Beta Function in String Theory

[nyc]

I've heard that that origin of String Theory was in Gabriele Veneziano's analysis of the Euler beta function in relation to the strong force. I was wondering if anyone could refer me to a paper or derivation describing how this function ended up describing particles as strings.

 

[Naty1]

Here's what Brian Greene in FABRIC OF THE COSMOS has to say about that:

In 1968 Gabriele Veneziano...was trying to understand the the strong nuclear force by studying thje results of high energy particle collisions produced in atom smashers...He realized that the 200 year old formula...seemed to match data on the strong nuclear force with precision. ..his briliance was to recognize unusual patterns in the data and to makwe the unanticipated link to a formula dervied centuries earlier for purely mathematical reasons.

Perhaps someone will post a specific paper for you...a hint MIGHT be that the beta function is related to probability distributions...and particle collision scattering is described by the S (Scattering) matrix...Gerard d'hooft used the S matrix to refute Stephen Hawking use of the $ (dollar) matrix Hawking used in his famous black hole "Hawking radiation" claims...but unless these clues lead someone else to more specific suggestions I'd hold off heading in any of this directions as I am guessing at connections...

Yet another hint is that string theory was originally developed to describe the strong force...and it initially failed...and then rather set aside for a time until it was discovered a spin 2 particle was suggested within its mathematics...meaning a massless particle with the spin required for a quantum of gravity...I see John Schwarz and his collaborator Joel Scherk apparently discovered this extension..and it was realized string theory math might be a forum for all forces..and string theory was reborn...

 

[arivero]

The formula for "Veneziano amplitude" in terms of the gamma function (and thus beta function if you wish) is easy to find. I think that the historical development was first the empirical Chew-Frautschi plot, then its interpretation by Regge, then the ansatz of "Veneziano Amplitude", then the Lagrangian for it, then the interpretation of this lagrangian as the one of a string. This last step was simultaneus in a a lot of papers, at least there is one from Nambu and other one from Susskind.

Veneziano's amplitude ansatz was not, as far as I know, inspired by strings, but just by the "democratic bootstrap" movement, where the different scattering channels were expected to have some symmetries. After the string was identified, people published a lot of diagrams showing that the symmetry was due to the "strip" appearing in Feynman diagrams if each particle was suppossed to be a string.

 

[arivero]

and then rather set aside for a time until it was discovered a spin 2 particle was suggested within its mathematics...meaning a massless particle with the spin required for a quantum of gravity...I see John Schwarz and his collaborator Joel Scherk apparently discovered this extension..and it was realized string theory math might be a forum for all forces..and string theory was reborn...

Actually, this could have been a hypothetical discovery path in an alternate universe, only that it did not happened. Point is, Riemann curvature tensor can be recognized as a indexed collection of bidimensional objects (so it colapses to the Ricci tensor for most uses), this is what curvature is about: cutting by planes and measuring the curves intersecting the cut. So some research on quantum gravity could have tried to use extended objects to explore space time, and then located string theory.

I would not say that the discovery of spin 2 caused a reborn, if you look at the dates, the work of Scherk and Schwarz is late 1974; string theory itself is 1968-1970, the fermion version is 1971. It could be said the contrary: the incorporation of gravity almost killed the interest by string theory, and it was not reborn until the suggestion of heterotic compactifications aiming to reproduce the standard model.

 

[Naty1]

The last paragraph in my post #2, disputed by arivero, comes from pages 338-342 of Brian Greene's FABRIC OF THE COSMOS...Although I was around at the time, I have no personal knowledge of the historical development...

 

[humanino]

The main problem for string theory in the early 70's is the proof of renormalizability of Yang-Mills theories by t'Hooft together with asymptotic freedom. Once QCD was discovered, interest faded quite a bit. The proposal from Scherk and Schwarz saved string theory from being completely abandoned, and I agree with arivero that this should not be called a "reborn". You might be interested in The Early Years of String Theory: A Personal Perspective

[...]
the success of QCD made the effort to formulate a string theory of hadrons less pressing
[...]
Scherk and I proposed to interpret string theory as a quantum theory of gravity, unified with the other forces. This meant taking the whole theory seriously, not just viewing it as a framework for deriving GR and Yang–Mills theory as limits. Our paper was entitled "Dual Models for Non-Hadrons"
[...]
For the most part our work was received politely — as far as I know, no one accused us of being crackpots. Yet, for a decade, very few experts showed much interest.

edit
also, check in the future (not yet available)
http://www.scholarpedia.org/article/Veneziano_amplitude

 

[arivero]

The last paragraph in my post #2, disputed by arivero, comes from pages 338-342 of Brian Greene's FABRIC OF THE COSMOS...Although I was around at the time, I have no personal knowledge of the historical development...

Yes, I dispute it just because of this rewritting of history. Really Brian Greene was not there neither, it was Michael B. Green who did the formal revolution of superstrings. Fortunately we have Spires:

http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+A+GREENE%2C+BRIAN+AND+DATE+BEFORE+1990&FORMAT=www&SEQUENCE=ds

http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+A+SCHWARZ+AND++a+Scherk+and+DATE+BEFORE+1995+&FORMAT=www&SEQUENCE=ds

http://www.slac.stanford.edu/spires/find/hep/www?rawcmd=FIND+A+SCHWARZ+AND+A+green+AND+DATE+BEFORE+1995&FORMAT=www&SEQUENCE=ds

You can see that:

the paper on gravity is from 1974: "Dual Models and the Geometry of Space-Time". There is also the preprint of 1975 "Dual Model Approach to a Renormalizable Theory of Gravitation", *reprinted* in 1986 but never under referee. It seems it was submitted to the 1975 Gravitation Essay Contest.

the 1984 paper, from Schwarz and Green, is related to the paper on anomaly cancelation for superstrings.

Brian R. Greene enters action in 1985, his main interest is...Realistic three generation GUT models from superstrings! He is not interested in gravity at all.

And really, the superstring revolution comes not only because of the review of schwarz (1982, sort of sequel to polyakov 1981 landmark) and the anomaly cancelation, but because of the almost simultaneous (1985) discovery of the heterotic string, http://prl.aps.org/abstract/PRL/v54/i6/p502_1 ... by David J Gross, a student of Chew himself!

Received 21 November 1984

A new type of superstring theory is constructed as a chiral combination of the closed D=26 bosonic and D=10 fermionic strings. The theory is supersymmetric, Lorentz invariant, and free of tachyons. Consistency requires the gauge group to be Spin(32)/Z2 or E8×E8.

What happened in between 1974 and 1984? That people was trained in supergravity and supersymmetry, so they were willing to accept the "strings as gravity" justification because they had already accepted sugras. But more important was that E8xE8 had a lot of playroom for GUT groups.

(my own opinion is that most of this E8xE8 is due to quantisation of the flavour/family symmetry, and then it is wrong to try to use it for the gauge Yang Mills forces).

 

[tom.stoer]

(my own opinion is that most of this E8xE8 is due to quantisation of the flavour/family symmetry, and then it is wrong to try to use it for the gauge Yang Mills forces).

But there is no difference: QCD is a gauge theory of color not affecting flavor; the weak interaction = the SU(2) part of the GSW theory is a gauge theory of flavor not affecting color. Simply look at the fermion indices the symmetry is acting on. That's why some people call the GSW model QFD = quantum flavor dynamics.