I've heard occasional mentions of the term "bootstraps" in connection with the S Matrix. I believe it applies to an old approach that was tried in the 1960s, whereby - well I'm not sure - but it sounds like they tried to compute the S Matrix without the interaction picture/perturbation theory approach that we currently use.
I'm aware that the approach was abandoned, but my question is: how was it envisaged to work ? What was the input to the calculation supposed to be and how did the calculation proceed ?
I know it's got something to do with analyticity properties in terms of the momenta, but that's all I know....
For example, from the wikipedia article:
Chew and followers believed that it would be possible to use crossing symmetry and Regge behavior to formulate a consistent S-matrix for infinitely many particle types. The Regge hypothesis would determine the spectrum, crossing and analyticity would determine the scattering amplitude--- the forces, while unitarity would determine the self-consistent quantum corrections in a way analogous to including loops.
For example - I can't really understand how you would hope to compute the scattering amplitude just given crossing symmetry and assuming analyticity
Answer
This approach most definitely does work, it just doesn't give the fundamental theory of the strong interactions, it gives string theory. String theory was originally defined by Venziano's bootstrap formula for the leading term in an S-matrix expansion, and the rest was worked out order by order to unitarize the S-matrix, not through a field theory expansion. This is good, because string theory isn't field theory, and it can't be derived from a field theory Lagrangian, at least not in the usual way. The result can nowadays be interpreted in terms of string field theory, or nonperturbative AdS/CFT constructions, but it is a new theory, which involves infinite towers of particles interacting consistently without a field theory underneath it all.
The idea that S-matrix died is a political thing. The original 1960s S-matrix people found themselves out of a job when QCD took over, and the new generation that co-opted string theory in 1984 mostly wanted to pretend that they had come up with the theory, because they were all on the winning side of the S-matrix/field-theory battle. This is unfortunate, because, in my opinion, the most interesting physics of the 1960s and early 1970s was S-matrix physics (and this is a period that saw the greatest field theory work in history, including quarks and the standard model!)
Chew Bootstrap
A bootstrap is a requirement that you compute the S-matrix directly without a quantum field theory. In order for the theory to be interesting, the S-matrix should obey certain properties abstracted away from field theory
- It should be unitary
- It should be Lorentz invariant
- It should be crossing invariant: this means that the antiparticle scattering should be described by the analytic continuation of the particle scattering
- It should obey the Landau property--- that all singularities of scattering are poles and cuts corresponding to exchange of collections of real particles on shell.
- It should obey (Mandelstam) analyticity: the amplitude should be writable as an integral over the imaginary part of the cut discontinuity from production of physical particles. Further, this cut discontinuity itself can be expanded in terms of another cut discontinuity (these are the mysterious then and still mysterious now double dispersion relations of Mandelstam).
This is a sketchy summary, because each of these conditions is involved. The unitarity condition in particular, is very difficult, because it is so nonlinear. The only practical way to solve it is in a perturbation series which starts with weakly interacting nearly stable particles (described by poles of the S-matrix) which exchange each other (the exchange picture is required by crossing, and the form of the scattering is fixed by the Landau and Mandelstam analyticity, once you know the spectrum).
The "Bootstrap property" is then the following heuristic idea, which is included in the above formal relations:
- The particles and interactions which emerge as the spectrum of the S-matrix from the scattering of states, including their binding together into bound states, should be the same spectrum of particles that come in as in-states.
This is a heuristic idea, because it is only saying that the S-matrix is consistent, and the formal consistency relations are those above. But the bootstrap was a slogan that implied that all the consistency conditions were not yet discovered, and there might be more.
This idea was very inspirational to many great people in the 1960s, because it was an approach to strong interactions that could accommodate non-field theories of infinitely many particle types of high spin, without postulating constituent particles (like quarks and gluons).
Regge theory
The theory above doesn't get you anywhere without the following additional stuff. If you don't do this, you end up starting with a finite number of particles and interactions, and then you end up in effective field theory land. The finite-number-of-particles version of S-matrix theory is a dead end, or at least, it is equivalent to effective field theory, and this was understood in the late 1960s by Weinberg, and others, and this led S-matrix theory to die. This was the road the Chew travelled on, and the end of this road must be very personally painful to him.
But there is another road for S-matrix theory which is much more interesting, so that Chew should not be disheartened. You need to know that the scattering amplitude is analytic in the angular momentum of the exchanged particles, so that the particles lie on Regge trajectories, which give their angular momentum as a function of their mass squared, s.
