## Sunday, October 24, 2010 ... //

### Richard Feynman and the birth of quantum gravity

Our theory of nearly everything (TONE) describes the forces in terms of an effective quantum field theory in which a few dozens of matter particle species interact through four fundamental forces - electromagnetism, the weak force, the strong force, and gravity.

Feynman and the four forces

Richard Feynman received his Nobel prize for the path integrals and especially their application to Quantum Electrodynamics; no doubt, he's been one of the most important physicists who have built our understanding of electromagnetism at the quantum level.

Together with Murray Gell-Mann, he refined Fermi's four-fermion interactions. These chaps added the right tensorial structure - namely the vector-axial-vector structure - to the indices in the four-fermion interactions. They were sure that they were right even though a famous experimenter claimed that his team had proved that the interaction was of a scalar-tensorial type. That was Feynman's key contribution to the weak force.

Obviously, Feynman has also contributed to the strong nuclear force - by his discovery of the partons, an alternative description of quarks (and gluons) that builds on their appearance in the scattering of hadrons. Feynman's contribution to the strong force was immense, too.

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However, TONE requires one more force, namely the force of gravity. Did Feynman contribute something? Well, I think that his contributions to this realm of knowledge are usually underestimated. He was one of the key people who made the birth of quantum gravity possible. What do I mean? Let's look at some lessons of the history of general relativity (GR) in the 1960s.

History of gravity from the late 1950s

As early as in the late 1950s, people began their attempts to apply quantum reasoning to gravity. Note that in the 1950s, their understanding of the nuclear interactions was something between tiny and none - so they assumed that only gravity and electromagnetism were real and important forces. Electromagnetism had already been successfully upgraded to the V2.0 quantum version so gravity was supposed to be the next step.

What did they mean by the task of understanding gravity in the quantum language?

In 1950, the Bergmann group as well as Paul Dirac analyzed systems with constraints. They ended up with the first-class and second-class constraints. In 1959, Paul Dirac fully rewrote GR in the "canonical form", as a Hamiltonian theory, using the constraints. Why did they do it?

Denial of the importance of renormalization

Well, those people didn't "believe" (i.e. didn't understand) renormalization so they were focusing on tree-level physics - physics given by Feynman diagrams without loops, if you use this terminology. But they thought it was important to rewrite GR in the Hamiltonian form and use this form to calculate the tree-level scattering amplitudes. They thought it was an amazing task.

Paul Dirac has made huge contributions to quantum mechanics, the unification of its early forms, and the description of the Dirac field, of course. However, quantum field theory including loops was just too conceptually hard for him, so he has never made any contribution to quantum field theory beyond the Dirac equation - which belongs to the classical part of QED today.

In fact, as recently as in 1975, Dirac wrote:
I must say that I am very dissatisfied with the situation because this so-called 'good theory' does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it is small — not neglecting it just because it is infinitely great and you do not want it!
Too bad, Sir. Depending on how you organize the renormalization procedure, the "infinite" terms are either properly added to compensate similar but opposite terms in the classical part of the calculation; or these "infinitely looking" terms are actually small or zero in the physical sense.

At any rate, Dirac's opinion that the procedure was invalid has clearly been shown incorrect and this fact became much more obvious with the birth of the renormalization group which showed, among other things, "why" the seemingly ad hoc procedures work. But the agreement of the renormalized amplitudes with the observations was there decades before the renormalization group and this empirical evidence simply couldn't have been ignored.

Dirac, despite of his complete misunderstanding of quantum field theory at the loop level, was the smartest person among the "canonical" people. All the other people who focused on the "canonical quantization" of GR were mostly Dirac's irrelevant inferior appendices. And there are still many people around today who are doing nothing else beyond just promoting Dirac's confusions about renormalization.

The people who understood their discipline in the 1960s already knew that the future of physics wasn't going to be about some curved or constrained phase spaces - the type of "small" modifications of the simple physical systems that Dirac et al. understood at the beginning. In fact, important quantum field theories may work with totally flat and unconstrained phase spaces. The new physics was hiding elsewhere and renormalization of the loop amplitudes was a key part of the story.

