Thursday, March 23, 2006

Unity of strings

One of the features of string theory that the non-experts often misunderstand is its unity and inevitability of its conclusions. People outside the field often believe that the string theorists are "inventing" different objects and structures studied under the name of "string theory".

The reality is very different: our conclusions about the mathematical structure that we continue to call "string theory" (despite the importance of non-stringy objects that has been appreciated in the least 10 years) are rigid, unchangeable, and inevitable. More importantly, it is impossible to "buy" some parts of string theory and abandon the rest.

String theory is undoubtedly an important subject to be studied as a topic in mathematics. As Edward Witten has said, mathematics of the 21st century will be dominated by string theory. But does string theory describe the real Universe more accurately than the previous theories of physics? The answer to this question is either Yes or No. It cannot be something in between.

As a theory of this Universe, string theory is either completely relevant or completely irrelevant.

In this text I want to avoid technical details while I want to convey the basic idea about the unity and "objective existence" and uniqueness of the mathematical structure. In order to achieve this goal, consider one of the descriptions of string theory that you like. For example, start with the maximally supersymmetric four-dimensional gauge theory whose gauge group is SU(N). The number of colors N, the coupling constant g, and the theta-angle are the only parameters that can be changed in this Yang-Mills theory.

This gauge theory is just a quantum field theory, a beautiful extension of QCD, and all theoretical physicists, with the possible exception of Bert Schroer, are convinced that quantum field theories like this one do exist. In 4,000 papers or so, starting with the breakthrough by Juan Maldacena, it has been established that for a large number N of colors, the four-dimensional gauge theory effectively develops a new dimension - something that we call "holography" because it is somewhat analogous (but not identical) to the holograms in optics.

You can find objects that are effectively isolated from each other much like objects in five-dimensional space. As the AdS/CFT correspondence has taught us, the SU(N) gauge theory is equivalent to supergravity on a five-dimensional anti de Sitter space.

Actually, the number of dimensions is more than five. Thousands of papers have shown that the physics of the four-dimensional gauge theory is actually ten-dimensional if the number of colors is large. If the number of colors is large, it is more convenient to describe physics of the gauge theory using ten-dimensional supergravity.

More precisely, it is not just supergravity. Supergravity per se does not make any sense at the quantum level; it is not renormalizable. In other words, it does not predict physics at very short distances. When you ask how does physics at very short distances in the ten-dimensional space that emerges from the four-dimensional gauge theory look like, the answer is that the local physics is exactly given by type IIB string theory. All string-theoretical objects that have been argued to exist and studied for years can be found in the gauge theory. In many cases, we can explicitly construct them from the gauge-theoretical variables. The set of such objects includes excited strings, wrapped branes, black holes, and others. These insights have been derived, checked, re-derived, and re-checked in hundreds of papers. It is completely unreasonable to deny some aspects of type IIB string theory and their importance for a complete understanding of gauge theories.

For a large number of colors, physics of the gauge theory looks like type IIB string theory in flat space which is described by certain equations - equations that generalize the field equations of supergravity to the stringy level. Field theory allows fields to take many values and general relativity allows the spacetime to have many shapes as long as they solve the correct field equations - Maxwell's equations, Einstein's equations, and their generalizations.

By basic principles of locality - a feature whose validity can be more or less proved in the relevant contexts - it is guaranteed that the Universe in which one coordinate is periodic is just as allowed by the equations of type IIB string theory as the infinite Universe. Both of them solve the same equations equally well. If one of the spatial dimensions has the shape of a circle, string theory predicts new objects - such as wound strings - whose existence is as well established as the existence of particles with momentum. It is completely unreasonable to deny the existence of these new states.

If the circle of spacetime described by type IIB string theory shrinks to zero, you may show that physics of the ten-dimensional type IIB string theory becomes equivalent, via the so-called T-duality, to physics of another kind of string theory, type IIA string theory, on another ten-dimensional spacetime: the 8+1 large dimensions are shared but the last, circular coordinate is completely different in the type IIA picture than it is in type IIB picture.

