## Thursday, July 20, 2006 ... //

### Evaluating extreme approaches to the theory of everything

One of the general strategies of some thinkers in theoretical physics is based on attempts to heavily extend the relevance of an existing concept or method. The hope is that all other difficult concepts become unnecessary or they will follow from a single, generalized and stretched idea. So one takes an existing procedure or a concept - a conceivably deep idea - and hopes that a more beautiful and a more powerful version of it will define and describe the whole Universe or all of string theory (or at least a much larger body of knowledge).

I will treat the "whole Universe" and "all of string theory" as synonyma because I am not aware of any controllable framework that would allow me to separate them sharply. Nevertheless, it is obvious that some of the ideas below will depend on our available knowledge of string theory more vitally than others.

One thing is clear: the principles below have almost nothing to do with the actual current research in string theory or the main ideas that drive the current research. Instead, the principles are possible speculations of string theorists as well as non-string theorists for the future. A certain Manhattan blogger has misunderstood this very basic point. It will be very difficult for him to correct his article because the critical error already appears in the title of his article. Well, the title of his blog is based on a profound ignorance, too, so it is difficult for him to write something reasonable on his whole blog.

In the bulk of this text, I will maintain a somewhat critical tone. Indeed, it is likely that all the big ideas below - or all ideas except for one - will turn out to be wrong, useless, or shallow. Even the most successful one may look as a very sketchy caricature of the truth in the future. Nevertheless, a loophole can emerge and revive one or two ideas in the list and lead to a significant progress in the future. Because the sentences so far could sound a bit abstract, the reader may look at the subtitles written in the bold face below before she decides to swallow or not to swallow the entire text.

The world is based on a local field theory after all

The effective quantum field theories have been highly successful. They are predictive because physics at low energies can be uniquely determined by a small number of parameters - the relevant and marginal couplings. Gravity does not seem to fit in this framework. Many string theorists would be happy to have a description of string theory that would make the locality in spacetime as explicit as possible (and it would probably be background-independent in the stringy sense, too) but there are many problems and no-go theorems that make such an approach difficult. The existence of a scale - the Planck scale - seems to be very important for gravity and the existence of useful fixed points seems unlikely. The Weinberg-Witten theorem makes it very rather inconceivable that the graviton can be a composite particle and that gravity can be emergent. String theory perturbatively requires infinitely many fields to be formulated as a field theory and the classification of its interactions according to the dimensions seems useless. The UV and IR physical phenomena are mixed and related in string theory which contradicts the one-way character of the renormalization group. String theory - and quantum gravity - simply seem to be different from effective field theories.

The world is determined by a gigantic global symmetry

In physics, we encounter two basic types of symmetries: global and local symmetries. Global symmetries are actual transformations that map one configuration (state) to a different configuration (state) whose physical behavior is however identical. Examples are the Lorentz group or supersymmetry in non-gravitational theories, the electric charge in descriptions of electromagnetism that are not based on gauge theory, or the approximate symmetry of isospin in the context of the strong interactions. An important difference between the global and local cases is that the physical states actually don't have to be invariant under the transformations of global symmetries. This means that the representation theory of the global groups is very important because all representations can occur in the physical spectrum. This situation profoundly differs from the case of gauge symmetries under which the physical states must be invariant: the trivial singlet representation is the only one that is directly relevant for physics.

Many people have proposed that there exists an extraordinary group G that is the global symmetry of reality and all properties of reality follow from the properties of G. What does it mean for G to be a symmetry group? Traditionally, it meant that all of its elements (and henceforth generators) commute with the Hamiltonian. In the context of special relativity or general relativity on fixed backgrounds, some symmetries - such as the boosts and their Lorentz generators - don't have to commute with the Hamiltonian. But they still generate a simple enough algebra. If we understand the term "symmetry" in this fashion, the main objection to the big idea is simply that the world does not seem to have large global symmetries beyond those that we know. The uniqueness of single-particle states with a given value of the momentum and the spin implies that there don't seem to be any additional labels associated with basis vectors of a nontrivial representation of a large unknown group G. Experimentally, all states seem to be singlets under an exotic and large enough group G. If all states transform trivially, you should better say that the symmetry does not exist unless you want to proliferate angels on tops of all needles.