The irritating gathering

However, the "canonical people" continued their business that was increasingly physically vacuous. One of the peaks of this misguided enterprise was the 1962 Warsaw Conference on the Theory of Gravitation where the canonical treatment was a "big topic", too. Richard Feynman attended that conference and he summarized his impressions as follows:
“I am not getting anything out of the meeting. I am learning nothing. Because there are no experiments, this field is not an active one, so few of the best men are doing work in it. The result is that there are hosts of dopes here (126) and it is not good for my blood pressure. Remind me not to come to any more gravity conferences!”
This letter to his wife was a non-technical description but the reasons why Feynman realized that the 126 other participants were dopes were much more specific. In fact, in the same year, he proved something that looks totally obvious to us today - it's a part of our terminology - but it was not obvious to the people in 1962. He proved that the ensemble of all tree-level scattering amplitudes contains the same information as the classical scattering of classical gravitational waves!

His certainty that the quanta of the gravitational fields are the spin-two gravitons - fully analogous to the quanta of the electromagnetic field, the spin-one photons (an inevitable point that most physics fans fail to understand even today) - was of course extremely important for him to be willing and able to derive any results that were more nontrivial than the very existence of gravitons.

Fine. So he knew that the tree-level amplitudes encoded the classical theory. It followed that if you go through all this formalism with first-class and second-class constraints, you translate all the things into the ugly Lorentz-symmetry-obscuring quantum formalism, and you compute the scattering amplitudes, you will only get data that are directly transformed from the classical behavior of gravitational waves according to classical GR!

If you actually want to predict the results of hypothetical or doable experiments, which is the ultimate application of any laws of physics, the calculations will give you nothing that you couldn't get from classical GR. (Recall that those other people didn't accept loop calculations so the tree-level amplitudes were everything that was possible or acceptable for them.)

By a relatively simple conceptual argument, Feynman was able to show that their "dreamed about" calculation of the scattering amplitudes in what they considered to be "quantum gravity" was just a tautological reshuffling of classical GR. From a physical perspective, it contained nothing new or valuable at all!

The facts about the graviton amplitudes that are obvious to every student of string theory today - that the loops are needed to go beyond the classical level - had to be "discovered" by someone and it was Feynman. Only with his classification of the amplitudes, it was sensible to ask what quantum effects actually do to the physics of GR.

In the mid 1980s, after the first superstring revolution, Feynman didn't believe that string theory was the only way how to "skin the cat" i.e. how to regulate the divergence of quantized GR. As of 2010, it still seems that he was wrong on this issue. String theory remains the only known computational framework to regulate the loop amplitudes in the framework of "tree-level plus loop" amplitudes that Feynman introduced and that the string theorists just refined.

Although, truth to be said, the previous sentence is only true with the modern, "extended" definition of string theory that includes all the vacua described by string theories of any kind or M-theory, compactified on any acceptable background with any consistent branes and fluxes: string theory has become a much richer theory which is no longer just a "theory of strings". From a different viewpoint, we could say that string theorists have indeed confirmed Feynman's expectation that there are many ways to "skin the cat" - a whole landscape of ways, in fact. It's plausible that Feynman - if alive - would interpret the current situation as a confirmation of his expectations. ;-)

The notion that a consistent theory of quantum gravity has to give you a unitary S-matrix and its tree-level approximation simply encodes classical GR is due to Feynman. In a Feynman-centric presentation of physics, one can say that the string theorists have essentially changed the "shape of the wooden earphones" or Feynman diagrams only. ;-)

The good ghosts

Unlike those "canonical" guys including Dirac, Feynman actually understood quantum field theory at the loop level: a new generation of physicists (than Dirac) was apparently needed for this extra step, much like a new generation of physicists (than Einstein) was needed to go from classical relativistic physics to quantum mechanics. With his modern knowledge, Feynman could try to attack general relativity at the loop level. And indeed, he actually began with this essential job in 1963; the work was partly published a year later.

The discovery that Feynman did in 1963 was very important: he tried to apply Gupta's rules how to deal with the local symmetry (the recipe: just ignore all the unphysical polarizations of gauge bosons) - something that has previously worked in electrodynamics (and photons) - to GR. And he was able to show that Gupta's rules broke down!

They didn't work because unitarity was violated already at the one-loop level. (Feynman, using some awkward combinatorial methods, had already some de facto mistakes at the two-loop level but these things are really getting harder too quickly.)

But Feynman went beyond these negative insights. He was able to figure out that the discrepancy may be fixed if one adds new fermionic fields. These fermionic fields are needed to reconstruct the gauge symmetry and/or unitarity at the quantum level in any theory whose local symmetries form a non-Abelian group: that includes both GR and Yang-Mills theories, aside from more exotic theories (with p-forms and other things: local supersymmetry is another example; however, in supergravity, one usually needs the extended and more obscure BRST formalism called the Batalin-Vilkovisky or BV formalism).

