I know that similar questions have been asked before, but none of them is exactly what I would like to know.

I want to have a general idea of what string theory is, without having to read several hundred pages first. So, if you had to give a short formulation of string theory, what would it be?

Here are examples of what kind of formulation I am looking for. In quantum mechanics the states of a system are modeled by elements in a Hilbert space (or a projective one), the observables by certain operators acting on that space. The dynamics of a system is governed by this equation. The possible values an observable can have are the elements of the spectrum of the corresponding operator and so on. For general relativity one can say that the set of all events is described by a four-dimensional manifold with a certain metric satisfying certain equations. Free bodies move along geodesics and so on. With a bit more detail and precision of course.

What is the mathematical description of the basic objects and notions? What are the states of a system described by (mathematically)? Same question for the observables. What are the equations and so on? If these questions are not meaningful and string theory is radically different, then how is it in string theory?

Bump: I don't think the question is so boring and uninteresting so that people don't want to spend the time to answer it. At least comparing to many other questions it seems reasonable. Also I don't think that there isn't anyone who can answer it. In fact I am convinced that there are may people that can write an answer, which would be appreciated.

Edit: After searching around, I found that some of my questions are answered in Zwiebach's book. I am guessing that it can be found in most textbooks. The question remains is there a relatively short straightforward non-popular overview article (that a mathematician would find readable)?

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    $\begingroup$ Any chance this might be a duplicate of physics.stackexchange.com/q/2 $\endgroup$ – David Z Aug 24 '11 at 17:21
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    $\begingroup$ No, I am not interested on how to prove it, whatever that means. Nor am I interested in a online popular 'explanation'. I want to see a short formulation with strict definitions. $\endgroup$ – MBN Aug 24 '11 at 18:49
  • $\begingroup$ Not hundreds of pages but only 64.5 pages (of useful stuff, since the first page is not useful, and the last few pages are too popular-sciencish): arxiv.org/abs/hep-th/0207249 $\endgroup$ – Abhimanyu Pallavi Sudhir Jun 17 '13 at 5:12
  • $\begingroup$ Witten in Physics Today 2015: dx.doi.org/10.1063/PT.3.2980 $\endgroup$ – Qmechanic Nov 20 '15 at 22:01
  • $\begingroup$ @Qmechanic: Nice read! $\endgroup$ – MBN Nov 21 '15 at 8:49

String theory is a perturbation theory of quantum gravity starting with perfectly linear Regge trajectories self-interacting in a consistent bootstrap. Bootstrap means that the interaction of the trajectories is only by exchange of other trajectories, so that the system is self-consistent, or, in 1960s terminology, that it pulls itself up by its own bootstraps.

The best way to learn what string theory is, is to get a copy of Gribov's "The Theory of Complex Angular Momentum", and learn the basic principles of Regge theory. You don't have to learn the Reggeon calculus covered later (although it is interesting), just the basic principles. The point of this theory is to understand spectral properties --- S-matrix states, not detailed microscopic field theory, which breaks down at the Planck scale. The S-matrix is valid at any scale, it is the fundamental observable object in relativistic quantum mechanics, when you don't have point probes.

In QCD, you can make little black holes and use them as point probes. You can even use electrons as point probes, without going to the trouble of making a black hole. This shows that the scale of QCD is not the appropriate scale for string theory, but nevertheless, string theory was discovered by trying to find a consistent bootstrap at this scale. This is very fortunate historically, and required vision and persistance.

Linear Regge trajectories can be understood as string-like excitations. They are the quantized states of an extended relativistic string described by a Nambu Goto action. But the string action by itself doesn't tell you anything about interactions. The interactions of these objects is only by exchanging states found in their own spectrum, with the condition that S-channel exchange is dual to T-channel exchange. They have no other interactions. This defines a dual string theory, the kind people study.

States and observables

String theory is a quantum mechanical S-matrix theory, so the state variables (in an asymptotically flat space) are the following:

  • A classical configuration of fields at infinity, which defines the background.
  • A finite number of incoming particles in a quantum superpostion of plane waves, which define the in-state at minus infinity.

The dynamical law is to produce the out-state, given these ingredients. There is only one observable,

  • The S-matrix between in and out states.

That's it. Every other observable has to be extracted from this one, by some trick. This is extraordinarily difficult, but in practice, there are some simplifications.

  • The classical field configuration is a background which must be consistent with the string theory itself. This gives classical equations which the background field must satisfy. These come from the condition that the string is still conformally invariant in the background, so that the $\beta$ function is zero. These equations define the classical allowed backgrounds, and from this you can extract the classical dynamics at infinity, which is what you use almost all the time, or the approximate quantum field theory description, which is what you use almost all the rest of the time.
  • You can take a field theory limit, and start telling quantum field stories of particle propagation. Almost all the work in the 1980s was based on low-energy supergravity approximations, sometimes with higher order effective action corrections. Such a description can be thought of as an approximation scheme to string theory, by adding higher order string corrections to a quantum field theory, to get a string-corrected effective action. This way of dealing with strings is philosophically least challenging, but it doesn't seem to me to be a very convergent process. The string theory is not a quantum field theory, after all.

