Gabriele Veneziano, “Dual Modes to Strings” - LNS46 Symposium: On the Matter of Particles
VENEZIANO: Well, it's a great honor and immense pleasure for me to give this talk today at NS 46. And just to be able to recall those marvelous four years that I had spent at the Center for Theoretical Physics between '68 and '72. And I say that both my wife and I still remember those years that some of the very best, perhaps the very best of our life. And of course you can say that was youth, or the birth of our first child here. The discovery of such a great country, but it's also the fact that we came back to Europe [INAUDIBLE] with a feeling of having enriched ourselves and to have made so many nice friends who might really thank very much today. In fact, I would like to conclude this by thanking, most of all, the person who twisted my arm and brought me here. And with whom I had a beautiful collaboration. [INAUDIBLE] who will arrive today to join us in the celebration. The subject of my talk-- let's see [INAUDIBLE]
The subject of my talk is from Dual Models to Strings. And, okay, that subject will be covered in this first five points. And then, I hope to have some time to tell you about how things have developed since that period. I arrived at MIT in the beginning of September '68, but a few months earlier, in the summer of '68, I had worked at what became known as the dual resonance model. And I want to start my story from there. Now I get often upset when reading accounts of early days of string theory. You hear, when [INAUDIBLE] has just opened Bateman and found the beta function sitting there, and that was the start of string theory.
Now, I think this is an important lesson. This was a rather long-lived model. As you will see, it died and resurrected again. And I guess the reason for that is that it was some good blending of phenomenology and of theory. So let me go over the main phenomenological inputs of the dual resonance model and then the theoretical ones. Those days, people working on hydronic interactions knew that the spectrum of hydronic state was extremely reached. Many, many resonances were being discovered almost daily or weekly. Now those resonances were found to lie on almost three Regge trajectories, which I will say more in a moment. And then, another property that [INAUDIBLE] and Schmidt found is that the resonance description of a strong interaction collision was complimentary. You would say dual, duality.
Now the description given in terms of the cross-channel exchange of wretched old finally the quantum numbers of the resonances, which were being discovered, were all consistently non-exhaustive by this. Meaning that their quantum numbers could be describing them so. [INAUDIBLE] bar or three quark systems. Now, to those phenomenological inputs, we try to have some good theory. One was to impose causality in terms of good analytic properties of scattering amplitudes. The second property was exact crossing symmetry. And finally, we took the perhaps, bold idea to work in narrow resonance approximation. An approximation in which resonances have zero width, don't decay, and therefore [INAUDIBLE] instead of being described in terms [INAUDIBLE] is described in terms of simple pulse.
So that was the beginning of the dual resonance model. And as I arrived at MIT, I started immediately to look into more properties of those models. And the first thing to do, for us was clear, it was to count the states. In other words, what physics was lined behind those holes in the scattering amplitudes of this model. Well, first of all, we knew that the pulse were equally spaced in the squared mass. In other words, the mass of the resonances were given by this very simple formula, where capital n is an integer, and alpha prime and alpha zero were two constants to be fitted to the data. Or pictorially, could plot the angular momentum and the squared mass of your states, and they would all line on straight line trajectories. These are the [INAUDIBLE] trajectories. But still the question was, how many states where they're at level, N. And the answer to this question would be answered, in principle, easily. In practice was a different story by using a fundamental property of resonances known as factorisation.
By looking at the amplitude for any initial channel to produce any final channel near the pole, we could try to write it in terms of the product of the copying of the initial state to some resonance times the coupling of the same resonance to the final state times the resonance propagator, which was just the pole. And if you could succeed to represent, indeed, for every initial and final state, that amplitude, in terms of a finite sum of factorisable contributions like that. Then the number of terms that sum contained was nothing but the level degenerates. Now, of course knowing that the resonances have higher and higher angular momentum gave us already some hint that there had to be degeneracy.
