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Dualities and equivalences

Most of the misunderstandings of physics and myths have a characteristic left-wing flavor. The following flawed concepts can often be identified as the primary obstacles that prevent many people from comprehending certain important ideas in theoretical physics:

  • social arguments used to find the answers to scientific questions
  • egalitarianism of theories, ideas, or even numbers

The first error appears whenever someone tries to guess a correct answer to a physical question according to various social features of the proponents of the different answers - which is a typical approach of postmodern social scientists. In the 1930s, some German physicists would discard a physical theory because its discoverer was Jewish. Most of us are no longer doing the same mistake but many of us are doing similar mistakes. Some people think that only famous or highly-paid people can be right. Other people think, on the contrary, that it is only the outsiders who should be correct and those who have been believed to be right so far must have been cheating. A third, "neutral" group of people believes that all people talking about physics should be right equally frequently. Then there are people who believe that only tough physicists may be right, and other people who believe that only polished or sissy liberals can be right. Needless to say, all these and similar ideas are scientifically absurd. Correct scientific ideas can indeed emerge in all environments. They are more likely to occur in some environments but one can never use the environment as a universal criterion because the probabilities that the right ideas occur in all environments are nonzero. All these arguments are unscientific.

The second error is based on egalitarianism of ideas, theories, formulae, or numbers. This error can also be described as the celebration of ignorance. Imagine that a schoolkid is asked what is Ohm's law. The kid answers "U=RI" or "U=R/I" or "U=I/R" or "U=1/IR", and perhaps also "U=R+I". The teacher won't be too satisfied and the kid won't have the opportunity to become terribly proud about her ignorance.

However, there are cases in which there is no teacher with the desired authority around. Our colleague from Santa Barbara argued that because we have not seen an evaporating black hole yet, all real positive numbers are equally good candidates for the entropy/area ratio. One quarter is just one possibility among infinitely many - just like ln(3)/sqrt(2)pi. Well, it is just one possibility among infinitely many from the viewpoint of a person who does not know how the semiclassical calculations are performed. More than 30 years ago, these calculations have shown that the correct constant is certainly one quarter, and independent microscopic calculations in string theory always confirmed this number. There is no doubt that "S=A/4" for macroscopic black holes. You can guess that once the colleague from Santa Barbara writes that she is a wiser, brighter, more tolerant, more cultural, broader, or a better human being because she considers all numbers to be as good as "1/4", my blood starts to boil. I just can't stand this behavior. If someone is ignorant about something and she admits it, it's OK. But if someone starts to paint her ignorance as a virtue or even the essence of humanity, it becomes a small disaster.

Similar egalitarian myths lead many other people to believe that the probabilities that string theory and loop quantum gravity are correct must be comparable. Fifty percent of people should be working on each, and so forth. Some people just can't understand reality. The fact that the probability that string theory is correct exceeds the analogous probability for loop quantum gravity by dozens of orders of magnitude is something "unfair" for them, and they just don't want to accept such a fact because it does not agree with their egalitarian sentiments. It's just like the "unfair" fact that Bill Gates is richer than others.

It is thus refreshing to see anti-egalitarian myths in action. At last, one can see some diversity here. ;-) The myth I will be talking about below is the misunderstanding of the meaning of "dualities" and "equivalences" in physics. The particular discussion that led me to write this text was about the gravity dual of the phenomena at RHIC - i.e. the gravity dual of QCD. Because this particular gravity dual is not exactly uniquely and exactly known, let me talk about a similar theory where the gravity dual is known perfectly: the N=4 gauge theory in 4 spacetime dimensions that is dual to type IIB string theory on the product of a five-dimensional anti de Sitter space and a five-dimensional sphere of the same radius. If the program to search for the gravity dual of QCD succeeds perfectly, the situations will be perfectly analogous. Consequently, we can talk about N=4 gauge theory without a loss of generality.

An anonymous reader suggested - and interpreted a comment by Prof. Mark Srednicki - that the dual gravity calculations in AdS/CFT are just mathematical tricks, but the "real" physics is still physics of gauge theory. He or she proposed an "analogous" example with a pendulum and an electrical circuit. The fact that they follow the same equations does not mean that we can say that an electrical circuit is a pendulum. Fair enough.

