Backdoorstudent asked on sci.physics.research:
- 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. ;-)