## Monday, October 25, 2004 ... /////

### Beauty of string theory

• What's so "beautiful" or "elegant" about string theory?
• I ask this seriously and respectfully. And I apologize if it seems like a troll. I always feel uncomfortable when I hear physicists make statements about beauty. Who here thinks reality is ugly? Interestingly, I do not hear mathematicians speak like this as often as I do physicists. So what is it that string theorists find so beautiful? Brian Greene did not convey it to me. Sorry and thanks.

Of course, I don't believe that if Brian failed, I can succeed. ;-) The feeling of beauty in physics is something caused by very objective and rational properties of the physical theory, but finally it is an emotional feeling, and if someone just does not have these emotions, it's hard to convey them.

But let me try to answer it anyway.

The issue of beauty in theoretical physics is very subtle, and I will have to explain what this beauty may be, but also what this beauty cannot be.

First of all, the laws behind the Universe are not dumb. If a physicist talks about beauty, she never thinks about "simplicity" of the kind that an average teenager who hates math classes at school might appreciate. The Universe is not that simple, and its rules just can't be dumb.

As Einstein said, the laws of the Universe should be as simple as possible, but not more.

In physics, this naive notion of simplicity is often replaced by symmetry. Symmetry has something to do with beauty, and it is something that constrains the physical system. If your face has a symmetry, you need to know one half of it only. If a theory respects some symmetries, it proves that it is special among other conceivable theories.

In older theories than string theory, some symmetries must be assumed, and this reduces the number of free parameters and arbitrariness. For example, the Standard Model has less than 30 parameters that describe the strength of various interactions (and masses) that are compatible with the given symmetries - gauge symmetries and the Lorentz symmetry. In this counting, string theory is maximally constrained - it has no adjustable parameters. This rigidity is one of the first justifications of beauty. Even though the spacetime symmetries don't play such a central role in string theory, the theory is more constrained than the most symmetric theories of the previous kinds.

There are many types of symmetries known in physics - rotational symmetry, U(1), SU(2), or SU(3) gauge symmetry, E_8 symmetry, supersymmetry, conformal symmetry, and so forth. I deliberately chose continuous symmetries only. All of them can be found in string theory, but they always seem to be tiny reflections of something much more beautiful. String theory is something that can start as a small package, however a package that contains so much good stuff. Moreover, the symmetries can transmute into each other as you walk along the stringy "landscape" (I mean moduli space). They can be spontaneously broken, unbroken, enhanced, confined.

Symmetric theories don't necessarily have to be simple, in the naive sense. Eleven-dimensional supergravity is, in some sense, the most (super)symmetric field theory, but its Lagrangian is pretty long. A real physicist does not care whether it's long or not; a physicist is always
ready to spend an hour by writing a Lagrangian. That's not a big deal for her, and such superficial questions as time and money don't matter. The beauty inside is more important, and eleven-dimensional supergravity has 32 supersymmetries and other symmetries. There is no rule that beautiful objects must fit one line.

Eleven-dimensional supergravity is a part of string theory, a low-energy limit of M-theory in 11 dimensions. There are many potentially beautiful theories in physics, and all of the good ones seem to be connected within string theory. This union is not artificial, and it is another reason that makes it beautiful. You usually find out that string theory can have moduli (exactly massless scalar fields, some sort of dynamical parameters), and as you change them, the different theories with different symmetries transform into one another in an exactly controllable and unique way.

Nevertheless, I don't really think that we view the symmetries as the most important reason why string theory is beautiful. Maybe string theory's power to naturally include all types of essential and "robust" physical phenomena and derive them from a modest starting point may be a more accurate reason behind our claims about "beauty" in string theory. Of course, this point will not be appreciated by an enemy of reductionism. ;-)

If someone is not impressed by the fact that a formula (e.g. the Lagrangian of QED) can explain a large number of physical situations, including chemistry and animals, as well as the sunset, she can never understand why the physicists think that string theory is beautiful. From this perspective, string theory is the ultimate achievement of reductionism - everything is included in a theory that uniquely and naturally follows from the assumption of a one-dimensional object with meaningful interactions (or from other possible starting points, and string theory now has many). The elementary particles and interactions of the Standard Model are reduced to something even more fundamental - something that probably cannot be reduced further.

But I believe that one thing is perhaps even more important for the beauty of string theory: the way how it avoids all potential problems.

If you "glue" a random theory of some type and you try to quantize it, you will be led to many different kinds of diseases that will make the quantum theory unusable. Classical symmetries will be destroyed by quantum effects (anomalies). Physical quantities will be expressed by divergent integrals, and sometimes the divergences cannot be eliminated, even if you use the best tricks (non-renormalizable theories). You may encounter ghosts and negative probabilities.

