# What are the details around the origin of the string theory?

It is well-known even among the lay public (thanks to popular books) that string theory first arose in the field of strong interactions where certain scattering amplitudes had properties that could be explained by assuming there were strings lurking around. Unfortunatelly, that's about as far as my knowledge reaches.

Can you explain in detail what kind of objects that show those peculiar stringy properties are and what precisely those properties are?

How did the old "string theory" explain those properties.

Are those explanations still relevant or have they been superseded by modern perspective gained from QCD (which hadn't been yet around at the Veneziano's time)?

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My experimental thesis was using Regge poles to search and explain resonances seen :). Regge poles are still there. Their value as an organizing substructure was undermined when the eightfold way first showed the value of symmetries, and then of course the SU(2)xSU(3)xU(1) success as the standard model erased them from memory . I have always thought of all string incarnations as the multiple uses of the harmonic oscillator solutions, which are really the first symmetric function in a series expansion of any potential. –  anna v Aug 19 '11 at 5:16
@anna: The Regge theory is enjoying a remarkable comeback. It was unfairly maligned. People are now discovering that S-matrix theory is the best approach to perturbative supergravity, and the mysterious structure of QCD Regge poles is again active in AdS/QCD. Regge theory wasn't a fad. –  Ron Maimon Aug 19 '11 at 8:41

in the late 1960s, the strongly interacting particles were a jungle. Protons, neutrons, pions, kaons, lambda hyperons, other hyperons, additional resonances, and so on. It seemed like dozens of elementary particles that strongly interacted. There was no order. People thought that quantum field theory had to die.

However, they noticed regularities such as Regge trajectories. The minimal mass of a particle of spin $J$ went like $$M^2 = aJ + b$$ i.e. the squared mass is a linear function of the spin. This relationship was confirmed phenomenologically for a couple of the particles. In the $M^2$-$J$ plane, you had these straight lines, the Regge trajectories.

Building on this and related insights, Veneziano "guessed" a nice formula for the scattering amplitudes of the $\pi+\pi \to \pi+\rho$ process, or something like that. It had four mesons and one of them was different. His first amplitude was the Euler beta function $$M = \frac{\Gamma(u)\Gamma(v)}{\Gamma(u+v)}$$ where $\Gamma$ is the generalized factorial and $u,v$ are linear functions of the Mandelstam variables $s,t$ with fixed coefficients again. This amplitude agrees with the Regge trajectories because $\Gamma(x)$ has poles for all non-positive integers. These poles in the amplitude correspond to the exchange of particles in the $s,t$ channels. One may show that if we expand the amplitude to the residues, the exchanged particles' maximum spin is indeed a linear function of the squared mass, just like in the Regge trajectory.

So why are there infinitely many particles that may be exchanged? Susskind, Nielsen, Yoneya, and maybe others realized that there has to be "one particle" of a sort that may have any internal excitations - like the Hydrogen atom. Except that the simple spacing of the levels looked much easier than the Hydrogen atom - it was like harmonic oscillators. Infinitely many of them were still needed. They ultimately realized that if we postulate that the mesons are (open) strings, you reproduce the whole Veneziano formula because of an integral that may be used to define it.

One of the immediate properties that the "string concept" demystified was the "duality" in the language of the 1960s - currently called the "world sheet duality". The amplitude $M$ above us $u,v$-symmetric. But it can be expanded in terms of poles for various values of $u$; or various values of $v$. So it may be calculated as a sum of exchanges purely in the $s$-channel; or purely in the $t$-channel. You don't need to sum up diagrams with the $s$-channel or with the $t$-channel: one of them is enough!

This simple principle, one that Veneziano actually correctly guessed to be a guiding principle for his search of the meson amplitude, is easily explained by string theory. The diagram in which 2 open strings merge into 1 open string and then split may be interpreted as a thickened $s$-channel graph; or a thick $t$-channel graph. There's no qualitative difference between them, so they correspond to a single stringy integral for the amplitude. This is more general - one stringy diagram usually reduces to the sum of many field-theoretical Feynman diagrams in various limits. String theory automatically resums them.

Around 1970, many things worked for the strong interactions in the stringy language. Others didn't. String theory turned out to be too good - in particular, it was "too soft" at high energies (the amplitudes decrease exponentially with energies). QCD and quarks emerged. Around mid 1970s, 't Hooft wrote his famous paper on large $N$ gauge theory - in which some strings emerge, too. Only in 1997, these hints were made explicit by Maldacena who showed that string theory was the right description of a gauge theory (or many of them) at the QCD scale, after all: the relevant target space must however be higher-dimensional and be an anti de Sitter space. In AdS/CFT, much of the original strategies - e.g. the assumption that mesons are open strings of a sort - get revived and become quantitatively accurate. It just works.