Where the Regge trajectories hit an integer angular momentum, you see a particle. The trajectory interpolates the particle mass-squared vs. angular momentum graph, and it gives the asymptotic scattering caused by exchanging all these particles together. This scattering can be softer than the exchange of any one of these particles, because exchanging a particle of high spin necessarily has very singular scattering amplitudes at high energy. The Regge trajectory cancels out this growth with an infinite series of higher particles which soften the blowup, and lead to a power-law near-beam scattering at an angle which shrinks to zero as the energy goes to infinity in a way determined by the shape of the trajectory.
So the Regge bootstrap adds the following conditions
- All the particles in the theory lie on Regge trajectories, and the scattering of these particles is by Regge theory.
This condition is the most stringent, because you can't deform a pure Regge trajectory by adding a single particle--- you have to add new trajectories. The following restriction was suggested by experiment
- The Regge trajectories are linear in s
This was suggested by Chew and Frautschi from the resonances known in 1960! The straight lines mostly had two points. The next condition is also ad-hoc and experimental
- The Regge slope is universal (for mesons), it's the same for all the trajectories.
There are also "pomerons" in this approach which are not mesons, which have a different Regge slopem but ignore this for now.
Finally, there is the following condition, which was experimentally motivated, but has derivations by Mandelstam and others from more theoretical foundations (although this is S-matrix theory, it doesn't have axioms, so derivation is a loose word).
- The exchange of trajectories is via the s-channel or the t-channel, but not both. It is double counting to exchange the same trajectories in both channels.
These conditions essentially uniquely determine Veneziano's amplitude and bosonic string theory. Adding Fermion trajectories requires Ramond style supersymmetry, and then the road to string theory is to reinterpret all these conditions in the string picture which emerges.
String theory incorporates and gives concrete form to all the boostrap ideas, so much so that anyone doing bootstrap today is doing string theory, especially since AdS/CFT showed why the bootstrap is relevant to gauge theories like QCD in the first place.
The highlight of Regge theory is the Reggeon calculus, a full diagrammatic formalism, due to Gribov, for calculating the exchange of pomerons in a perturbation framework. This approach inspired a 2d parton picture of QCD which is studied heavily by several people, notably, Gribov, Lipatov, Feynman (as part of his parton program), and more recently Rajeev. Nearly every problem here is open and interesting.
For an example of a reasearch field which (partly) emerged from this, one of the major motivations for taking PT quantum mechanics seriously was the strange non-Hermitian form of the Reggeon field theory Hamiltonian.
Pomerons and Reggeon Field theory
The main success of this picture is describing near-beam scattering, or diffractive scattering, at high energies. The idea here is that there is a Regge trajectory which is called the pomeron, which dominates high energy scattering, and which has no quantum numbers. This means that any particle will exchange the pomeron at high energies, so that p-pbar and p-p total cross sections will become equal.
This idea is spectacularly confirmed by mid 90's measurements of total p-p and p-pbar cross sections, and in a better political climate, this would have won some boostrap theorists a Nobel prize. Instead, it is never mentioned.
The pomeron in string theory becomes the closed string, which includes the graviton, which couples universally to stress energy. The relation between the closed string and the QCD pomeron is the subject of active research, associated with the names of Lipatov, Polchinski, Tan, and collaborators.
Regge scattering also predicts near beam scattering amplitudes from the sum of the appropriate trajectory function you can exchange. These predictions have been known to roughly work since the late 1960s.
Modern work
The S-matrix bootstrap has had something of a revival in the last few years, due to the fact that Feynman diagrams are more cumbersome for SUGRA than the S-matrix amplitudes, which obey remarkable relations. These partly come from the Kawai-Lewellan-Tye open-string closed-string relations, which relate the gauge-sector of a string theory to the gravity sector. These relations are pure S-matrix theory, they are derived by a weird analytic continuation of the string scattering integral, and they are a highlight of the 1980s.
People today are busy using unitarity and tree-level S-matrices to compute SUGRA amplitudes, with the goal of proving the all-but-certain finiteness of N=8 SUGRA. This work is reviving the interest in the bootstrap.
There is also the top-down and bottom up AdS/QCD approach, which attempts to fit the strong interactions by a string theory model, or a more heuristic semi-string approximation.
But the hard bootstrap work, of deriving Regge theory from QCD, is not even begun. The closest in the interpretation of a field theoretic BFKL pomeron in string theory by Brouwer, Polchinski, Strassler, Tan and the Lipatov group, which links the perturbative pomeron to the nonperturbative 1960s pomeron for the first time.
I apologize for the sketch, but this is a huge field which I have only read a fraction of the literature, and only done a handful of the more trivial calculations, and I believe it is a scandelously neglected world.
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