For instance, the most naive expectation would be that the degenerates would be 2N plus 1. 2J plus one. Or probably higher because we knew there were also states of lowers being lying down. But the answer we got by the explicit check of factorization was absolutely surprising. So what we and [INAUDIBLE] got was the degeneracy was drawing enormously like an exponential of the square root of N, therefore recalling this formula as an exponential of the mass at large N. Okay. That made quite a bit of sense. In fact it agreed with some model put forward by [INAUDIBLE] much earlier, to explain some spectra of secondary pions. In hydronic collision. Of course this maximum temperature later on was reinterpreted in [INAUDIBLE] as the phase transition to the [INAUDIBLE] plasma. But that's much later. And that formula was also-- that exponential spectrum and the concept of the maximum temperature was also applied by [INAUDIBLE] and Steve Weinberg in those days to the early universe.
So the first discovery was this extreme richness of the spectrum of the dual resonance model. And what came next was that this very rich spectrum could be represented in a very simple way, which goes as follows. It's quite simple. You take [INAUDIBLE] vacuum, call it zero, and you [INAUDIBLE] with [INAUDIBLE] creation operators. So you introduce the usual [INAUDIBLE] daggers of the harmonic oscillators. you introduce them in several species each harmonic oscillator has a Lorentz index new going from 0 1 to 3 and then other integer n, which just takes over the actual number and you pose some sort of standard computation relations between creation and destruction operators with a little catch that you have to put a funny minus sign when they [INAUDIBLE] concern the time like components. And that was simply by lowering [INAUDIBLE] variance.
Then a genetic state, which could achieve that factorization was simply the general state that you get by applying, repeatedly, creation operators on the back. So they were labeled by just occupation numbers. The usual thing. And the mass formula for such a state was given by this operator. The mass [INAUDIBLE] operator was just the occupation number. Except that they were weighted by this integer, N. That explains the [INAUDIBLE] of the huge degeneracy as coming from the fact that you have infinitely many oscillators, and N goes to infinity. And that the frequencies of oscillators are all multiple of the fundamental frequency that this N sitting there.
And now that already smells of a string. We know that this string has many vibration modes, and they are multiples of the fundamental frequency. And Holbert Nielsen, Lenny [? Susskind ?] and many others said, aha, but that must be true. But that was not coming here. Let me, rather, go back to this problem with the relativistic invariance. Relativistic invariance of the theory, imply as I said, at the time-like oscillators a [INAUDIBLE] zero N were giving states of negative norm when acting on the vacuum. Because of this minus sign in the commutator. This negative [INAUDIBLE] state violated the probability conservation, and we call them the ghosts. Okay. They were bad states, which could ruin, completely, the theory.
Then what I could call ghost hunting started. And what does that mean? Well, it means that fortunately, the full set of states that I presented in the previous transparency was redone. It was sufficient to give factorization, but not necessary. Why? Because some linear combination of those states were decoupled from any i and any f. Call them the spurious states. States which really decoupled and which you can dispose of. Then the real question was, does one have a positive-norm Hilbert-space after the removal of the spurious states and not before? And in fact, some ghosts-- those associated in fact, with the first or simply, time-like component of the first oscillator, wherein they found to be spurious by [INAUDIBLE] myself. But that was hardly enough, because we had an infinity of such time-like creation operators, which were all producing negative [INAUDIBLE] states.
At that point, Virasoro made a very important discovery. He found that for a very special value of alpha [INAUDIBLE] not one of those two parameters in the mass formula. In fact, for the value of [INAUDIBLE] one, the number of spurious states increased dramatically. In fact, to put it in formulae, very simply, why [INAUDIBLE] and I found that a certain operator, which now is called l minus 1 acting on any state, and which contained the first harmonic oscillator. Gave spurious states. Virasoro noticed that for alpha 0 equal one, any l minus n, an infinite set of such operators acting on any state was producing spurious states. So this was a very important generalization, which opened the way to the possibility-- not the proof-- but the possibility that ghosts could be completely eliminated thus making this theory well-defined and consistent.
Well, to prove the no ghosts theorem took a little while. And it was very important, at this point, to develop a formalism, which would be as efficient, as elegant, as compact as possible. And this is the period that I would like to call Algebras, Vertices, and Fields. It started, actually, with a work by Gilozzi, which is not often recalled. '69. He noticed that if you take that operator, l minus one, that created the spurious states, the mass operator, which we defined as l zero. And the third operator, which also has some meaning [INAUDIBLE] l plus 1. These three operators satisfy the [INAUDIBLE] algebra of so21, which is a known compact version of ordinary three dimensional rotations. In fact, I mean, if you see that at this time you may be a bit sleepy-- I certainly am-- you may think that this is the algebra of angular momentum. But of course this sign is wrong, and this sign is also wrong if you try to compare with the algebra of angular momentum.