More dynamical formulations

There are more dynamical formulations than the S-matrix theory, and more honest formulations than the "effective-action" string-corrected quantum field theory. These honest formulations are due to Mandelstam, Kaku-Kikkawa, Banks-Fischler-Susskind-Shenker, and Maldacena-Witten-Gubser-Klebanov-Polyakov.

  • When you absolutely need to use string theory itself, instead of quantum field theory with effective action corrections, you can move to a dynamical picture where the string tells a story which is local in a version of spacetime. In such a picture, the string theory can be thought of as a normal quantum theory, not an S-matrix theory. The states are defined by superpositions of configurations, jsut like any other quantum theory. The Mandelstam description of strings is one such picture, and because it exists, one could go to a second quantized string field theory, by defining creation and annihilation operators for the string states. So string field theory de-S-matrixes the S-matrix theory. But it is defined on a light cone, and it is technically complicated. But in such a picture, the basic states of string theory are quantum superpositions of light-cone configurations of strings.
  • In 11-dimensional matrix theory, you have a point-particle description in which you can define the state space and evolution again like any other quantum field theory--- as superpositions of noncommutative matrix model configurations, with a normal quantum dynamics in 0+1 dimensions. This is the easiest state space to formulate.
  • In asymptotically AdS backgrounds, instead of incoming/outgoing particles and an S-matrix, you have a full honest to goodness quantum field theory's worth of information at the boundary of space-time. This quantum field theory maps in a not-completely-understood way to the interior description, but the state-space and dynamics are obvious--- they are just like any other quantum field theory.

Notice that the descriptions of the state space is entirely different in the different formulations! This is important, both because their mutual self consistency is an insanely stringent consistency constraint, which has zero chance of being satisfied unless there is a consistent gravitational theory behind it, and also it gives completely different types of observables in different asymptotic backgrounds.

There is no unified way of defining the state space on all backgrounds at once. Each type of background has its own formulation. It requires physical intuition to move between the pictures, and there is no way to communicate the results to a mathematician without communicating the physical picture, because the theory isn't 100% complete.

Literature and Misconceptions

The best review for me was Mandelstam review from 1974. It is very important to learn these old-fasioned ideas, because otherwise you will have all sorts of nonsense in your head about what you can do to strings.

  • Dual strings do not describe statistical polymer properties. It doesn't work, they aren't those types of strings. Polymers interact by self-intersection, strings don't.
  • Dual strings don't work to describe vortex lines in a quantum field theory, although this idea was one of the ways in which they were discovered, by Nielsson. The vortex line picture was simultaneous with the flux-line picture, but the flux line picture is now known to be correct, while the vortex line picture, as far as I know, has no precise version beyond that the string is extended in 1d, like a vortex line. If you make an effective theory of vortex lines in a scalar/gauge field theory, they will interact in crazy non-string ways. In gauge theories with a gravitational dual, like N=4 gauge theory, you probably can make the string be a vortex line, so Neilsson's idea is not altogether wrong (I think), but then some duality will have to link up the vortex and flux line.
  • Dual strings have no deformations: you can't make dual strings interact at collision points, you can't make them attach or detach other non-string objects. They are either a theory of everything or a theory of nothing.
  • Dual strings only allow S-matrix probing. You can't calculate off-shell behavior, meaning you can't describe their detailed dynamics in space-time. You can formulate their world-sheet behavior in space and time. There is string field theory, which attempts to take strings off shell, but now we know the right way to do this is AdS/CFT, although string field theory is still very important.
  • The only way to touch strings to classical objects is to fiddle with asymptotic values of fields. You can't probe them using local quantum fields, because they generate their own local fields. You can make them move in a classical background, but their dynamics determines the quantum properties of all the backgrounds.
  • You can also make them interact with branes, but this is a surprise, and it only works because the branes are weak dual black holes, where strings can partially fall through. It doesn't work for arbitrary surfaces, and there are strong constraints on which branes are allowed.

If you learn the old-fasioned string theory of the 1960s and 1970s, you can understand the rest of the stuff. If you don't, you can't.