In the context of string theory, we have a more conceptual argument against a large global symmetry underlying the whole world: global symmetries should not exist at all. One can prove that this is the case perturbatively: for every current of a global continuous symmetry, one can also construct a vertex operator for a corresponding gauge boson that proves that the symmetry is local. Moreover, this local character of all exact symmetries in string theory seems to hold nonperturbatively especially because all known symmetries can be related to weakly coupled perturbative symmetries by dualities, and the latter must be gauge symmetries.

There exists another way how to interpret the term "large global symmetry". It could be such a huge group or algebra that it would contain virtually all of the observables. Imagine the algebra of all operators in your theory - something that people in algebraic quantum field theory like to think about. With this understanding, you also face problems. This algebra is either completely unconstrained or it is over-constrained. The algebra of operators in a quantum field theory is not an organizational principle that can determine some physical questions that you actually want to be answered. It is at most a language to present insights that have been found by other tools. For every possible quantum field theory, there is an algebra and such an algebra is clearly not unique. We don't learn anything from such a huge algebra in which most commutators are zero. On the other hand, there are very large algebras that are much more unique - such as W_infinity algebras. Such algebras lead to strong constraints on physics. In fact, these constraints are far too strong: they typically make your theory integrable so that the interactions essentially vanish. I think that there exists no new, yet unknown intermediate case that would allow interactions but that would lead to an algebra and a theory with finitely many undetermined parameters. The Coleman-Mandula theorem and its ramifications that allow for supersymmetry seem to enumerate all possibilities.

The world is all about a gigantic gauge group

Gauge theories have been very successful. Their tools have appeared many times in the Standard Model and their fundamental character was also codified by string theory. It is natural to think that the whole world is derived from a much larger gauge group or its generalization. The first steps in this direction were made by Grand Unified Theories and a more ambitious goal would be to extend the principle to include gravity and the rest of physics, hoping that it will also become more constraining. The cubic open string field theory may be sold as an infinite extension of the principles of gauge symmetries. It has a gigantic gauge symmetry: the parameter of a transformation is a string field and the transformation acts on another string field. The Yang-Mills transformations appear as a very small subgroup. The stringy gauge transformations carry a flavor of the whole theory. In some sense, the whole theory - of interacting open strings - follows. Note that open strings don't carry gravity either: open string field theory is another non-gravitational extension of Yang-Mills theories.

Nevertheless, this approach became less promising in the last 15 years. The main reason is that we have learned that the gauge symmetries are not fundamental in physics. More precisely, a gauge symmetry associated with a particular physical system is not unique. Even in the perturbative context, we can construct the physical space of a theory such as the electroweak theory without any gauge symmetries. Nonperturbatively, the "social scientific" character of gauge symmetries is even more obvious. Because of various dualities, two systems with two different "natural" gauge symmetries may be completely equivalent. The identity of the gauge symmetry depends not only on physics but also on the classical limit around which we expand and other choices in our formalism.

The list of such dualities includes S-duality - for example the equivalence between the SO(2N+1) and USp(2N) gauge theories with N=4 supersymmetry in d=4. It also includes equivalences between non-commutative and commutative non-Abelian gauge theories. The gauge-gravity dualities are yet another example. Yang-Mills theories with many colors are equivalent to gravitational theories whose "natural" local symmetry group includes diffeomorphisms. In the context of string theory, "natural" gauge symmetries often appear and disappear. The SO(32) heterotic string theory on a circle can be Higgsed by Wilson lines down to U(1)^16 and we can move in the moduli space so that a new enhanced E8 x E8 gauge symmetry suddenly emerges. This forces us to say that the points with both unbroken gauge symmetries belong to the same theory. Neither SO(32) nor E8 x E8 can be viewed as a "more fundamental" or "master" group. In fact, both of them are rank 16 dimension 496 groups and they are surely not subgroups of one another. It is also hard to reconcile these groups by embedding both into a larger group. The only natural candidate for such a larger group is the gauge group of the corresponding string field theory whose open string version has already appeared in this text.

