The key message of the article is that Bootstrap approach or S-matrix theory was fashionable approach to strong interaction at sixties before the emergence of standard model based on quantum field theory approach. QCD became the theory of strong interactions but the fact is that low energy hadron physics is still poorly understood.
After the fall-down of superstring models, the bootstrap approach has made a comeback and relies on the progress made in the understanding of scattering amplitudes of massless theories - in particular N=4 SUSY. These theories are unrealistic since most particles are massive and SUSY in the sense assumed has been excluded at LHC but twistor Grassmann approach and so called amplituhedron have led to the renaissance. Nima Arkani-Hamed is powerfully propagating this ideology. The bootstrap philosophy involves space-time symmetries Poincare invariance extended to conformal invariance in 4-D Minkowski space and in cosmological context restricted to special conformal invariance in de-Sitter space.
The article starts with an argument of Baumann. Particles with spin 1/2,2, 2 are highly unique in QFT context in the sense that higher spins do not allow QFT description. In super-string models the situation changes. What is argued that physics is inevitable. One can of of course ask "What about electroweak and color interactions?".
One can also ask what "inevitable" means. Every mathematical theory relies on axioms and those one cannot prove.
Sabine Hossenfelder talked about this in her blog . A more realistic notion would be inevitability forced by general mathematical conditions such as mathematical existence plus general ontological assumptions forced by empirical facts. For instance, the existence of space-time or of symmetries allowing a well-defined unique theory.
The spirit of bootstrap approach is technical. It is indeed nice to be able to calculate. It would be however much nicer is to start from problems and thought experiments and try to proceed to new principles. Calculations come when the time is ripe for them. General Relativity is wonderful example about this. The rise of superstring models followed by their fall-down should serve as a school example about what can happen when calculations replace thinking.
A. Renewed bootstrap program
The starting point idea is to look at cosmology from the point of view of particle physics where S-matrix is expressible in terms of correlation functions for fields. Now one would look for correlation functions for mass density. Non-vanishing correlation function would mean that the mass densities at different points do not reduce to constant density as in the standard model for cosmology.
Remark: What is somewhat ironic that there is flux of anomalies from cosmology and correlations between dynamics of galaxies are routinely observed in cosmological scales but theoreticians at the top of the academic hierarchy do not notice these findings at all! See for instance this.
There are a many assumptions. One might argue quite too many and not well-founded.
- Inflation is assumed although there is no real empirical support for it and it has severe conceptual problems. Inflation would guarantee that the correlations in short scales are scaled up to long length scales.
Remark: An alternative TGD based view would be that the long range correlations have much deeper content and related to quantum coherence of dark matter in arbitrarily long length sales.
- De-Sitter spaces serves as initial and final state of cosmology. Special conformal symmetries in 4-D sense are realized. This means that translations not symmetries anymore. Four-momentum is lost. Poincare symmetry must be assumed as a mere tangent space symmetry as in GRT. This could be seen as the basic explanation for why GRT cannot be quantized: conserved energy would generate unitary S-matrix but does not exist mathematically. This problem was the starting point of TGD.
Remark: De-Sitter space is not anti-de-Sitter space appearing AdS/CFT correspondence.
One could assume conformal symmetries and M4 but now one would have only massless particles but the fact is that also massive particles exist.
- Twistor Grassmann approach and amplituhedron are taken as starting point in the program of Nima Arkani-Hamed. The Starting point is applications of twistor Grassmann approach to massless theories in M4, in particular N=4 SUSY.
The first problem of standard twistor approach has been already mentioned. Only massless particles are allowed. The generalization of masslessness would be needed.
Remark: It seems that I am thinking negatively and just digging problems. My excuse is that becoming aware of a problem is the only manner to make progress.
- The idea is also to get rid of space-time and replace it with amplituhedron. This means however forgetting the basic assumptions. Twistors are a concept relying essentially on 4-D Minkowski space. They allow an alternative field theoretic description of massless fields in M4. One cannot say that M4 is just a scaffolding, which can be forgotten when the house has been built.
Second deep problem of he twistor Grassmann approach is that classical picture is missing. Amplitudes are not enough. One must perform also measurements and every measurement applies space-time picture: one does not measure just abstract momenta and spins of final states but their space-time correlates such as particle orbits, frequencies and wave vectors. The very existence of 4-momentum and spin require M4.
- One could weaken the idea about getting rid of space-time. Get rid of time only. This approach is of course in blatant conflict with the very ida of special relativity, which is also the starting point of also twistor approach.
Remark: Philosophically oriented physicist might wonder why one should get rid of time. Even worse, we have actually two times to get rid of: experienced time and geometric time. They are very different: the first one is irreversible, has no future and the moment "Now" is in preferred role. Geometric time is reversible, there is no distinction between past and future, and there is no preferred moment "Now". Why should we continue to identify them? The answer of an orthodox physicalistic theoretician is easy to guess: consciousness is a mere epiphenomenon and one can safely forget it when oen is trying to become conscious of theory of everything.