Both type II string theories in ten dimensions have the dilaton scalar field whose existence is implied by supersymmetry. Once again, it is completely crazy to question the existence of the dilaton. You may ask what happens if the field goes to infinity. The answer, proved in hundreds papers, is that type IIA string theory actually develops a new, eleventh dimension as the dilaton grows. You end up with the so-called M-theory whose long-distance limit is the eleven-dimensional supergravity. Again, it is absolutely unreasonable to question that this is what happens because it has been demonstrated by many arguments - arguments based on BPS objects as well as more complete arguments involving a matrix description of these vacua or other descriptions.

Remember that we started from four-dimensional gauge theory. If you allow the fields and parameters - the number of colors and the shape of the geometry of spacetime that demonstrably emerges - to change in any way, you just cannot avoid the conclusion that physics of M-theory is relevant for processes in situations in which some of these fields have been changed in one way or another.

There is a whole network, a very dense network, of dualities and other equivalences that allow you to get to any point of the "landscape" or any description of string/M-theory that we like and study. For example, all string theories as well as M-theory can be compactified on Calabi-Yau manifolds. It is simply because the shape of the Calabi-Yau manifold solves the equations of motion whose form can be derived otherwise, and moreover all conceivable anomalies that could invalidate the background cancel, and it is therefore impossible to argue that these solutions are illegitimate or ignorable. They exist as a part of a demonstrably important mathematical structure. Their existence can't be denied just like the existence of 101 in the set of prime integers cannot be denied.

Similar reasoning established in hundreds of other papers shows that the statements about branes, fluxes, and their relations are unquestionable, too.

In biology, the creationists often invent some "barrier" that cannot be penetrated by the natural process of evolution. A critic of string theory who decides that the network of string theory background can be "cut" at some place is equally unreasonable. In reality, these critics of string theory often like to pretend that they are more reasonable than the creationists. Such an idea is, of course, a complete nonsense. They are exactly as unreasonable as a creationist who argues that the chimpanzees' brains are the maximal ones that can evolve naturally, without a divine intervention. More importantly, the arguments showing that you cannot cut string theory into pieces in order to deny some of its pieces are rigorous and unquestionable arguments - while the individual mechanisms proposed to explain various evolutionary steps are usually shaky and uncertain.

The punch line is that William Dembski (who sometimes promotes "good" microevolution but not "bad" macroevolution) and Peter Woit (who also likes to cherry-pick things in string theory) not only use the same blog design but they are also more or less equally unreasonable. The only reason why some people XY say that Peter Woit's approach is more scientific than William Dembski's approach is that Peter Woit is, much like XY, a left-winger. There is no rational justification for his denial of "large parts" of string theory, as long as he wants to accept the importance of some other parts.

When you study the equations describing physics of string/M-theory in various limits and using various descriptions accurately enough, you can also derive that objects such as the black holes and branes can be created from colissions of other objects in these theories whose existence has already been established. You can't create large D2-branes in actual experiments that can be paid from the U.S. budget, but in the Universe governed by the strict rules of string/M-theory, such a process is possible in principle. We are talking about a process that can be done, at least in thought experiments, and the fact that it can be done is mathematically established.

Centuries ago, we did not know that we can create particles moving in extra dimensions and/or D-branes. We did not know that we could build nuclear power plants either. The claims that nuclear energy is impossible seems rather awkward these days because we have seen certain experimental evidence that nuclear energy exists. We have not seen any experimental evidence for additional dimensions and similar concepts but there exists extremely strong theoretical evidence. The fact that ten dimensions secretly exist in the maximally supersymmetric four-dimensional gauge theory is just one among many examples to see that all consistent quantum theories of gravity are probably related to those 10- and 11-dimensional string/M-theories that we study. If someone says otherwise (for example, if she decides to argue that the N=4 gauge theory in four dimensions does not have 10 dimensions in the large N limit), she should have some rational arguments. General anti-scientific skepticism is not a rational argument. A superficial statement that extra dimensions seem religious to someone is not an argument either. These comments have no scientific value, which should be viewed as an explanation why the people in quantum gravity don't listen to people like Peter Woit or Lawrence Krauss: the two Gentlemen have nothing substantial or quantitative to say.

