## Monday, June 15, 2009 ... /////

### Bert Schroer on the worldsheet "metaphor"

A reader has pointed out an entertaining article on the arXiv. It was written by Bert Schroer and its name is

Remarks on the world-sheet saga.
As Jacques Distler has independently figured out on Clifford Johnson's blog, Schroer's general "reasoning" goes as follows:
1. Only 0-dimensional particles are acceptable building blocks in physics. So string theory must be a theory of point-like particle fields with infinitely many components, too.
2. It follows that one can never include winding modes even if the target space is not simply connected.
3. It follows that there is no T-duality. In fact, nothing that string theorists ever talk about exists, and everything is just a metaphor. String theory must be replaced by the Schroer caricature of string theory. The latter contains no strings and is as meaningless as any paper that Mr Schroer has ever printed. In Schroer's logic, it follows that string theory sucks.
Up to some moment, it is funny to see such extremely tiny, frail, modest, and perhaps somewhat excessively old brains that are nevertheless still able to hold such a huge chunk of self-confidence, surpassing its expected size by a few orders of magnitude, even though they are clearly not large enough to incorporate elementary concepts such as the winding strings, worldsheets, T-duality, on-shell amplitudes, or path integral.

For Schroer, all these notions and many others are the "ultimate sins of theoretical physics". On the other hand, for every other physicist who actually knows what has happened in the last 50 years, they're just "theoretical physics" - in fact, pretty much all of it. Schroer also thinks that his paper is a contribution to Strings 2009 (see the subtitle). I guess that the organizers of Strings 2009 should hire a psychiatrist for the scenario that Mr Schroer visits Rome next week and appears at the proceedings. They may be needed to deal with this - whatever is the politically correct name for a psychopath.

Is there at least one true proposition in Bert Schroer's rant?

Reality

Well, in reality, perturbative string theory is completely well-established and the results such as T-duality are proved theorems that are as rigorous as any good results in theoretical physics have ever been. Are strings just a metaphoric description of many point-like particle species?

Yes and no. A string can behave as one of an infinite family of possible vibration modes or particle species. This is the viewpoint that e.g. string field theory tries to take as seriously as you can get. In string field theory, physics is described by a "string field" which may be thought of as a collection of infinitely many point-like fields. However, there are some limitations of this picture:
1. In string theory, the properties of this whole infinite tower of states is constrained or derived from the same string - one governed by the worldsheet physics. The idea of a "generic" collection of point-like particle fields would have no explanation for this infinite collection of constraints. The interactions of all these fields are determined by the same worldsheet physics, too. That's the first sense in which the worldsheet is very real.
2. String theory has its own rules to derive the particle content in more complicated situations, e.g. with curved extra dimensions. The whole spectrum of masses has to be recalculated for each background.
3. If the spacetime is not simply connected, winding strings are added and have to be added - for reasons that can be proved in many seemingly inequivalent ways: winding modes must be a part of the story. T-duality is one of their key consequences.
4. The actual physics of string theory leads to somewhat modified rules how to compute loop diagrams. They slightly differ from the rules that one would expect in the case of infinitely many point-like particles. This difference is important for the short-distance finiteness of string theory.
Let me say a few more words about these points.

A constrained tower of states

Fermi's four-fermion theory of beta decay is non-renormalizable. We know that it is just the low-energy approximation of a more accurate and well-behaved theory, a spontaneously broken gauge theory, where the force is mediated by the exchange of vector bosons.

Similarly, the short-distance problems of quantized general relativity have to be cured by a similar package of new physics. In this case, one needs an infinite number of new particle species to contribute to the graviton scattering.

However, an infinite collection of new arbitrary fields would depend on an infinite collection of parameters such as masses and couplings. That would be as bad as a non-renormalizable theory because an infinite number of unknown parameters could never be fully determined. However, string theory does something different: it determines the masses and interactions of all these new massive states that are needed to regulate the divergences of gravity.

So in some sense, you may always parameterize string field operators as many field operators that create point-like particles. However, if you want to know anything about their masses, interactions, and other physical properties, you need to appreciate that they arise from a quantized string or a quantized worldsheet.

The more general visualization of string fields as a collection of many point-like particle fields would be unable to see all the constraints that follow from the worldsheet equations. For example, the worldsheet is Lorentz-invariant. Locally, a piece of worldsheet doesn't allow you to choose a preferred time direction: its spatial and temporal coordinates have properties that are related by the symmetry.

While it's always possible to decompose a periodic function into the Fourier modes or a string field into the infinitely many component fields, such a procedure obscures virtually all key physical properties of the stringy objects - their actual inner architecture.

Background-dependence of the spectrum

Another point I mentioned is that the spectrum of the point-like particle fields depends on the background. In the realm of "cracked" or "broken would-be popular physics", the background dependence is often presented as a bad thing. But physical consistency requires many things to depend on the background. And the dependence on the background is necessary for making predictions, too.