All these problems always miraculously disappear in string theory. It's like in a good movie that keeps you excited, nervous, but eventually leads to an unexpected (but reasonable) happy end. It's like the Superman who can save the city in time by an unexpected move - except that in string theory, we can prove that these unlikely events are *facts*. You may want to invent an "easier" approach than string theory to make the integrals convergent, but such choices will always introduce new problems - such as anomalies (or more generally, some breaking of gauge symmetries). String theory just seems to be the only framework where all these problems - anomalies and divergences - are avoided. It's the only movie with a real happy end. Also, you must think for a while to see why the end is really happy - string theory is not like one of the cheap movies. It requires you to think, and the beauty can only be appreciated if it works through your mind for some time.

Peter Woit finds it unacceptable to work with more than 4 coordinates, so he will prefer movie directors that claim that a movie should only contain 4 points. He may like these movies, but they are really cheap movies. You know that good movies should really have several dimensions. The movie of string theory is 10 or 11-dimensional, depending on the way how you look at it. From some point of view, it is 12-dimensional, and from a more general point of view, it is infinite-dimensional. Yes, the higher-dimensional geometry itself is beautiful, too. It's what distinguishes a sophisticated 3D sculpture from a naive 2D cartoon.

But let me return to the miraculous power of string/M-theory to eliminate inconsistencies.

What we're thinking about is the infinite ocean of "ugly" theories. Each of them suffers from a problem. And string/M-theory marches on an infinitely thin road (or string) stretched above this ocean, and its calculations always miraculously combine in such a way that the predictions are unique, and they fit together. The detailed features always turn out to be "right" so that the result makes sense, even though a single "error" would make the theory meaningless.

Finally, string theory is beautiful because of dualities. Take five things that you like - for example, your girlfriend, your favorite bird, a photograph with a sunset above the ocean, your favorite food in a French restaurant, and your new car. ;-) Now imagine an object ST that can be observed from five different directions, or in five different ways of thinking. From one vantage point, it will look like your girlfriend, and so forth.

You may think that it is impossible - if something looks like your girlfriend from the left, it can't look like a car from another direction. Someone may come with a similar argument in string theory. Nevertheless string theory always brings a set of miracles that make these different pictures compatible, and therefore it can look like five (or more) different beautiful things simultaneously.

String theory is able to change an object to a different object or phenomenon smoothly; it is free of any unpredictable singularities. Every time something becomes too singular or sharp and one starts to be afraid that a disaster is looming, string theory always predicts some new objects and phenomena that regularize physics and make it as smooth as before. There are many different tricks how a disaster may be avoided in a movie - and all of these types of tricks seem to be contained in string theory.

OK, the beauty is a combination of symmetries and their interplays (something that Einstein knows well from his theories of relativity, and something that underlies the Standard Model too); inevitability and uniqueness of the predictions - the absence of any adjustable and arbitrary parameters; cancellation of divergences and anomalies and the unexpected character of these cancelations; equivalences between different ways to look at the theory that eventually turn out to be totally compatible; its natural unification of virtually all other important phenomena and concepts in quantum field theory and general relativity; its connections to structures in mathematics that are also called "beautiful" - for example those associated with higher-dimensional geometry (mirror symmetry).

Yes, some mathematicians do not talk about "beauty" as often - many of them, in fact, really enjoy if their research is really dry. ;-)

#### snail feedback (9) :

>If you "glue" a random theory of some type and you try to >quantize it, you will be led to many different kinds of diseases >that will make the quantum theory unusable. Classical symmetries >will be destroyed by quantum effects (anomalies).
...
>All these problems always miraculously disappear in string >theory.

Well, this is not quite true.
Anomalies cancel out in string theory if you choose the right
number of dimensions (25 in bosonic, 11 in superstrings etc.)

This is _not_ more "beautiful" than some anomaly cancellations within the standard model, like triangle anomalies which disappear if all constituent's charges sum to 0.

My criticism of string theory on the "beauty" front is basically this. I don't mind working with an arbitrary number of dimensions. But to choose it on the basis of anomaly cancellations is not really good enough for a theory of everything.

"Anomalies cancel out in string theory if you choose the right
number of dimensions (25 in bosonic, 11 in superstrings etc.)"

d=26 (bosonic) and d=10 (superstring). d=11 for supergravity theory behind it.

BTW, why is it said that higher spin field theories are inconsistent? My understanding is that there are two reasons:

* Simple interactions theories non-renormalizable. But as Weinberg argues in Chapter 5, free-field theories for arbitrary spin are well-defined, and likewise at least perturbatively one can make sense of them.

* Fermion and gravitational anomalies (e.g. Alvarez-Gaume and Witten). But interactions with scalar ok?

Are there any others?

The reason I am asking is that spin-2 QFTs are also 'problematic', but clearly gravity exists.

*Could it be that there are higher spin fields that we have not observed? Or has that been definitively ruled out?

* Is it fair to say that string theory *predicts* that there are NO MASSLESS FIELDS WITH SPIN >2? If one such particle discovered/inferred, string theory is proven wrong?

Second contributor: Thanks for having corrected the critical dimensions of the first contributor! ;-)

To the first contributor: Yes, you must require the Weyl symmetry in perturbative string theory, and it is just "another" anomaly, in a sense, and therefore it is equaly beautiful. But once you satisfy this single requirement, you automatically cancel all divergences and all types of anomalies in spacetime, which is beautiful, especially if you see how nontrivial these cancelations look like from a spacetime perspective.