So, you also found this so21 algebra underlying those operators. And then we said, [INAUDIBLE] in a paper we published in a special volume of analog physics dedicated to the late Amos de-Shalit whom I had, by the way, the great pleasure of meeting-- and the wife, [INAUDIBLE] before coming here. We decided to try to take the full set of Virasoro operators, l minus n, l zero again, and then some new operators, l plus n, and try to generalize this result. And then we obtained-- obtained should be quote-on-quote, the algebra. I mean, I'm still reading that paper and eating my-- biting my fingers. We proposed that was the correct generalization of that algebra to the Virasoro case. Okay.
In fact, before the paper appeared in print, the late Joe [INAUDIBLE] then, I think at Berkeley, pointed out that we had made a stupid mistake and that this algebra has an extra C number sitting on the right hand side. A kind of [INAUDIBLE] term, and that the correct algebra was this term plus the certain C over 12. N cubed minus n, times the [INAUDIBLE] delta and minus n. And that c turned out to be d, or the number of space-time dimensions in which we could consider the theory. So you see the crucial difference. And, I mean, I think a [INAUDIBLE] this teaches me an important lesson. I mean, we couldn't guess anything else than this. This of course we could use this [INAUDIBLE] algebra correctly if n and m are 0, 1, or minus 1. The stupid thing is that we could have done the explicit check. And we trusted so much our intuition not to do it, and this was obviously a big mistake.
Anyway this is in fact, the famous algebra of the Virasoro operators, usually referred to as Virasoro algebra, but they want to point out that it was discovered-- well half at MIT and half at Berkley. Now, the next thing was the following. In the work of [INAUDIBLE] and [INAUDIBLE] and myself-- sorry I didn't mention the names earlier-- this is the work where the operators were introduced. The scattering amplitudes of the dual model were expected-- were written as expectation value of a product of vertices and propagators in analogy with field theory, all constructed out of these harmonic oscillators. But we know, in field theory, the harmonic oscillators come as the Fourier coefficients of expansion of a field, phi of x. Now, as it has consistently happened in string theory and the dual model, you had to work backward.
We had the operators, but we didn't have the fields. So we had to work backwards, but that was not too hard. And we [INAUDIBLE] for [INAUDIBLE] we introduced an analogous field, which was Fourier coefficients where precisely the harmonic oscillators that had been used. Well, that field had this funny appearance. Now, we didn't call it x mu, we call it q. But in any case, it had to do with some position.
Now sometimes position is called x, sometimes it's called q. In fact, the first piece is like a position operator of a normal particle. The spin is nothing but the momentum operator of a normal particle. And this is the new piece, but we did not know what that new piece meant. Anyway, we had this field. Now, out of this field, one can construct the so-called vertex operators, and in terms of the vertex operator, finally the [INAUDIBLE] could be written simply as the expectation value of a product of these operators, precisely like in [INAUDIBLE] formalism, you write [INAUDIBLE] amplitude for n particles. You relate it to the expectation value of a product of local fields.
So this formalism led to the possibility of starting farther the theory and to its extensions which I mentioned here. In particular, [INAUDIBLE] used it to construct physical states through which, finally, it proof of the no ghost theorem came about by Richard Brower, and independently by Goddard and Thorne. They proved that there were indeed no negative norm states and the theory provided the number, d, of space-time dimensions was less than or equal to 26. Let me jump here in another development, loops, namely corrections to this narrow resonance approximation to give resonances a width were constructed and Lovelace was the first one to point out that actually, this theory had to be in the equal 26. So the number of space-time dimensions in which these objects leave had to be quantized, if you want.
And then the other extension, which is very important, is the one by [INAUDIBLE] They constructed a dual resonance model with both [INAUDIBLE] and [INAUDIBLE] And you may know, or not know, that that's where supersymmetry was first found, at least in the west before it was applied to field theory. Okay.
Now, once you have a field-- as I told you, so we went from the creation and destruction operators on to a field. Once you have the field you can ask what is the field theory underlying the dual resonance model. Now, [INAUDIBLE] the answer is simple, but of course it took the genius of Nambu and [INAUDIBLE] to guess the right field theory. And the right field theory was the one of a relativistic string.
