Today on the arxiv Oswaldo Zapata wrote an essay on issues about fact and belief systems in superstring theory. Naturally, Peter Woit decided that this was really important and wrote a whole article about it. Here I will collect a few recollections that serve as a rebuttal/complement to some of those discussions. Mostly, I feel implicated by some of this discussion since at least one of my works (together with Juan Maldacena and Horatiu Nastase) is mentioned as changing the history of an idea from belief to fact.
The whole issue is to quantify the following statement: the AdS/CFT correspondence is a true fact.
So what is one supposed to make of this? There is no formal proof of that statement. There is no theorem in mathematics that lets one go from quantum gravity on AdS spaces (whatever that is) to a dual field theory that is generally strongly coupled. The coupling constant being strong means that we are short of tools to provide reliable calculations for all applicable quantities that are relevant for the discussion. We are expected to show this not for just a few calculations. We are expected to show the complete equivalence of all observables and phenomena between these two theories.
So clearly, the AdS/CFT is not a fact in the mathematical sense. However, many practitioners consider it a fact. How so? There is overwhelming circumstantial evidence in its favor. If you allow me a little digression in the theory of probabilities, there is natural place for belief. This is at the level of prior distributions in Bayesian statistics. (Here I am taking the point of view of Jaynes’ book on Probability theory).
We can give the statement: The AdS/CFT is true some probability p of being correct. At the same time, the converse The AdS/CFT is false, would be assigned a probability of 1-p. This is our belief (bias). A subjective probability quantifies how much value we give to each of the two statements.
At issue is what value should p attain before we can comfortably state that the first statement is correct. For a mathematical proof the only allowed value is p=1. However, providing such a mathematical proof is certainly beyond what we konw how to do so far, so instead we have to ask what is the value that one could reasonably asssign to p given the current knowledege and how close to being a fact the statement really is.
If one begins life as a skeptic (circa 1997), one would start with a value where p is of order 0.0001 lets say. Now, a paper on November of 1997 claims that such a correspondence is true and provides various pieces of evidence for it. Given such an evidence, one should upgrade the probability p to reflect such evidence. However, we do not know how to quantify the usual posterior distribution for p (mostly because quantifying the conditional probabilities in the Bayesian formalism is not feasible in this case). However, it is clear that p should have increased. Here, it depends on various theoretical biases how much one decides to increase p. Basically, we have to evaluate the various conditional probabilities based on a subjective estimate.
For me, personally, the value of p became about 0.5 then. Of course, it was not immediate. Reading the first paper on the subject was really difficult at the time because I did not know what to do with that information. Fortunately, Gubser, Klebanov, Polyakov and Witten provided a dictionary to compare calculations. This made various tests computable. Using properties of various protected quantities they were able to match the supergraviity spectrum with a corresponding spectrum of operators in the dual field theory. Again, quantifying the advance in p is hard for the same reasons.
I would say then that p~ 0.8. Why? Because supersymmetrically protected quantities provide partial evidence, but by no means are a complete test of dynamics. However, given the match of an infinite series of data, it would be unfair to say that p didn’ change. Even after many papers came out, one could claim that in 2002, p was between 0.95-0.99. Again, that is a bias on computing conditional probabilities. Should one claim that p~0.999 that wouldn’t change much the argument.
Once a conditional probability implies that something is almost a certainty, one should be allowed to call it a fact (that can be subject to revision if future evidence against it is found). In physics we can never do better than this. In mathematics, it stays a conjeture until it is proved. This is the heart of the matter.
Our paper provided another `infinite series’ of evidence that the AdS/CFT correspondence was correct. This evidence did depend on the coupling constant of the Yang Mills theory and was very non-trivial (yet very computable once we understood what was the correct calculation to do). I remember that when we were discussing the paper before it was released with our collleagues at the Institute for Advanced Study, there was general skepticism on the claims we were making over the lunch table. This all changed the day after it was published.
For me, the AdS/CFT correspondence is a working assumption. I would claim that the current value of p is about 0.99999, but that is just my personal bias in the end. It certainly is very close to one.
After all, over 6000 papers have been written on the subject, and although puzzles have appeared along the way, there has not been any real evidence that the correspondence is fundamentally wrong. As a matter of fact, it is so hard to wrap one’s head around it that the idea has gained an enormous amount of traction from peoplee trying to understand what it means. Many of us feel this is a very profound statement cutting to the heart of how does a consistent theory of quantum gravity really look like as a fundamental theory.
I have spent many of the last few years trying to get my teeth into the hard problems of providing a better understanding of the strong coupling limit of Yang Mills theory on it’s own. I have been using the AdS/CFT correspondence to check various approximations in field theory and surprisingly they turn out to match a large number of phenomena. So depending on what I believe to be more true on a given day, I can take the results to mean that they provide evidence for the AdS/CFT correspondence, or evidence for the strong coupling expansions I am advocating.
What am I to do about this ? My techniques provide various self-consistency arguments, but there is no independent proof in the mathematical sense. I woud claim that the picture I have been working on is very compelling. Others have different approaches that use different assumptions (integrability for example) and have been able to predict the outcome of various perturbative calculations with this information used as a working assumption. The whole structure seems to be incredibly tight and really hard to penetrate.
I think the field has moved from the question: Is this really true? to How does this really work? and the validity of the statement can be taken as a fact (with the usual ex-provisos). So, is there no room left for doubt? Quite the contrary: doubts are one of the driving engines of progress. There are healthy skeptics and then there are flat out denialists and contrarians. I count myself on the first camp: I’m not yet satisfied with the evidence and I will keep on prodding the AdS/CFT correspondence until I am satisfied. As for denialists, I think they start with p=0 and will never update p according to evidence. They are fanatics, not scientists. The contrarians will claim p=0.5 until a proof is found.
Finally, there is the issue of communicating this information to the public. How does one do this? I really don’t know. But maybe these discussions will help.