If we summarize, gauge symmetries seem to be a rather good description of spin 1 particles (or spin 1 and higher in string theory where the internal motion of strings can add extra spin) and they determine their interactions in the classical, weakly-coupled limit. The power of the principle of gauge symmetry seems to diminish in the very quantum, strongly-coupled regime. The interactions of fields with spin below 1 seems to be undetermined. Moreover, different classical limits of the same quantum theory - those related by dualities - typically have different or very different gauge symmetries. In the context of string theory, large gauge symmetries appear in the language of string field theory. These large symmetries are generated by the conformal symmetry on the worldsheet whose range of validity is perturbative once again. The precise realization of the string field gauge transformations is not unique and their classification is as difficult as a classification of conformal field theories. This is enough to see that the "grand principle" of a large gauge symmetry won't tell us much more than what we already know.

The world is the cohomology of a master operator

This paradigm is closely related to the previous one because the BRST operator can be interpreted both as the exterior derivative on a generalized "master manifold" as well as a BRST operator imposing a gauge symmetry. It is necessary to say that some people exaggerate the fundamental character of BRST operators. They're a trick to deal with gauge symmetries at the quantum level. You can always present any physical Hilbert space as a cohomology of a BRST operator - for example as the cohomology of the "Q=0" operator. Note that all states are annihilated by Q and no non-trivial states are of the form "Q lambda". Talking about a BRST operator as a fundamental structure is a vacuous proposal because we just move the bulk of the question to the question What is the kinematical Hilbert space from which the BRST cohomologies are constructed and how does Q act on it?

When I mentioned cohomologies, we could also mention equivariant cohomologies and other notions. These concepts routinely occur in the research of orbifolds, D-branes on orbifolds, and other objects and many string theorists know them extremely well. Peter Woit recently wrote that equivariant cohomology should turn out to be extremely deep and far-reaching. Woit often writes the same thing about the Dirac equation: he presents himself as a confused teenager who has not yet digested the Dirac equation or equivariant cohomology and who has many irrational ideas how far-reaching these concepts could become. Meanwhile, others have studied these structures in detail: the Dirac equation has been around for 80 years. These structures have transmuted into routine parts of the research in theoretical physics and their mythical fog has largely evaporated. Peter Woit's suggestions are so vague that I think it is a waste of time to try to add the correct link to his text even though it could be another good example how the "alternatives" of string theory look like: they're about a superficial understanding of 80-year-old concepts. A superficial understanding of anything is always a good starting point for irrational religions.

The world is a huge spontaneously broken group

You could also imagine that the fundamental principle underlying the real world is a particular spontaneously broken group or an unknown confining group. These two cases may probably be treated together because confinement and Higgsing are dual to each other. Roughly speaking, confinement of a group G is equivalent to the Higgsing of its S-dual group. If you choose the language of confinement, you can see that a hypothesis about a new confining group is a hypothesis about a new substructure - one that is analogous to the parton substructure of the nucleons. You will still have to figure out what the substructure is and what the corresponding confining group is. While it is always a possibility that such a new confining symmetry waits in the heart of the matter, you should notice that until you actually say what the symmetry is, you have not made any progress. Moreover, confining groups are gauge groups after all and all of the previous criticisms of "master" gauge symmetries, especially the non-uniqueness of a gauge group for a given physical system, still hold. The hypothetical existence of new confining groups is similar to the belief of many people - starting with Vladimir Lenin - that the hierarchy of substructures of matter is infinitely long. This belief simply seems to be incompatible with quantum gravity that does not allow any structures shorter than the Planck scale.

Discrete world

I have discussed the paradigm of a discrete world many times in the past. I think that it is the preferred idea of most crackpots. It seems as en extremely shallow idea to me - one that people choose because their mathematical skills and their imagination are heavily limited rather than strong. Truly fundamental ideas cannot be discrete. Discreteness is always an emergent feature of a physical system. In the old quantum theory, discreteness was postulated but the "new" quantum mechanics has derived a more accurate version of the previous insights from a formalism based on continuous objects. Such an evolution is bound to happen in all similar situations in the future.

