Feynman Diagrams and Beyond

The Spring 2009 IAS newsletter is out, available online here. It includes the news that the IAS is stealing yet another physics faculty member from Harvard, with Matias Zaldarriaga moving there in the fall.

The cover story of the newsletter is called Feynman Diagrams and Beyond, and it starts with some history, emphasizing the role of the IAS’s Freeman Dyson. It goes on to describe recent work on the structure of gauge theory scattering amplitudes going on at the IAS, emphasizing recent work by IAS professor Arkani-Hamed and collaborators that uses twistor space techniques, as well as Maldacena’s work using AdS/CFT to relate such calculations to string theory. Arkani-Hamed (see related posting here) says he’s trying to find a direct formulation of the theory (not just the scattering amplitudes) in twistor space:

We have a lot of clues now, and I think there is a path towards a complete theory that will rewrite physics in a language that won’t have space-time in it but will explain these patterns.

and explains the relation to AdS/CFT as:

The AdS/CFT correspondence already tells us how to formulate physics in this way for negatively curved space-times; we are trying to figure out if there is some analog of that picture for describing scattering amplitudes in flat space. Since a sufficiently small portion of any space-time is flat, figuring out how to talk about the physics of flat space holographically will likely represent a real step forward in theoretical physics.

One IAS member who is also working in this area is Emil Bjerrum-Bohr, a great-grandson of Niels Bohr, and the newsletter has an article about him and the various members of the Bohr family who have been at the IAS at one point or another.

For one more piece of news related to Feynman diagrams, Zvi Bern et al. have a new paper out where they explicitly construct the four-loop four-particle amplitude, for N=8 supergravity, and show that it is ultraviolet finite in both 4 and 5d. This provides yet one more piece of evidence for the ultraviolet finiteness of N=8 supergravity. Remember all those claims made for string theory that it is the only way to tame the short-distance fluctuations of a quantum theory of gravity?

Update: One of the authors of the four-loop paper wrote to me with some comments about it, which he gave me permission to post here:

I just wanted to point out what I see as two of the interesting things with this calculation:

1) Honest four-loop QFT calculations in (massless) gauge and gravity theories are now possible, if not exactly trivial. This isn’t just “big fancy computers.” Sure, computers help with the book-keeping of the calculation, but no computer in the world could have accomplished this by naively marching through Feynman diagrams (just look at the size of the expression of 3-graviton Feynman rule in your favorite gauge, and do vertex counting on the number of distinct graph topologies). Rather, this is due to advances in understanding how to manipulate lower-loop and tree-level scattering amplitudes to get (complete) higher-loop scattering amplitudes.

To understand how powerful this is, consider the following: the construction of the four-loop four-point N=4 super-Yang-Mills amplitude required (as input) nothing more complicated than the Parke-Taylor expressions for MHV three-, four-, and five-gluon scattering amplitudes in four dimensions — not even requiring the (very nice) recursion relations for higher point trees mentioned in the IAS piece above. (Verification, of course, required more 🙂 ). If you’ve seen the Parke-Taylor expressions you’ll know how simple they are! The construction and verification of the four-loop N=8 supergravity amplitude requires only knowing the four-loop four-point N=4 super-Yang-Mills amplitude.

Even had we not gotten the nice result regarding the tame UV behavior, getting to the point where these types of calculations are doable is I think important in its own right, and possibly even more important in the long-run. I should probably point out that these types of approaches can and are being generalized to more physical theories, like the exciting high-multiplicity one-loop QCD work going on.

2) Maybe there’s a perturbatively finite (point-like) QFT of gravity in 4D. This is exciting as it suggests that QFT could be a more powerful framework for describing the universe than people have been giving it credit for recently. We do believe that, if it is perturbatively finite, it will be so due to some previously unrecognized symmetry or dynamical mechanism that once understood should greatly improve our understanding of gravity. There does seem to be some connection with the very good scaling behavior of tree-level pure-graviton amplitudes in theories related to Einstein-Hilbert gravity.

That being said, we really don’t have anything to say about its non-perturbative behavior. Really. Nothing at all. It absolutely could require non-perturbative completeness from string theory. It could already be non-perturbatively complete in a way that’s best described by a string theory in certain regimes (emergent string theory if you like). Maybe it only works with higher-dimensional invisi-pink elephants. I really don’t know; it’s not what we’re after right now. I certainly encourage people to consider working on non-perturbative N=8 questions if they’re curious!

Not to be overly contrarian, but I wouldn’t characterize any of this as a blow against string theory, and I don’t think most string theorists see it as such. String theorists have, on the whole, been very supportive of this line of research (even if it might mean a small technical modification of certain sentences in the introduction of certain texts 🙂 ). Besides one of our collaborators (Radu) also being a practicing string-theorist, we’ve met with a lot of support from all sorts of people who appreciate calculation, and are honestly curious about the results. Besides, there have been very strong string-theorists actively working on understanding this from the string-side. In terms of community support, i.e. not just good individuals here and there, Zvi’s been invited to talk at Strings ’09, Lance talked at ’08, and I think Zvi talked at ’07 if I remember correctly.

I, of course, can’t help but flinch a little when people glibly say string theory is the only way to talk about gravity (which is manifestly wrong, e.g. the CFT side of the AdS/CFT *duality*). Most thoughtful string-theorists I’ve met who say something similar, however, are using it as a shorthand for a much more long-winded statement which is accurate. Namely they’re compressing a statement regarding the level of understanding we’ve gained about gravity and gauge theories and non-perturbative solutions through string-theoretic analysis, which we haven’t from anywhere else. As we can see by my comment here, there are perils to giving in to long-windiness, so I tend to refrain from giving them too hard a time about it. There is trouble of course when similar statements are mindlessly parroted by the thoughtless, but the thoughtless tend to generate grief generically in any case.

John Joseph M. Carrasco

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3 Responses to Feynman Diagrams and Beyond

  1. bla says:

    no comments? this N=8 SUSY being finite idea looks interesting, I’d appreciate more info on this. Is it really a blow to string theory?

  2. Peter Woit says:


    Evidence for the finiteness of N=8 supergravity has been around for a few years now, I first wrote about it here:


    One reaction to this possibility from string theorists is to argue that N=8 supergravity has problems non-perturbatively. Another is to basically just ignore all evidence that there are QFTs with sensible perturbative expansions and keep on repeating the argument that “string theory is the only known way” to get a finite theory of quantum gravity.

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