A New (or Old?) Way to Make Gravity Consistent with Quantum Mechanics?
In 1916, Albert Einstein proposed his general theory of relativity, which describes the gravitational force in terms of curved space-time. This theory has successfully predicted gravitational lensing of light, relativistic time delays, black holes and gravitational radiation (the latter has only been detected indirectly so far). Yet all of these tests have been at the classical level, ignoring the effects of quantum mechanics. For several decades, physicists have tried to extend Einstein's general theory of relativity into the quantum realm. Experimental tests are hard to come by here, so theoretical consistency has been the coin of the realm. Recently, a few different theoretical investigations have suggested that an old theory of quantum gravity might actually be consistent, contrary to the prevailing wisdom over the intervening decades.
Early attempts to make consistent quantum theories of gravity assumed that all particles were point-like, with no extended structure. Such an approach has worked extremely well for the other three known forces, the strong, weak and electromagnetic interactions. The quantum theory of these forces, the Standard Model, has survived intense experimental scrutiny for three decades.
However, theories of quantum gravity with point-like particles always seemed likely to be inconsistent, and intrinsically unpredictive, due to nonsensical infinities. These infinities arise because the gravitational force on a particle is proportional to its mass or energy. In quantum corrections, virtual particles can carry very large energies. Summing over the possible energies gives an infinite result. In contrast, the electromagnetic force on a particle is proportional to its electric charge, which does not increase with energy. So quantum corrections in electromagnetism are much better behaved than those in Einstein's theory of gravity.
There are infinities in quantum electrodynamics, or QED, but they are mild enough that they can be removed systematically through a process called renormalization, leaving behind finite predictions. Richard Feynman, Julian Schwinger and Sin-Itiro Tomonaga received the 1965 Nobel Prize for understanding how to do this in QED, paving the way for experimental tests of the theory with incredible precision, better than a part per billion. Similarly, in the 1970's, Gerard 't Hooft and Martinus Veltman demonstrated that the more complex theories of the strong and weak interactions could also be renormalized, work for which they were awarded the Nobel Prize in 1999. Precise experimental tests of the Standard Model have since been carried out at the Stanford Linear Collider and elsewhere, to the accuracy of parts per thousand.
It has been known for twenty or thirty years that most quantum gravity theories with point-like particles are not renormalizablethat is, the infinities cannot be removedbecause the gravitational force is proportional to the energy. A way around this problem has been provided by superstring theory, which postulates that particles are not point-like, but actually objects extended in one dimension, like little loops of string. Because strings are fuzzy and spread out, their interactions at very short distances are much weaker than those of point-like particles. This cures the quantum gravity infinities. However, the notion that particles are no longer point-like is a radical one, so it is worth exploring whether there are any alternative ways to cure the infinities.
My collaborators, Zvi Bern, John Joseph Carrasco, Henrik Johansson, David Kosower, Radu Roiban and I, have been studying a point-like theory of quantum gravity in which the infinities, or divergences, are cancelledat least in partby a great deal of symmetry. The symmetry we use, called supersymmetry, relates force-carrier particles called bosons to matter particles called fermions. We want to use the maximum amount of supersymmetry we can, which turns out to be eight times the amount that might be found at the Large Hadron Collider. The theory is called N=8 supergravity, and it was constructed by Eugene Cremmer, Bernard Julia and Joel Scherk in the late 1970's.
No-one knows how to put the Standard Model into this theory, but we are interested in whether it could be a first example of a sensible quantum gravity theory with only point-like particles. N=8 supergravity was studied for a while in the 1980's, and then fell out of fashion. As the noted historian of science Stephen Hawking discussed in 1994, "Reports of the death of supergravity are exaggerations. One year everyone believed that supergravity was finite. The next year the fashion changed and everyone said that supergravity was bound to have divergences, even though none had actually been found."
In a recent paper, we looked at how the force-carriers of gravity, called gravitons, scatter off each other in N=8 supergravity, through the third order of quantum-mechanical corrections. This was the first order at which infinities had been predicted to appear in the 1980's. However, we found no infinities. Not only that, we found novel cancellations not predicted by arguments based on supersymmetry. Certain of these new cancellations will persist to still higher quantum-mechanical orders. If we can understand all the patterns better, we might be able to show directly that the theory has no infinities at any order. Some other lines of reasoning, due to Nathan Berkovits, Michael Green, Jorge Russo and Pierre Vanhove, seem to point toward the same result. Their methods are less direct than ours, but they extend in principle to higher orders than our current results. One argument studies graviton scattering in a ten-dimensional superstring theory which contains within it N=8 supergravity. If one can extrapolate certain properties of the stringy scattering to N=8 supergravity, then this theory should be finite through the eighth quantum-mechanical order. A second argument, for finiteness to all orders, rests on postulated properties of a mysterious eleven-dimensional theory called M theory, which contains N=8 supergravity as well as several ten-dimensional superstring theories.
In summary, a new calculation of ours, together with other recent investigations, suggests the real possibility that an old theory of quantum gravity with point-like particles might be finite and consistent after all.
—Lance Dixon, March 29, 2007
Above image: A diagram providing a "window" into N=8 supergravity at the third quantum-mechanical order.