A discrete description of the world and its objects is nothing more than a method to divide objects and their relationships into boxes but it can never explain why the objects are what they are, and why they interact the way they interact (infinitely many undetermined parameters of the loop quantum gravity Hamiltonian represent a textbook example of the ignorance). I can offer one more framework to explain why I consider discrete physics to be junk science. Visualizing the world in terms of discrete fundamental building blocks is a sort of literary criticism. Discrete physics is a soft social science after all. Much like the feminist scholars who believe that the penis is a social construct or a creation of words, discrete physicists believe that everything in the Cosmos follows from similar words, characters, or finite mathematical structures. But this is how science has never worked and it will never work so in the future. Words, letters, and tables with discrete entries will always be just a description of physics that has existed before. Words or discrete entries may sometimes describe your perceptions but they will never explain them. You need mathematics - deep enough mathematics which means mathematics with continuous structures - to be able to answer all questions "Why".

Among all theories of a certain kind, the theories whose all building blocks are discrete span a subclass that is not more mathematically consistent than the rest. This subclass is not more predictive than a generic theory either and it has no other physically attractive features either. Discreteness is a postulate whose only virtue is that it allows mathematically challenged people and social scientists to create a skewed psychological idea how the real world looks like. I don't think that simplifying life for intellectually challenged people is the main goal of Nature.

Categories, 2-groups, 3-groups

The relation between category theory and physics has been discussed on this blog, too. One of the "positive" topics was the way how category theory is believed to be relevant for analyzing some properties of D-branes that go beyond K-theory. Category theory, the generalized abstract nonsense, may be a way to rigorously organize the concept of "analogies" and other notions that are useful for thinking in general. But it is not suited for physics exactly because it has very little to say about the continuous objects and procedures how verifiable real numbers can be predicted. After all, the characteristic picture of category theory is a combinatorial graph, a very discrete object. The non-trivial content is really hidden in the detailed properties of the nodes and links. The graph itself - and category theory as such - is as unnecessary for a physicist as a book XY whose only goal is to explain the ordering of chapters of another book UV and their relationships.

Here I want to say a few words about the gerbes and related issues. Many people have struggled with a generalization of non-Abelian gauge symmetry to the case of p-forms, something that could be relevant for the (2,0) superconformal field theory in six dimensions, among other applications. The results so far seem to be a failure to me. One should realize that there is no reason why such manipulations with symbols should generate a Lagrangian that is relevant for the (2,0) theory. Because the theory has no coupling that can be adjusted to a small value - its characteristic coupling is essentially one - there is no reason why there should exist a classical limit that could be described by a classical Lagrangian. It is not a proof that such a Lagrangian cannot exist either but it is a reason for doubts.

What I find even more discouraging is the fact that the people who believe that new papers about these gerbes and 2-groups etc. are very deep - for example, our colleagues from the String Coffee Table such as Urs Schreiber (with John Baez) - mostly ignore the actual results of string theory. I think that such an approach just can't lead to any progress even if you invest centuries and hundreds of people. We know very well, albeit indirectly, that a theory that should have physics analogous to the "gerbes" exists as a particular decoupling limit of string theory. The precise properties of such a (2,0)-like theory that can be derived from string theory are the best available tool - because we can't use experiments here - to decide which new ideas about the gerbes are good ideas and which ideas are bad. If someone chooses not to use such a tool, it is almost guaranteed that her new papers will be just new random combinations of the old symbols and there won't be any real progress. Progress in science and not only science critically depends on the tools that can separate better ideas and structures from the worse ones.

The previous paragraph also clarifies my style of reading these papers. The abstract has so far been always enough to see that these fundamental gerbes papers make no quantitative comparison with the known physics - i.e. physics of string theory - and for me, it is enough to be 99.99% certain (I apologize for this Bayesian number whose precise value has no physical meaning) that the paper won't contain new interesting physics insights. More seriously, I believe that the theories analogous to the (2,0) theory are rather rare while the authors often seem to think and claim that they can construct such theories as easily as classical gauge theories. Because I think that most of these theories can't exist, it leads me to the opinion that the whole papers must be wrong at the quantum level.