The first contributor also wrote: "But to choose it on the basis of anomaly cancellations is not really good enough for a theory of everything."

I don't know how to interpret this sentence so that there would be something true about it. Do you want to work with theories that suffer from gauge anomalies? I don't because these are nonsensical theories. I want to work with anomaly free theories, and if there is a type of theory where the anomaly cancelation implies the whole physics more or less uniquely, it is certainly the type of a theory that I would be interested in even before I knew anything about string theory. One principle implies everything else, and everything makes sense - it's beautiful.

For the second contributor, concerning higher spin theories: the reason is much more simple - gauge invariance. Forget about renormalizability. You don't need it. If you have a field with a spin higher than two, then it certainly contains some polarizations that have negative norm - for example, the components of a tensor with an odd number of timelike indices.

These components would behave as ghosts, and lead to negative probabilities. You must therefore have a local symmetry that decouples these components. It's U(1) or Yang-Mills symmetry for vector fields, and diffeomorphism covariance for spin 2 fields.

Higher spin fields can also have a gauge invariance that would decouple these modes. However, if you try to write down gauge-invariant expressions, you won't find any. Much like the Yang-Mills invariance implies that you must produce invariant combinations of the covariant field strength F_{\mu \nu} and much like diff invariance implies that you must write your action in terms of the Riemann tensor and its derivatives, the gauge invariances for higher-spin fields are so constraining that you won't be able to write down *any* interactions at all.

Just try it. For example if you have a symmetric tensor with 3 indices - a simplest spin 3 field P - the gauge invariance is something like \delta P_{abc} = d_a lambda_{bc} symmetrized over bc, roughly speaking. Try to construct gauge-invariant expressions, analogous to the field strength. You won't find any. For example, the variation is canceled if you consider d_a P_{bcd}, antisymmetrized over a,b,c,d, but this of course cancels because it is still bcd symmetric.

In other words, the global part of these gauge symmetries - required symmetries to kill the ghosts - leads to new conservation laws that are so constraining that the S-matrix is guaranteed to be more or less trivial if you impose them.

The only theory to write massless spin(1) fields is with U(1) or Yang-Mills invariance; the only consistent interacting theory of massive spin(1) fields is Yang-Mills theory broken by Higgs mechanism; the only theory for
massless spin 3/2 is supergravity (a theory with local SUSY), and the only theory for spin 2 is a generally covariant theory. Nothing beyond that exists because any interactions contradict the required gauge symmetries.

The only real loophole for *massive* higher spin theories is the whole perturbative string theory - where all the required gauge symmetries are linked in a very specific way.

Yes, if you find that there exist massless fields with spin above two, string theory will be ruled out (and the rest of the late 20th century theoretical physics too). ;-) Note that the gauge invariance argument above is a spacetime argument, but it is also confirmed by the fact that you never finds such massless fields with higher spins in string theory.

Hi Lubos,

An interesting post. I've posted some of my thoughts on this on my weblog.

Peter

Just out of curiosity, regarding the comments concerning the non-existence of massless higher spin theories, is there any special reason you disregard those theories which actually do exist (at least classically)?

I'm thinking of the Vasiliev-type theories, which have been written down in at least 3 and 4 dimensions (and, I think, partially in 5D).
Admittedly they require an infinite tower of higher spin fields, a negative cosmological constant (thus no S-matrix), and are formulated in (IMO) obscure variables, making it difficult to identify the physical fields. But still, the equations of motion are non-linear, gauge invariant, and gives the expected linearized theory around an AdS vacuum.

Jens

Hi Jens!

I apologize, and I wanted to list these theories as another loophole, and say that these loopholes always have an infinite number of fields, like string theory.

Sasha Polyakov likes these theories, and he's invented a AdS/CFT duality for them - one that does not look as convincing as Maldacena's examples, but nevertheless it's fun. The dual is the O(N) vector model.

Best
Lubos

"For the second contributor, concerning higher spin theories: the reason is much more simple - gauge invariance...."

Thanks a lot for the clear explanation!

Warnin: I am a mathematician and my stringy comments are from that point of view.

and I never met one who ENJOYED being dry.

For higher spin interactions, there ae not only the Vasilliev-tupe but also Behrends-Burgers-vanDam-type.

Some of us think Witten's Fields Medal was justified by his Morse theory. Notice two of the other Fields Medals
that year provoked the joke they were QFM - Quantum Fields Medals

The influence of this kind of physics is not all directly from
quantum field theory - you might as well list also Feyneman `path' integrals once mathematicians stopped trying to make them rigorous and instead found alternate proofs for results suggested by path integrals.

The Batalin-Vilkovisky machinery was motivated by quantization problems but turned out to have a considerable mathematical impact classically.

String field theory (as opposed to string theory) has had a non-quantum invluence in mathematics, including string topology of Chas-Sullivan.

Mily Lubo\v s,
Aha, Pilsen -so perhaps your rants are Pilsener influenced - or do you prefer the real Budweis?

jim