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    $\begingroup$ OK, but do you have anything to say about my question!? $\endgroup$ – MBN Sep 9 '11 at 22:06
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    $\begingroup$ I see your point--- I did answer the question, in a way, but only if you already have an idea of how S-matrix theory is formulated. This is not common knowledge anymore, unfortunately. I'll add a brief description, but it's explained much better in the references. $\endgroup$ – Ron Maimon Sep 10 '11 at 0:09
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    $\begingroup$ I see you have expanded your answer, but it is still not waht I was asking for. Also, I find it very hard to take your words seriously, because you have discribed yourself as self taught and you think that the profesional are incompetent!!! And you have your one theories which you spread online!!! All that sounds like a discribtion of a crackpot. Don't take this as an atempt for a persanal insult, I am just saying how it looks form aside. $\endgroup$ – MBN Sep 21 '11 at 12:33
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    $\begingroup$ @MBN: I answered your question. If you do not wish to understand string theory because you don't like my credentials, then it is better not to learn it. People must never judge physics by credentials of the author, there's been too much of that recently. $\endgroup$ – Ron Maimon Sep 21 '11 at 15:40
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    $\begingroup$ @MBN: About the professionals being incompetent, I wish it weren't so, but it is an unfortunate fact of life. I gave you the best answer you are going to get, and it is completely precise. "Math style" is ridiculous--- you just mean "more symbols". That will never help you. The space of states is defined in the answer, it is the S-matrix "in" space, which is a perfectly well defined Hilbert space. The dynamics is the "out" state. The relation is the S-matrix. Nobody else will ever tell you this (although some know it), because S-matrix theory has been politically "out" in physics since 1975! $\endgroup$ – Ron Maimon Sep 22 '11 at 12:25

Ron Maimon's answer is an interesting one, though somewhat polemical, and maybe more suited for someone who wants to think about string theory for themselves, as opposed to just seeing it in its present form.

If MBN isn't satisfied with Zwiebach's textbook - which is just about the most economical path to string theory that is possible, if you're learning it as physics - then I think the formalism to investigate is two-dimensional conformal field theory. This is the formalism used to describe the scattering of superstrings in flat space. Through "target-space duality", the space-time history of a string (its worldsheet) moving through a ten-dimensional background space, is reconstrued as a two-dimensional space-time populated by fields which correspond to the position vectors of the points along the string. Through conformal rescaling, this infinite two-dimensional history (which may split and rejoin itself - these are string interactions) is mapped onto a closed Riemann surface with special points corresponding to the states of the strings at past and future infinity. There are operators located at these points on the Riemann surface, which represent the asymptotic quantum states of the strings.

In quantum field theory, an S-matrix is a "scattering matrix": ultimately it tells you the probability of arriving at future infinity in a state |out> having started at past infinity with the state |in>. In field theory, you might obtain these probabilities by summing over Feynman diagrams corresponding to particle interaction histories. In string theory, instead you have a sum over Riemann surfaces, which are topological equivalence classes of string interaction histories. So you may need to understand perturbative quantum field theory first. If you understand that, the topological expansion of the string S-matrix in terms of the sum over Riemann surfaces should become much more comprehensible.

  • $\begingroup$ This is a good enough answer, but the question asked for the model of physical reality in string theory. To understand this, it is essential to understand that it is S-matrix theory, that the only observable allowed (in asymptotically flat space) is the S-matrix, and that this is a good framework for quantum gravity which includes field theory only in the low energy space-time exists limit. $\endgroup$ – Ron Maimon Sep 18 '11 at 22:11
  • $\begingroup$ Tanks. Zwiebach's book is fine. It is just very long and I was looking for a shorter summary. $\endgroup$ – MBN Sep 21 '11 at 12:28

CERN has a useful series of academic training lectures, available as video and slides(pdf)

Have a look at http://indico.cern.ch/conferenceDisplay.py?confId=52733

Prof. Constantin Bachas, "String Theory for Pedestrians (1/3)"

from february 2009, the most recent one. There are a large number of similar lectures earlier in time. CERN believes in training its academics :) .

My even more pedestrian summary is that: string theory is an attempt to describe and derive the microcosm of elementary particles as measured experimentally up to now, within the mathematical framework of relativistic strings ( and more). A bonus is the objective of unifying all four forces acting on these particles in one mathematical formulation.

  • $\begingroup$ This seems to follow the exposition of a typical introductory textbook, just shorter, which is good. But it doesn't address my questions. Mathematically what are the states, observables and so on. They are used in the text, but I found no definition whatsoever. Should I think of string theory as a type of quatum field theory? For example along the line of QED but instead of Maxwell's equations we quantize relativistic wave equation? My impression was that ST is not just a specific QFT. $\endgroup$ – MBN Aug 24 '11 at 14:56
  • $\begingroup$ @MBN Did you reach the last slide where he summarizes? Relativistic string solutions are the equivalent of the solution of Klein Gordon and Dirac equations and describe towers of particles. Making a quantum field theory of this will be a meta level on these solutions, the way QFT is a meta level on the Dirac and Klein Gordon solutions, turning them into operators. $\endgroup$ – anna v Aug 24 '11 at 16:48
  • $\begingroup$ Yes, but it is not a formulation of string theory, it is more of a general comment. I would like to see some definitions. $\endgroup$ – MBN Aug 24 '11 at 18:47
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    $\begingroup$ @MBN I think there are consistent definitions in the first lecture linked. The level of complexity has gone up, and defining string equations and the solutions pertinent to particle physics cannot be contained in a paragraph without being gibberish, imo. For example, Hilbert spaces would be gibberish for somebody who only knows classical electricity and magnetism at the introductory level. The tools and language have to be built up to a data base, and that is what the academic lectures attempt. $\endgroup$ – anna v Aug 25 '11 at 3:19

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