In fact, inspired by the relativistic point, these people wrote in action for a massless relativistic string in a very simple form. The [INAUDIBLE] go to action, classical action, for thing is nothing but-- I'm sorry this is a bit messed up-- this doesn't belong to the formula. Minus t times the integral over the area element, or if you want, times the total area swept by the string, which is this line as it sweeps the surface doing its motion. So you have to imagine this sheet being embedded in D, capital D, dimensional space-time. Here, represented by x1, x2, x3, but there are many more. See. And the action of number [INAUDIBLE] that was simply proportional to the area spanned by-- swept by the string. In more mathematical form that area is related to the [INAUDIBLE] that I was talking about, by this formula, where alpha and beta can take values 1 and 2 and simply label the coordinates on this two dimensional surface. So, never mind about the details.
Now, this turned out to be a system with constraints. Namely, this D coordinates x mu of the string-- I'm not really all physical. In the sense that only the motion orthogonal to the string is physical. You see a motion of the string, which leaves this surface and change in motion in the plane-- in the surface has no effect and so it's unphysical. I put theory this is what eliminates some of the bad degrees of freedom, which were giving us the problem. And in fact, the algebra of the classical constraints of the system was nothing but the algebra of the Virasoro operator that [INAUDIBLE] and I had. The one without [INAUDIBLE] numbers.
You see, again, this moving backward. Okay. We had a quantum theory, and now through this work, one is trying to reconstruct the classical theory of which what we had in our hands was the quantum [INAUDIBLE] And that is very interesting because classically the angular momentum and the mass of such a system, is continuous, of course. And it turns out to satisfy a simple relation. The angular momentum of any string has to be bound by alpha prime times m square where alpha prime, just our old friend, Rachel Sloan, is related to this parameter, T, of the action of this inverse relationship. Now, however things change drastically at the quantum level.
At the quantum level, j, of course, is quantized as everybody knows, and so is this mass [INAUDIBLE] Less evident but it is a fact. Now, in fact you can impose, now, the validity of the constraints of only-- of the classical [INAUDIBLE] state over to the quantum level and that was the fundamental work of Goddard, Goldstone, Rebbi, and Thorn '73, who found this way, [INAUDIBLE] quantization. That now, at the quantum level, the correct results were that the angular momentum is less than alpha prime m square plus alpha zero h bar. I put back h bar, which was one, in most of the previous formulae. And then, of course, they found also that the algebra [INAUDIBLE] operators had this extra central charge from the string point of view.
And the consistency, and I mean imposing the [INAUDIBLE] of the quantum level required both alpha zero equal one and the central charge to be 26. Therefore, required [INAUDIBLE] equal 26, and this was, from the string point of view, but in perfect agreement with the [INAUDIBLE] condition, alpha zero equal one with Lovelace results about loops with a no ghost theorem and everything. Okay.
And, well, let me skip this for a second. With this work we can say that the dual string theory had reached full maturity. Okay. The circle was closed, but I also want to point out this very interesting conclusion, which as you will see, is good and bad at the same time. And this is that states with zero mass and no zero angular momentum, which are excluded for classical string because of the inequality, j less than m square, are allowed-- even necessary- at the quantum level. Okay.
This is a very important point, which we'll see means the death of the old string. It means life for the new one. In fact, what do I mean by this? That you see, the string has reached complete maturity was a perfectly elegant theory, but of course, as it should be the case, nature had its last word-- has to have the last word on whether it is good or bad. And unfortunately, there were three basic properties of this beautiful, consistent theory, which represented as many death sentences for the old hydronics string. One of the massless states, as they say, at the quantum level you get this massless states-- spinning massless states. And we know, if there is one feature of strong interactions, we know for sure, is the strong range nature. The excessive number of space-time dimensions. We were forced into that, again, by quantization. And it was an obvious problem. And third-- less evident-- is the momentum cutoff that there was at a [INAUDIBLE]
Now this last third property looped phenomenologically good at first. In fact, [INAUDIBLE] was based precisely on the idea that there was a cutoff, an exponential cutoff, on high PT. But, at least these three sets of experiments, the SLAC experiments on [INAUDIBLE] elastic collisions, and which we will hear tomorrow, the CEA experiment, e plus, e minus two hadrons showing a remarkably constant, r. Finally the ISR experiments at CERN on large angle hadron-hadron collision, large PT, hadron-hadron collisions. They all reveal the presence of point-like constituents within the hadron's and there were none in the string. So this third point was really pretty bad.