Third quantization, fourth quantization, infinitely many quantizations

Quantum field theory is often presented as a result of the second quantization in the textbooks. In reality, what we do is the ordinary first quantization applied to the classical concept of a field. However, a classical field seems to be mathematically similar to a wavefunction - the result of the first quantization in mechanics. They have a very different interpretation but they include more or less the same "amount of numbers" in them. The Maxwell field is a vector (or 2-form) field and the additional Lorentz indices slightly distinguish it from the simplest wavefunction. However, a classical string field in string field theory is described by a function or a functional of the very same variables as a first-quantized wavefunction of a string. In this sense, the term "second quantization" can be justified quite rigorously in string theory. There is also a reason to say that the last, second quantization of string field theory is actually a third quantization. It's because a classical configuration of a single classical string is already described by a function X(sigma) which approximately contains as many numbers as as a wavefunction of a particle on a line. This classical string is quantized to obtain the single string Hilbert space which is then quantized again in string field theory. You can see that there are really three quantizations going on here but it is a matter of terminology and ideology.

Every quantization creates more continuous objects. How do I measure how much continuous objects are? Discrete objects are, in some sense, zero-dimensional. The higher dimension of space you deal with, the more continuous your treatment is. You can jump to an infinite dimension to obtain "very continuous" objects: such an infinite jump is a part of the quantization procedure. The nature of "infinity" that counts the dimension may be subtle. Structures can be more infinite-dimensional than others, especially if they result from multiple quantization procedures. You could also imagine that the right description of a system is obtained by quantizing a system infinitely many times. There could also exist fixed points under the quantization "functor". The last word in the quotation marks was picked to show a relation with the previous "big ideas" based on category theory. Nevertheless, I am not aware of any way how to construct a predictive theory out of these superficially attractive concepts of multiple quantizations.

The worldvolumes are spacetimes of other string theories, and so on

The paradigm of multiple quantization is also closely related to another "big idea" that is probably the most favorite of mine in this whole list. Perturbative string theory shows that the fields in spacetime are not yet fundamental: they are described by states of a more fundamental theory that lives on the two-dimensional worldsheet. Now, the two-dimensional worldsheet is described by a two-dimensional gravitational conformal field theory. Although gravity can be more or less described by a local field theory in less than four dimensions - because it has no real physics in it - you could still argue that the right way to describe a gravitational theory should be in terms of string theory. The worldsheet should be a spacetime of another string theory. And perhaps, this step could continue infinitely many times.

That's an idea that was first clearly articulated by Michael Green around 1987. The underlying string theory was identified with one of the string theories with extended worldsheet supersymmetry - N=2 or (2,1) strings - especially in the subsequent papers backed by names such as Ooguri, Vafa, Kutasov, Martinec, and a few others. I've spent a lot of time trying to reconcile these ideas with the mysterious duality by Iqbal, Neitzke, and Vafa that relates del Pezzo surfaces and toroidal compactifications of M-theory. With the help of technically gifted colleages such as Natalia Saulina, it was eventually possible to see that some very particular schemes of mine how this whole system of ideas could work are simply impossible. The critical step was to decide whether a certain class of worldsheet theories with N=2 supersymmetry can exist or not. This is an example how people who are more experienced with a certain kind of thinking can abruptly clarify ideas that others could try to study for years. For example, if the people in loop quantum gravity suddenly decided to listen to other physicists, we could explain them in 5 minutes why their multi-decadal efforts are hopeless and why many of their big conjectures can be clearly answered "No" instead of remaining in the limbo of wishful thinking: it's enough to turn a sufficiently powerful brain on to find the correct answer "No" in a few minutes.