But the situation in AdS/CFT or any other duality in string theory is very different. In the case of the pendulum, we can find many other physical features that distinguish it from the electrical circuit. A pendulum should have a certain design, people with common sense would say. A more sophisticated way to distinguish a pendulum from an electrical circuit is based on the fact that the pendulum oscillates because of gravity, and the oscillations thus emit gravity waves made of gravitons whose spin is two, which you can measure in principle, while the circuits emit electromagnetic waves, streams of spin one photons. At any rate, there are other details that distinguish the two cases, and the two systems only share the rough differential equations. We have tools to say: this is a pendulum, not an electrical circuit.

But in the AdS/CFT correspondence, there are no other details that could tell you whether your world follows physics of gravity or physics of the conformal theory. Any phenomenon on one side has an equivalent description on the other side. It makes no sense to ask which of the two descriptions is correct if they are fully mathematically equivalent. It is equally meaningless as asking whether the German or French translation of the European constitution is correct. (In the latter case, both of them are wrong, of course.) It is equally meaningless as asking whether the Schrodinger picture or the Heisenberg picture is correct. These frameworks use slightly different concepts or intermediate equations, but they lead to exactly identical predictions of those physical observables that can actually be measured.

If such an equivalence is obvious, we would not spend hours with it; we would say that the "two" things are simply one thing. Equivalences in modern theoretical physics are only interesting because they're not so obvious. We like to use the word "duality" because the two descriptions of the same physics originally look like two different theories. Obvious examples include strong-weak S-dualities, large-small T-dualities of compactifications, large-weak-strong-small U-dualities mixing many parameters, and especially the gauge-gravity and AdS/CFT holographic dualities relating a gravitational theory with a non-gravitational theory defined on the boundary of the gravitational spacetime.

In each case, it is simply meaningless to ask which of the two descriptions is correct and which of them is just a trick. If they're exactly equivalent - giving exactly the same predictions for physically measurable quantities such as cross sections - they are equally correct. Because they're equivalent, it means that if one of them is a relevant set of laws for XY phenomena, the second of them is relevant, too. If one of them is irrelevant, the other is also irrelevant. If one of them is valuable, the other must also be valuable. And so on, and so on.

One of the descriptions might be more useful in one context or another, but usefulness and the truth are different things. For a theoretical physicist, they are very different things.

In the particular case of the gravity dual of QCD, it is indeed important to note that the gravity dual is not just a "trick" but the "real" theory of the phenomena in different variables. If you are optimistic and imagine that our colleagues will find an exact gravity dual of the Standard Model, such a geometry will literally be a part of the geometry that is relevant for the real world in string theory. Because the Standard Model is not valid at arbitrarily high energies, the geometry dual to the Standard Model will have to be cut and modified to obtain the geometry describing everything - it will have to be glued to a smooth Calabi-Yau manifold, to mention the most typical example, that will be responsible for the physics at the Planck scale.

At any rate, if the program is fully quantitatively successful, one will obtain a background that carefully follows the rules of string theory. By string theory, I mean the same theory that other string theorists study, with the same strings, branes, black holes, low-energy fields, their interactions, transitions, moduli spaces, and everything else. The same string theory that is most likely the correct description of everything in the real world including the ordinary gravity in four nearly flat spacetime dimensions.

In the case of the gauge-gravity duality or any other exact duality in string theory, claiming that one side is true and the other is either wrong or just a "trick" is simply a misunderstanding of the meaning of the concept of dualities. Dual theories are exactly equivalent, no one has any right to discriminate against one of the dual descriptions, and when we talk about string theory, we always mean the same string theory that always contains quantum gravity, and always contains other forces and objects. The gravitational duals of gauge theories are backgrounds of string theory. String theory in this context or any other context is no trick but reality, and once we accept that the rules of string theory are relevant for some questions in physics - such as the dual description of gauge theories - the rest of string theory logically follows from the same rules. Within the established theoretical insights of string theory, there is no way how to build walls of political correctness within string theory that would separate things that can be done and that are valuable from those that aren't. Only people who misunderstand the internal structure of string theory - for example a certain controversial blogger at Manhattan - propose to build such walls.

And that's the memo.

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