And now at that time, QCD came about with quarks, gluons with asymptotic freedom, which could justify the naive parton model. Could we present the data, and so on? And a dual string approach-- though beautiful, consistent, and whatever you like-- had to concede defeat. I'm sorry. I had to erase this. I was [INAUDIBLE] for the other side of the coin, but they talked about strings also. Switch the subject. Now, well the reason probably why I'm here is that this was really not the whole end of the story for strings.
So, I want to come to strings of the 90s. And so this means a jump of 20 years. And I will go quickly through those 20 years, but just by mentioning one point. That the bold proposal of Scherk and Schwarz in 1974. As I said strings, at the quantum level have inevitably massless spinning states. Now such states do appear in nature, though not as hadrons. So their idea was, why not use strings for something else? Electromagnetic interactions? Weak interactions, gravity, because other varieties of strings have massless particle of spin. Two for instance. And also, why not for strong interactions, but at the quark gluon rather than at the hadron level.
So, while that proposal went quite unnoticed for some time, meanwhile, the standard model got really established on firmer and firmer experimental grounds as we know. But, okay, something which is often forgot, it failed in one respect. Could not incorporate gravity, okay. It's not yet a theory of quantum value. On the other hand, as John Schwarz puts it, quantum theory clashes with gravity and string theory can't even exist without it. And furthermore, seems to be a consistent theory of quantum gravity. Also of course there was the famous breakthrough of Greenan Schwartz in 1984. And all this plus this frustration about standard quantum field theory being unable to cope with quantum gravity, brought back strings in the front line.
So in my last five minutes, I want to give you a little taste of how string theory has changed. Has made its goals extremely bigger. Okay. We'll see that not be an easy job to check that. In fact, the 1990 version of the string action is lightly different from the one of Nambu and [INAUDIBLE] although one can prove its [INAUDIBLE] equivalence of the two. It looks more like general relativity, two dimensions. And, in fact, following the work of several groups, the new action, which is usually called the [INAUDIBLE] action due to the work of many groups. Contains our old friend x of z, but also contains a new friend. A new person. A two dimensional metric. That's why I call it general relativity two dimension.
These are, for instance, is just the [INAUDIBLE] curvature or the scalar curvature associated with this [INAUDIBLE] Okay, these are the fields to be quantized. You'll see, it's really an evolutionary point of view. Namely, you have to think of the position operator of the string and the two dimensional [INAUDIBLE] as the quantum field to be quantized or integrated over, if you like, [INAUDIBLE]
By contrast, the fields, which I underline here, [INAUDIBLE] I use this capital [INAUDIBLE] you know because [INAUDIBLE] measure. Five and others are functions of x This is not indicated and I really like the ordinary space-time fields that quantum field theory usually deals with, and which quantum field theory, by definition, quantizes. Okay.
Now, and then [INAUDIBLE] two problems when dealing with gravity for instance. Now, the good news of this construction. Most of this is checked and true. I mean, maybe there is some slight wishful thinking, too. The good news is that the conditions of quantum consistency, namely the correct algebra of constraints, remember. That those which gave us alpha zero equal one and [INAUDIBLE] leads, actually, two equations of motion for the space-time time fields, which are the usual classical field theory equations plus short distance corrections. What does it mean? Einstein's equation for the gravitational field, or the [INAUDIBLE] equation, of the Maxwell equation. They come from the quantum consistency of that string action. Okay. Of that two dimensional general activity.
It's very interesting. Classical field theory follows from the first quantization of the string. Not from classical string, but from its [INAUDIBLE] Now what about second quantization? I mean, the reason why we quantize-- usually the electromagnetic field- and we try also to quantize unsuccessfully the gravitational field is because we want to have second quantization effect. Now the remarkable thing about string theory is that it doesn't seem to be any need-- and, in fact, it's probably wrong-- to quantize G and phi, the usual quantum field theory way. Like imposing local [INAUDIBLE] In fact, typical second quantization effects, like pair creation, running of coupling constant, what [INAUDIBLE] have you follow, simply, from first quantization of the string. Namely same first quantization, which gives the classical equation. So if you carry that farther by being careful about how you integrate over the two-dimensional metric, I cannot explain the details, that reproduces also the second quantization effect.
