY quantum field theory.

Hawking, I am sure, is not happy with the effective field theory approach. I don't really know what ``unsatisfactory'' means, as EFT's don't pretend to be anything that is fundamental.

 

This is what i am saying my friend. I made an error when posting. Itype that fast, i sometimes get mixed up...
The strong cannot be unififed with the electroweak is what i meant, and that electroweak was unified to 100GeV.

 

The strong, obviously with quarks and gluons through QCD.

Having obtained a renormalization for the strong and the weak, we needed to combine the two together. This is what is unacceptable, because there are a lot of problems... one being energy required.

 

Ok, as promised, here is the first in a series of essays (?) on what renormalization actually means, from someone who has renormalized things before, and does so on a semi-weekly basis.

The first thing to understand is that everything is an effective description. Take a painting, for example---some French Impressionist. If you stand from a distance, you can see trees, or lillies, or a bridge, or something. But the closer you LOOK at the painting, the less it looks like flowers or trees---it starts to look less and less focused.

This is what physics is, in a nutshell. We have some well-defined picture of what things look like when we are standing a long way away, and the closer we get to the painting, the less it looks like something we recognize.

Here is another example. Take a hydrogen atom. Bohr computed the energy levels of the hydrogen atom from fairly general assumptions. The Bohr model is predictive and accurate---one can measure the ionization energy of hydrogen to within a few tenths of electron volts, and it matches Bohr's prediction very well. The picture is a familiar one---electrons orbit the nucleus in nice orbits, like planets orbiting the sun.

But what happens if you try to extend the Bohr model outside of its range of validity? Bad things happen---one can easily generalize the Bohr model for arbitrary atomic number, called Z:

\(E_n = \frac{Z^2}{2 n^2}m_e \alpha^2 \).

Typically, we work in units where hbar = c = 1, to make life easier. If you want to know where all of the hbar's and c's go, look it up on Wikipedia. The first thing I want everybody to notice is that the only numbers with units are E and m. In some sense, this proportionality was inecitible---the electron's mass IS the only number with units in the problem...the proton is treated as stationary, and doesn't matter much. (This is akin to the mass of the earth dividing out of the calculation when we calculate orbital periods of satellites, or something.)

The second thing that I want people to be aware of is that this model is absolutely useless for pretty much anything other than hdrogen. As Z gets larger and larger, the agreement with experiment becomes worse and worse. Even if you look closely at hydgrogen (i.e. get closer to the impressionist painting), there are fine and hyperfine splittings of the energy levels, due to magnetic interractions between the proton and electron.

So let's sumarize. If you are looking at the hydrogen atom from ``far away'', the Bohr model is a great description. If you scrutinize the predictions, however, Bohr becomes less and less palatable, and eventually you have to throw the whole model out and build something new. Once you try to apply Bohr's model outside of a range where it was designed to be useful, you can no longer trust the predictions.

What does ANY of this have to do with renormalization? Renormalization is the process of looking at theories in the regime where they apply. Let's take an example which surfaces ALL the time in computing loop diagrams in QFT.

\(\int_0^{\infty} \frac{d^4p}{(2\pi)^4}\frac{1}{p^2}\)

Typically, one works in spherical coordinates, and ends up with something like

\(\int_0^{\infty} \frac{p^2dp}{4\pi^2}\frac{1}{p^2} \rightarrow \int_0^{\infty} dp = \infty\).

This is a phase space integral, which you are probably quite familiar with if you have ever done any calculations in statistical mechanics. The first thing that you should notice is that this integral is CLEARLY divergent. If you are a mathematician, the first thing you should notice is that this expression drives you out of your skin. If you like:

\(\lim_{a \rightarrow \infty} \int_0^a dp = \infty\).

Either way, the point is the same...this integral is divergent in every sense of the word. Now the pivotal question...WHY is the integral divergent? Clearly it is divergent because of the upper limit---what if the upper limit weren't infite?

Let's think about this statement for just a second, and what the integral means? The integral is a phase space integral---in other words, we are integrating over, say, photon momenta. Why should the photon be allowed to have infinite momenta? This corresponds to a wavelength of zero, and CLEARLY that isn't physical. How can light have zero wavelength?

I think I will leave yall with that question, for now. Discuss.

 

Good question. Not sure to be honest.

 

Was the question rhetorical Ben, because you haven't provided any answer...?

 

Well, kind of.

The point is that infinite energies correspond to unphysical degrees of freedom. This is what the integral above says---for example, if we are computing a simple process like electron + positron goes to muon + anti-muon. One can compute quantum corrections to the classical result, and end up with integrals like the one above.

The quantum corrections are loop diagrams: for example, electron plus positron can anihilate into a photon, which can turn into a muon and anti-muon, like this:

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The corrections are quantum mechanical effects that aren't taken care of in the classical limit. They look like this, for example:

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The quantum process here is the creation of a ``virtual'' electron positron pair in the middle---that is, the photon which is being exchanged turns into an electron and positron, which quickly anihilate back into the photon. They are called ``virtual'' because we cannot see them, but we know that they're there---we can calculate and measure their effects.

When we compute the effects of the virtual electron positron pair, we are left with integrals that look like

\(\int_0^{\infty} dp \rightarrow \infty\)

which we encoutered in the previous post. The problem is that these virtual processes DO occur, but the answers that we measure are finite. That is, this process contributes to something called Compton scattering, which we CAN measure very accurately.

Does this mean that all of QFT is wrong?

Of course not. The resolution is the fact that we are integrating over unphysical momenta---the integral goes all the way up to infinity, but there is clearly no way for the electron/positron in the loop to have infinite momentum.

The main point of all of this is that you can think of renormalization as a procedure to cut off an unphysical region of your integral. There are much more fancy procedures for doing this, but in general you could do something like this:

\(\int_0^{\infty} dp \rightarrow \int_0^{M} dp \),

where M is called the cutoff. The amazing thing you find is that the answer doesn't depend on M!!! So what you have done, is you have cut off a region of parameter space which is unphysical, and your final answer doesn't depend on how you did the cutoff in the first place.

This is why I don't understand Hawking's comments---he said (aparently) that ``Renormalization is not proven''. But there's nothing to prove---all that you're doing is limiting the theory to a place where you expect it to apply. For example, the Bohr model example that I used before. You wouldn't think of trying to compute the electron energies of uranium with such a model---it JUST doesn't apply. Similarly, you wouldn't think of computing QED with infinite photon energies---it just doesn't apply there.

 

I was used to dealing with infinities, when working with electrons... not so much photons, so as you can see, i find this exciting.

 

Personally, i think renormalization is a dud. Whilst it has been successful, i think something more sinister lurks behind these calculations.

 

Renormalization is just a way to deal with the fact that we don't know what the final theory is. It can't be ``a dud'' because it gives the correct answer.

 

An incomplete correct answer... correct in the sense that it seems to produce the correct answer. I have to admit, i think Hawking is correct on the non-provable side of it. No?