Although our Universe may be viewed as a family of relatively mild excitations around a particular point of the "landscape", and we have various semirealistic candidates which of the points lead to Universes that qualitatively agree with ours, the existence of all other points in the landscape and all objects in these other points of the landscape that are studied by the string theorists is a fact, not a fantasy or a matter of choice. In principle, if you had enough energy, you could expand the size of the "small" dimensions in a small region of space for a while, and study whether all the branes that the string theorists predict exist in this region.

If you compute these things more accurately, most of such experiments would result in a creation of a black hole that would not teach you much - except for physics of black holes - but you can surely view this fact as a technical limitation only. In principle, the values of fields that exist can be continued to new intervals and physics describing these distant points of the configuration space is relevant for a full understanding of our Universe.

The existence of these "vacua" - points in the landscape - and the precise identity of the allowed objects that can live within these vacua and the precise structure of their interactions is a well-defined mathematical question that has been fully answered in a significant portion of the vacua with at least 8 supercharges. Possible vacua of string theory without any spacetime supersymmetry may remain somewhat uncertain because it is harder to calculate accurately if spacetime supersymmetry is absent. But the principle is clear: when you study string theory carefully enough, all questions are eventually matched by clear answers, much like the question which numbers greater than 100 and smaller than 120 are primes. All string theorists agree about the answers much like all mathematicians agree about the list of primes between 100 and 120. This consensus is not a consequence of a big conspiracy but rather a manifestation of the objective features of a rather well-defined mathematical structure.

Mathematicians may have a hard time to say how many primes there are between 10^{90 million} and 10^{90 million} plus 2006. They don't have any fast algorithm to decide this question. String theorists may also find some particular problems difficult to calculate. In many cases, it is just a matter of technical difficulty. In other cases, we don't even know how we would decide the question even if we had a superpowerful supercomputer. But despite this partial (and ever decreasing) ignorance, the whole research of string theory indicates quite clearly that each meaningful question about the physics of string theory has exactly one correct answer - even though the answer may often be surprising. It has always been working in this way and there is no reason to expect that it won't be the case in the future.

If we suddenly decided that we know which "point of the landscape" captures the physics of the Universe around us - and which description of string theory is the most useful method to expand the calculable quantities - it is clear that this point of the landscape would become the most popular topic among all string theorists who want to maximize their contact with the actual experiments. But the rest of string theory will never go away. A "natural philosopher" will always know that the other interesting phenomena and objects found in string theory can be, in principle, reached by a sufficiently extreme change of the parameters of the environment - parameters that demonstrably can be changed. And this is why the rest of string theory, including the "unrealistic backgrounds", will always belong under the slightly generalized umbrella of physics.

Unless someone proves that all of us have been crazy and string theory is wrong in some mysterious way that is unimaginable for people like me. Someone else can prove that 101 is not a prime after all. Strange things can happen but because such a possibility is incompatible with everything that I call "rational and careful thinking", I can't provide you with any prediction what would happen to mathematics or physics if 101 were factorized or if string theory were proved wrong. I don't really know what it means.

If someone hypothetically proposes another consistent theory that can incorporate quantum field theories as well as gravity, and if this theory gets some support suggesting that it may actually be more relevant for the Universe around us than string theory is, we may get some material to discuss - why there exists such a rigid structure as string theory if something else describes the Universe. But because we don't know any such structure that is inequivalent to string theory - and frankly, we are convinced that the hypothetical existence of such a structure looks nonsensical - I would continue to say that these speculations remain outside of rational science.

String theory is not only tightly connected, as argued above. It is also unique. There is no way how can you "infinitesimally perturb it" without making the whole structure inconsistent. Every infinitesimal deformation of string theory that you can think of either makes the theory physically inconsistent - by violating the conservation of probabilities or by making the results infinite - or it can actually be interpreted as a deformation of degrees of freedom that have existed in the theory before the deformation and that were always allowed to be changed: any adjustable parameter is actually a scalar field whose value can be changed by actual experimenters. In a certain sense, string theory can't be an approximation of the truth. It must be either a true description of reality or a false description of reality and those who say otherwise are not even wrong.

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