Unfortunately, they have not decided to listen to other physicists so far which is why they're wasting one decade after another by efforts that can clearly never lead to anything interesting. At any rate, I am virtually certain that the value of the worldsheets-for-worldsheets idea exceeds the value of all loop quantum gravities and spin foams that have ever been studied. The only reason why we hear "loop quantum gravity" more often than "worldsheets for worldsheets" is a sociological one, namely that loop quantum gravity is vocally supported by a large number of crackpots and semi-crackpots who find, because of their limited mathematical intuition, the underlying dynamics of worldvolumes in string theory as well as many other ideas far too abstract.

If we talk about the spacetime vs. worldvolume descriptions in string theory, we can ask which of them is more fundamental. In the golden era of perturbative string theory, the worldsheet was obviously more fundamental. During the duality revolution, the fundamental string has become just one type of an object that happens to be the lightest one in a certain regime, namely the weakly-coupled regime. Nonperturbatively, the strings are as fundamental or non-fundamental as the branes that look like heavy solitons. Because the strings are no longer fundamental, their worldsheets cannot be quite fundamental either. In the 1990s, the focus has returned to the spacetime. We may also say that the holographic correspondence has codified a balance between the two approaches. The boundary CFT is a worldvolume theory and it is exactly equivalent - which also means equally fundamental - as the bulk spacetime description of gravity.

Bootstrap

It is therefore not so clear whether a hierarchical picture of theories generating each other is the best picture of the scheme of things. Various descriptions of physics and the objects could be equally fundamental and their precise form could be determined by a self-contained set of conditions. Another extreme big idea in physics is that everything follows from general physical consistency requirements. This idea has been proposed many times in the past. In the context of strong interactions, bootstrap turned out to be very wrong. We know that QCD is physics of fields that are as fundamental - and describable by classical Lagrangians - as the fundamental particles of QED. Moreover, there are many consistent, asymptotically free theories analogous to QCD and the general physical consistency constraints are certainly not enough to find the correct one.

However, bootstrap was successful in the case of two-dimensional conformal field theories. Many classes of these theories have been classified and some of them have been solved.

In the context of quantum gravity, many of us more or less secretly believe another version of the bootstrap. I think that most of the real big shots in string theory are convinced that all of string theory is exactly the same thing as all consistent backgrounds of quantum gravity. By a consistent quantum theory of gravity, we mean e.g. a unitary S-matrix with some analytical conditions implied by locality or approximate locality, with gravitons in the spectrum that reproduce low-energy semiclassical general relativity, and with black hole microstates that protect the correct high-energy behavior of the scattering that can also be derived from a semi-classical description of general relativity, especially from the black hole physics.

This belief is not influencing the explicit current research much - something that a blogger from Columbia University is completely unable to understand - but it is definitely a nice and plausible idea. The main obstacle that you must overcome to transform this idea into a valuable established paradigm is a yet unknown system of mathematical proofs that the general rules of quantum gravity above - and perhaps a few more that I have missed - do imply that physics must reproduce what we know from string theory. For example, prove that there must always exist a physical process in a theory of quantum gravity as defined above that creates a large region whose local physics mimicks one of the six maximally dimensional backgrounds of string/M-theory (11 dimensions, type I, IIA, IIB, HE, HO). I think that the statement that string theory and consistent quantum gravity is the same thing is almost certainly true - 95% of certainty - and I can even imagine that someone will eventually prove it.

The universal Hamiltonian is a Laplacian on a master manifold

Let me end up with one more attractive idea. The Hamiltonian describing all interesting physics of string theory and everything else could be as simple as a Laplacian on a master manifold with very special properties. Consequently, all the partition sums could be encoded in a master function picked by some very particular constraints. The U-duality group of M-theory and modular functions are certain to play an important role. This approach was elaborated upon by Ori Ganor a few years ago, and recently by Neitzke et al. in a paper that connects this religion with more-or-less established insights by Ooguri, Strominger, and Vafa about the black hole entropy's relations to topological string theory. Again, there are some technical reasons to doubt that this approach could describe "everything". But I think that these ideas are deep enough and there is no metaphysical reason why such ideas would have to remain wild speculations with limited impact forever.