So It's very nice. You have a field theory, in D, dimensions [INAUDIBLE] principle. Maybe some [INAUDIBLE] and you can do all of that by quantizing the two dimensional field. Now, the usual ultraviolet divergences obtained both in the gauge and in the gravity sector. This actually follows from the final string size that strings acquire from quantization. Strings are like harmonic oscillators and you certainly [INAUDIBLE] for the harmonic oscillator goes like this, [INAUDIBLE] Of course times the string tension, you get the fundamental length. Okay. This was the cutoff which gave us trouble for [INAUDIBLE] strings vis-a-vis of, you know, experiments. But certainly the cutoff is there, and now it's welcome instead of being-- it's welcomed because [INAUDIBLE] in quantum field theory mean inconsistency if the theory is not normalizable or some lack of [INAUDIBLE]
For instance, [? renormalized ?] parameters are arbitrary in field theory. Finiteness means predictivity. And in fact, strings can be shown to have really no free adjustable parameters. The parameters that enter in the theory are those which follow from the vacuum degeneracy, okay. Now, in fact, there are two vacuum parameters in string theory, which have dimensions of a speed and the length, and they are respectively the speed of light and this fundamental lane which turns out to be the Planck constant of the theory. There is this very interesting relationship in string theory between the Planck constant and the cutoff, which are completely unrelated in field theory.
Furthermore, that field phi, which is called the dilaton, controls the fine structure constant, Planck constant and therefore the two get related. And since they have different dimensions they're related also through the fundamental length. And other scalar fields, determined in principle, the fate of extra dimensions and through that, what is the gap scale and what is the symmetry there? Sorry, I'm very sketchy about this point. Just to give you a flavor of how much is at stake. Now there are some bad news, and these are related to what [? Herman ?] was saying. A theory which cannot be wrong, cannot be right. I think he said that, right?
And this problem, that the relation between the gauge couplings and the Newton constant tell us that the new strings are not larger than say 10 times the Planck length. So they will be 10 to the minus 32 centimeters [INAUDIBLE] And then for most purposes they look like points, and therefore how can we shoot them down? How can we really see a distinction between strings and points? You see, the old hydronic strings were 10 to the minus 13 centimeters big so they were giving effects at the [INAUDIBLE] scale and they could be ruled out simply on experimental grounds. And now, I think this is a bit related to what Alan [INAUDIBLE] was saying. When you talk about these scales-- when you talk about these energies-- probably there's only one hope, and this is cosmology or using the universe as a whole, as a way to test. And here is something a little bit in the same spirit.
You see, the theory has massless scalers. What I mean what by hope, I mean a hope to be able to shoot down the string, or maybe proof that it's [INAUDIBLE] Now, these are coupled to matter with typical gravitational strain. It's like a [INAUDIBLE] scalar, but with a small omega parameter, if you know what I mean. There effect could contradict the basic tests of general regulativity. Maybe that explains why [INAUDIBLE] feels heavier and heavier. But, I'm sorry. Now, they are massless because of the flat directions. You see, which implies [INAUDIBLE] arbitrariness in the vacuum expectation values and therefore the fundamental constants of nature. I told you, there are no fundamental constants. The only one come because the vacuum appears to be extremely degenerate in perturbation theory.
So, there is an undetermined [INAUDIBLE] on the value of some fundamental parameters. At the same time, as you know from the Goldstone theorem, there are massless particles which travel in this flat direction. This massless partical give problem. So, ultimately also, this vacuum [INAUDIBLE] is related to some symmetry which we start to understand. So it seems to me that the conclusion is that the new strings are also probably doomed unless some mechanism is found to break this huge, symmetry which keeps the vacuum degenerate and, therefore, keeps these unwanted scalers massless and therefore also limits the predictivity of them on the theorem. So, perhaps fortunately the new string game may turn out to be an all win, or all lose game. And I think that is how physics should be, and especially in the case of such a [? pretentious ?] theory. Thank you.