The article contains very misty argument suggesting how the time might emerge. One considers singularities of n-point function emerging as two or more points fuse. The result of fusion must be an n-1-point function, and this is rather strong constraint on n-point function. What has been found that near the fusion point the n-point function oscillates. Oscillation is a dynamical phenomenon involving time. Heureka: time has emerged! The elementary mistake is that one has a homotopy depending on some parameter analogous to time and one actually thinks that there is some-one performing this homotopy! A little bit of consciousness theory would have helped to avoid the error.
Remark: Holography suggests that S-matrix could allow construction in terms of 3-D n-point functions but also this requires the notion of space-time. In TGD framework this is indeed achieved in zero energy ontology (ZEO) solving the basic problem of quantum measurement theory and leading to a theory of consciousness by lifting the observer from an outsider to part of the system.
As a matter of fact, number theoretic vision about TGD leads to a cognitive representation as a purely number theoretic unique discretization of space-time and n-point functions have as arguments points of cognitive representations as points of 8-D imbedding space. The condition that the limit for co-inciding arguments gives lower n-point function is an important constraint.
- Super string theorists might also criticize the approach. 2-D conformal symmetry is an infinite-D symmetry and huge as compared to conformal symmetry of Minkowski space or de-Sitter space. It seems awkward to give up this symmetry also allowing massive particles as excitations of strings. Superstrings did not work but why not consider generalization of the 2-D conformal symmetry to 4-D one in some sense. The super-conformal symmetries find a huge extension to their 4-D analog in TGD framework. This is due to the fact that 3-D light-like surfaces allow extension of 2-D conformal symmetries so that light-like coordinate becomes a parameter of the 2-D conformal symmetry.
It is interesting to see that TGD could have been discovered already 4 decades ago and by making certain very general either mathematical or physical assumptions TGD is indeed inevitable. To see this consider the theoretical situation around 1978.
- Standard model and its symmetries had been discovered but not understood.
- Spontaneous compactication was invented already in Kaluza-Klein theories but was highly non-unique and much later led to landscape catastrophe and after that to swampland catastrophe. One would give up 4-D space-time except as approximation. This does not look nice.
- Super string model had been discovered and their huge 2-D conformal symmetries had been discovered. The possibility that particles are not point-like had been realized. The problem was that string world sheets are not 4-D and the higher-D space-time in which superstrings moved had to be 10-D. How to get 4-D space-time?
Why not replace strings with 3-surfaces in some higher-D imbedding space H? This was the innocent question, which I made around 1978. Space-time would be 4-D surface representing orbit of 3-D particle. One would obtain 4-D space-time and would get rid of spontaneous compactification. But in what space H space-time surfaces would live?
One wants Poincare symmetries so that one must have H= M4× S, S some compact space. This would solve the energy problem of GRT: this was actually second starting point of TGD. It is ironic that people still have failed to realize how deep this problem is. Space-time as surface would allow dynamical space-time since Poincare symmetry would be lifted to H.
Gravitational field would be obtained from the induced metric. S has some symmetries and it has spinor connection. Could one geometrize gauge fields in terms of induced spinor connection. Could spinors of H have standard model quantum numbers.
- What could be the choice of S? Could it have standard model symmetries and allow to realized standard model gauge fields as induced gauge fields?
Hawking et al had already discovered CP2. They had not realized that it has standard model symmetries: I had to discover this. Could one have S=CP2? Theory would be unique.
- But what about inevitability - at least in some sense? Is there something special in M4 and CP2? Hitchin had already discovered that S4 and CP2 are the only compact 4-D spaces having twistor space with Kähler structure. M4 has also twisor space with Kähler structure. See "Kählerian twistor spaces" of Hitchin. I learned about this article much later when I realized that TGD could allow twistor lift.
- Could one find some use for the existence of the Kähler structure in the twistor spaces of M4 and CP2? The existence of twistor space for general space-time surface is actually a problem of GRT. Could one imagine that one replaces space-time surfaces with 6-D surfaces in the 12-D product of twistor spaces of M4 and CP2 as 12-D imbedding space?
These 6-surfaces should have induced twistor structure as S2 bundle over space-time surface. This would require dimensional reduction. The action would be 6-D Kähler action by conformal invariance and exist only for M4× CP2! This action reduces to a sum of 4-D Kähler action and volume term in dimensional reduction required by twistor bundle property of 6-surfaces and volume term has interpretation as dynamically generated cosmological constant.
The theory would would be unique just by its mathematical existence!
- This would also solve the basic problem of the 4-D twistor approach: 8-D analog of masslessness allows 4-D massiveness.
For a summary of earlier postings see Latest progress in TGD.
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