Of course, meanwhile, around mid 1970s, it was also realized that string theory was primarily a quantum theory of gravity because the spin 2 massless modes inevitably exist and inevitably interact via general relativity at long distances. In the early and mid 1980s, it was realized that string theory included the right excitations and interactions to describe all particle species and all forces we know in Nature and nothing could have been undone about this insight later.

Today, we know that the original motivation of string theory wasn't really wrong: it was just trying to use non-minimal compactifications of string theory. Simpler vacua of string theory explain gravity in a quantum language long before they explain the strong interactions.

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+1, but a quibble--- I don't agree that "strings" by themselves demystify world-sheet duality, they just give it a clear picture. They explain duality if by "strings" you mean that there is only one open string diagram, but this is a duality assumption. Imagine scattering of physical rubber bands--- they can be exchanged in t-channel or s-channel, and you have to sum the two contributions, they are separate. New virtual rubber bands are produced at collisions at localized points, not by smoothly opening the topology like a string diagram. Duality says string aren't like rubber bands. –  Ron Maimon Aug 19 '11 at 8:34
Dear Ron, I don't think that your comment is right. Strings in string theory are "relativistic rubber bands" while normal rubber bands are made out of atoms that pick a special frame and obey a Lorentz-breaking world volume theory, but this is the only difference. Rubber bands, much like strings, would still respect the world sheet duality. Well, at least any rubber bands that could be continued to the Euclidean spacetime. In particular, you can't create a whole rubber band at a single point. Rubber bands are 1-dimensional so you may only create them 1 point/atom at a time, one after another. –  Luboš Motl Aug 19 '11 at 10:13
@Lubos: I don't think you are right for what people think of as ordinary "rubber bands". The free action might resemble a string action, but a naive collection of point particles with delta-repulsive interaction will have generic self-intersection interaction wherever the world-sheet self intersects, so their interactions cannot be given by a local worldsheet theory (although the free propagation might). If you look at how string theory gets around that, its by worldsheet duality, that the interactions are all due to exchange of strings, and this is an additional assumption (Chew's bootstrap). –  Ron Maimon Aug 19 '11 at 16:55
To expand a little--- in string theory, you use S-matrix asymptotic states as the objects which are exchanged between the strings (which for linear Regge trajectories turns out to be equivalent to summing over worldsheets), without asking for more fundamental things of which these things are built, and without allowing arbitrary interactions between the fundamental things to kick in at high energies. So it is automatically a theory of everything if it is a theory of anything. I think this was understood by the bootrstrap community to a certain extent even in the 1960s. –  Ron Maimon Aug 19 '11 at 17:16
Dear @Ron, 1-dimensional objects almost never self-intersect in spacetime of higher dimension of 3+1, e.g. $D=26$ or $D=10$, so this is a measure zero problem and you may completely forget about it. An even more obvious "measure zero" situation that may be ignored is when all atoms of a rubber end are located at the same point - which, by the way, can't happen at all if the rubber bands are made out of atoms such as the real rubber bands. The bulk of the dynamics of a string or a rubber band - it's totally analogous in both cases - is given by the dynamics of 2-dimensional world sheets. –  Luboš Motl Aug 19 '11 at 18:17

I liked this video, where Susskind explains how he and some friends discovered string theory in the context of meson scattering.

(This is just a very preliminary answer, the video is probably way below your level ... :-P, I will take it back as soon as it is superseded)

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You`re welcome :-))). He starts drawing a plot of the Regge Trajectories and in the course of the lecture nicely derives it from meson scattering, including the transformation to light cone coordinates step by step. I dont remember all of the details, will have to rewatch it when I busy myself more seriously with the whole string course of Lenny Susskind ... –  Dilaton Aug 18 '11 at 20:12
Perfect example of how deleting "thanks" comments lead to weird results . –  Dimensio1n0 Jul 30 '13 at 17:54
@Dimension10 yep, Marek wrote a very friendly nice comment to which I responded... –  Dilaton Jul 30 '13 at 18:31

Well I believe the original clue was Regge trajectories. It was observed that if you plotted mass squared vs. angular momentum for strongly interacting resonances, they tended to follow straight lines. This could be explained as the spectrum of rotating strings connecting massless particles.

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This is what I've heard as well. –  David Z Aug 18 '11 at 19:40
Interesting. What I heard about the origins was actually something to do with Veneziano and beta function. How does this relate? –  Marek Aug 18 '11 at 20:03
Okay, I think I can answer my own question. Since the spectrum lies on the Regge trajectories it means it can be interpreted as the poles of the beta function. I'd still want to have more details though (ideally a reference). –  Marek Aug 18 '11 at 20:16