QED Calculations: Progress Made in Last 25-30 Years

[ObsessiveMathsFreak]

I'm not sure if this is the right forum for this topic, so apologies if I got it wrong.

I've been reading the Feynman Lectures on Physics. In it, Feynman states that though Quantum Electrodynamics is highly successful, it is still extremely difficult to evaluate the equations to obtain a theoretical result to compare to experiment. I believe he said it was the integrals that were the difficult part(I imagine this would indeed be the case).

The Feynman lecture were written/given in the sixties I believe, but I've also seen videos of Feynman in Auckland University in 1979, where he again reiterate this fact, and even states that there are experiments for which no-one has been able to evaluate a theoretical result.

The question I would like to ask is; what progress has been made on evaluating such integrals in the last 25-30 years. Specifically, have computers and computer algebra systems helped to tame this task? Can anyone give an example of the integrals QED theorists are faced with, if indeed it is the integrals that are giving the trouble.

Is this aspect of QED still a serious problem, or is it simply a question of throwing more CPU cycles at the problem. Naive I know, but my real question is can the equations nowadays be beaten into submission?

 

[nrqed]

I'm not sure if this is the right forum for this topic, so apologies if I got it wrong.

I've been reading the Feynman Lectures on Physics. In it, Feynman states that though Quantum Electrodynamics is highly successful, it is still extremely difficult to evaluate the equations to obtain a theoretical result to compare to experiment. I believe he said it was the integrals that were the difficult part(I imagine this would indeed be the case).

The Feynman lecture were written/given in the sixties I believe, but I've also seen videos of Feynman in Auckland University in 1979, where he again reiterate this fact, and even states that there are experiments for which no-one has been able to evaluate a theoretical result.

The question I would like to ask is; what progress has been made on evaluating such integrals in the last 25-30 years. Specifically, have computers and computer algebra systems helped to tame this task? Can anyone give an example of the integrals QED theorists are faced with, if indeed it is the integrals that are giving the trouble.

Is this aspect of QED still a serious problem, or is it simply a question of throwing more CPU cycles at the problem. Naive I know, but my real question is can the equations nowadays be beaten into submission?

To get an idea of the state of the art, you could look up the work of Toichiro Kino****a of Cornell and his work on the calculation of g-2 (If I recall he completed the four-loop calculation). It's more tricky than just throwing CPU at it because of all the divergences involved. One has to take care of nasty overlapping divergences and renormalize things in a very clever way.
Not surprisingly, things are even more difficult in a bound state. I did a two loop calculation in positronium for my thesis and it was doable only because I used a clever technique developped by my adviser and which was applicable only because positronium is nonrelativistic.

A lop of people are working on NNLO (next to next to leading order) contributions in things like the top quark decay and other systems but usually the techniques are targeted at specific kinematical points.

 

[Feynman diagram]

Yeah, it's somewhat unfortunate that the fine structure constant isn't smaller than it is, and thus avoid us having to calculate third and forth order corrections ;)

 

[Haelfix]

Analytically not much progress has been made since the sixties. I know of only 2 counterexamples in 4d since then.

So people just keep doing perturbation theory, and yes the numerical methods have vastly increased in efficiency: Lattice methods, twistor methods, powerful algorithms for planar feynman graphs etc etc.

What is it now, they have QED down to 16 loops or something like that?

 

[nrqed]

Analytically not much progress has been made since the sixties. I know of only 2 counterexamples in 4d since then.

So people just keep doing perturbation theory, and yes the numerical methods have vastly increased in efficiency: Lattice methods, twistor methods, powerful algorithms for planar feynman graphs etc etc.

What is it now, they have QED down to 16 loops or something like that?

Are you sure? As far as I know, QED has "only" been done to 4 loops! (I am talking a bout a full calculation, including the finite pieces. Sometimes people may go a bit beyond if they are interested in just extracting the divergence structure for the renormalisation group analysis).

Sometimes, people wil talk about "eight order" or "tenth order" instead but this is referring to the powers of the coupling constant (roughly, the electric charge) but this is not the number of loops. For example, a one-loop calculation may be called either a fourth order calculation (because there are 4 powers of the coupling constant in the amplitude) or a second order correction to the tree level (because there are two more powers of e than the tree level).


I have never seen that, but maybe some people also give the powers of e appearing in the cross section or decay rate (in a measurable quantity). So maybe this is where the number 16 might have come from! A four-loops calculation would generate 8 powers of "e" relative to tree level, which when squared would give " a 16th order correction". Maybe that is the context in which you saw that number. But I am pretty sure that no complete 5 loops calculation has been done. When I was at Cornell, Kino****a (the world expert in g-2) had completed the 4 loops calculation and he did not intend to do the 5 loops, I think!


Regards

Patrick

 

[Haelfix]

Hi Nrqed. That sounds correct. Its been several years since I thought about this (probably dating back to when I was a grad student taking a class or somesuch), so its very possible my memory has transformed it into something erroneous (loop instead of order).

 

[Hans de Vries]

They are doing 10th order now, requiring 12 672 Feynman diagrams...

http://hussle.harvard.edu/~gabrielse/gabrielse/papers/2006/NewFineStructureConstant.pdf

Another nice overview is this one, although specifically on the tau's magnetic
anomaly:

http://arxiv.org/PS_cache/hep-ph/pdf/0701/0701260v1.pdf


Regards, Hans