In the case of curved extra dimensions of a Calabi-Yau shape, you may ask how many generations of quarks and leptons a heterotic string compactification has. The answer is that the number depends on topological invariants of the Calabi-Yau manifold, such as its Euler character (or the Hodge numbers).

If you only cared about the massless fields, you could determine their spectrum in the large dimensions by applying Kaluza-Klein methodology to the higher-dimensional massless fields. But more generally, the massive spectrum of string theory on a curved manifold is affected by its shape. You shouldn't imagine that you are putting a predefined theory of fixed point-like particles (and fixed fields) on the same manifold.

Winding modes

The most obvious example showing that such an idea would not be quite right are the winding modes. If the target space has non-contractible loops in it, strings can be wound around it "w" times. This is not just an arbitrary possibility that someone has added to make string theory richer (or more confusing). Quite on the contrary, all methods to calculate anything in string theory unambiguously imply that winding strings exist, have to exist, and their properties are completely determined.

To see why, you may imagine a compactification with one dimension whose shape is a circle. Consider a string that randomly vibrates. At some point, its shape resembles the combination of a winding string with "w=+1" and another, nearby one with "w=-1".

Because interactions have to be allowed in string theory, such a closed string must be allowed to split into two strings with the "w=+1" and "w=-1". In fact, the local character of the interaction ("crossing over") is identical to any other interaction where a closed string splits into two.

If you wanted to prohibit the creation of winding strings in this way, you would have to impose rules that would be checking whether wound strings are being created according to some reference frame. Such rules would have to be non-local and they would arguably be unnatural. They would probably lead to further inconsistencies. You could still consider such a theory with global restrictions. At any rate, string theory as string theorists know it is a different theory. It is described by local physics on the worldsheet. It makes sense and nothing is arbitrary in it.

The existence of winding strings is also necessary for us to derive T-duality, among other things.

One-loop Schwinger parameter

Finally, I mentioned that string theory leads to slightly different rules for loop amplitudes than a collection of infinitely many point-like particle fields would. Loop diagrams in field theory may be written as integrals over the Schwinger parameters - essentially lengths of imaginary world lines of the intermediate particles.

Similarly, loop diagrams in string theory are integrals over the so-called moduli (of Riemann surfaces) which generalize the Schwinger parameters except that the number of these parameters is larger by a factor of 2-6. These moduli describe the inequivalent shapes of the worldsheets with a nontrivial topology. The integrals can be re-imagined as integrals over the ordinary Schwinger parameters.
Except that the range of integration is not quite identical.

In field theory, the Schwinger parameter for a one-loop diagram would go from 0 to infinity, counting a length of a circle. In the perturbative theory of closed strings, this diagram would be replaced by a two-dimensional integral over the shape of the torus. If you assume that strings are just point-like particles with many species, the stringy integral over "tau" would cover the area defined by
|Re(tau)| < 1/2,
Im(tau) > 0.
However, the stringy domain of integration is smaller:
|Re(tau)| < 1/2,
Im(tau) > 0, |tau| > 1.
Note that the additional third condition removes the region with a small imaginary part of "tau": this region is (or was) the source of ultraviolet divergences for point-like particles. But in string theory, it corresponds to very thin (ultraviolet) tori which can be - by a rotation by 90 degrees - reinterpreted as very thick (infrared) tori.

One only has to sum over nonequivalent diagrams. Because this "modular group" (not just Z2, actually SL(2,Z)) proves that certain tori are equivalent, only a smaller "fundamental domain" is integrated over. It follows that all extreme regions of the integration space that could give rise to an ultraviolet divergence may be interpreted as extreme "infrared", long-distance regions. Once you prove that the long-distance limit of your theory is consistent, it follows that there can't be any short-distance problems, either: without a loss of generality, all hypothetical short-distance problems have been transformed to long-distance problems.

Summary

To summarize, the worldsheets are more real in perturbative string theory than virtually all laymen and beginners imagine. When the string coupling gets large, all the worldsheets and strings cease to lose their "monopoly" to be the fundamental histories or particles: other objects, such as D-branes, join them (or even supersede them). But as long as you study the weakly coupled string theory perturbatively, worldsheets of string theory are not just a metaphor: they are as real as the worldlines of point-like particles in the old-fashioned point-like particle field theories.

The rules to deal with the worldsheets generalize the methods to deal with the worldlines. And one may always try to "Fourier-expand" worldsheets into worldlines and reinterpret string theory as a theory of point-like particles. Up to some point, such a reinterpretation works but it completely misses the point of string theory - that it actually tells us everything about the structure and behavior of the infinitely many fields. And for technical calculations, the dogmatically applied point-like formulae can lead to results that have subtle errors in them.

And that's the memo.