Paradoxically, these extra fermionic fields that fix the problems with unitarity at the tree level are nowadays known as the Faddeev-Popov ghosts - because of a newer, 1967 paper about the quantization of Yang-Mills theories by the two Russian authors whose work was probably not quite independent of Feynman's findings. Under the new FP name, these fields have become a part of the modern BRST quantization in the 1970s. The Faddeev-Popov name is used for gravity, too.

Physics vs formalism: again

But I want to re-emphasize that it was Feynman who actually focused on the physics - rather than meaningless games with the formalism - and was able to separate what was new and what was not new in the perturbative calculations of the amplitudes. For him, physical theories were always gadgets into which you have to insert something and that spit a result that is observationally relevant. And he always wanted to know where the information was actually hiding.

I don't have to remind you that the mindless people who would just have fun with the meaningless "canonical" masturbations with the formalism but who wouldn't understand what it meant haven't quite died away. This community survived and is currently known as the "loop quantum gravity" community. Their ignorance about physics is exactly as big as the ignorance of their predecessors in the late 1950s; they haven't learned anything whatsoever during the last 50+ years. They're still the same hosts of dopes.

They still think that the formalism itself is physics even if they don't understand it; but it's not. Physics only comes out of the formalism if you know what the formalism means and how to deal with it. In that case, you also know what's important about it and what's not; what quantities depend on conventions and schemes and which of them don't; how you have to organize your calculations to determine particular answers at a given level of accuracy.

Hawking evaporation: the priority

To show the Feynman really understood what the individual terms in the amplitudes meant, let me recall a story from a theater show that included actor David Gross. According to that history, Feynman discovered the Hawking radiation - including the right formula - in 1972, i.e. before Stephen Hawking:
In 1972 before Hawking came out with the Hawking radiation formula, Feynman was meeting with Kip Thorne's grad students, Bill Press, Saul Teukolsky & Lightman. They discussed a recent calculation of shining light on a rotating black hole and getting more energy out then in at expense of decreasing rotational energy of the hole. They all went back to Lightman's office. Feynman said: "Hey this is like stimulated emission. So he went to black board and did a A & B coefficient model and then when angular momentum J of the black hole J -> 0 there was still "A" spontaneous emission and it was the later Hawking formula!
It's probably possible to calculate the Hawking's "spontaneous" radiation along these lines but I wasn't there to be sure that the correct "A" was actually written on that blackboard. ;-) Hawking is a great physicist who has made a huge discovery (and many other great disoveries) but I see no way to disprove that Feynman had everything he needed to get the result in 1972, too. So he could do it before Hawking, indeed. Feynman has notoriously refused to publish many of his important discoveries.

John Wheeler and the birth of quantum gravity

Another guy who did important contributions to quantum gravity in the 1960s was Feynman's former adviser, John Wheeler. In 1963, he realized that the Planckian regime of quantum gravity becomes very violent - and he coined the concept of quantum foam to describe what was going on at this shortest distance scale in physics. He also helped to develop the Wheeler-DeWitt equation in 1967 - although this equation has remained a "possible useful tool for the future" rather than something that can already be reliably exploited. And he gave the name to the "black holes" in 1967, too. That helped the people to focus on these important objects - much more so than when they were called "collapsars" which led some people to think that they were just some messy stars.

Needless to say, the lasting contributions of the other 126 participants of the 1962 Warsaw Conference were close to zero. So maybe, the early 21st century is not the first moment in the modern history when the overwhelming bulk of an otherwise respectable, high-IQ discipline was composed of dopes. But is it really necessary that people and societies make no progress in 50 years?

A huge portion of the community is still unable to learn the lessons that have been around for nearly half a century. They continue with their dumb canonical quantizations of GR, even though it's obvious that one can't learn anything about quantum gravity in this way, and they present their half-baked classical pictures how the Universe could have behaved around t=0 (or "before") even though it's obvious that such classical stories can't settle anything about quantum gravity and quantum cosmology, either.

The BBC program has shown that even in 2010, most of the people in the subfield are dopes. Unlike their predecessors in 1962, they're dopes who have learned nothing out of 50 years of failures of their predecessors.

It's sad but that's the memo.