Physicists Uncover Geometric ‘Theory Space’Natalie Wolchover wrote a story about bootstrap which contains lots of the right information – old basics as well as some cutting-edge research in recent years – and it's sort of incredible that there are no serious bugs in her text, as far as I can see.

*Her story also boasts this periodic animated GIF with 16 frames which is cool by itself.*

Werner Heisenberg really started the story in 1943 when he introduced the S-matrix – the evolution operator from the "minus infinite time" to the "plus infinite time" (within the framework of quantum field theory that Pauli, Jordan, himself, and others began to construct in the late 1920s and early 1930s) and conjectured that the right form of the S-matrix could follow from consistency conditions and nothing else. When lots of messy hadrons began to be discovered in the 1960s (and perhaps already in the 1950s – some of his quotes were "backdated" so it's not easy to give time stamps to every piece of this history), Heisenberg also conjectured that the consistency would dictate the properties of all particles and all of them would be some compromise between elementary and composite particles. By this belief, Heisenberg stood against a major industry in these two decades that was dedicated to the identification which particles were elementary and which were composite.

What he said about uniqueness couldn't be quite true because we know numerous theories – and their inequivalent S-matrices – which seem perfectly consistent so some conditions have to be added. But the idea was out.

Wolchover's story only begins in the 1960s when the physicist whom she called "charismatic" and the "most handsome physicist" – who made her doubt about her orientation – revived the idea and declared the new epoch of "nuclear democracy". Geoffrey Chew is 92-year-old now but he loves his skateboard and one of the amazing things he can do is to "pull himself up by his own bootstraps" while on the skateboard. ;-)

Half a century ago, he conjectured that particles could do the same.

Wolchover's article continues by telling you that Chew has impressed many people by calculating the mass of the rho meson using the original bootstrap methodology. However, it turned out to be an isolated achievement and nothing else could have been done with the method. Instead, QCD was born to explain the strong nuclear force. And QCD is as anti-bootstrap as you can get. It defines elementary particles – quarks and gluons – in the most conventional constructive or reductionist way. Quark and gluon fields are elementary, hadrons are composites of quarks and gluons. Period.

There's no room for self-referring vague half-elementary, half-composite particles. Things are made out of the elementary QCD fields (and their quanta i.e. the particles). The theory is created constructively and its consistency may be shown in the same way that was applied to simpler but analogous constructive theories such as QED before.

However, Alexander Poljakov started to realize in the 1970s that bootstrap may be particularly powerful to understand the universality of models that are invariant under the scaling of distances, such as the low-energy (conformal) limit of the two-dimensional Ising model (the statistical physics of a static magnetic plate). The physical behavior in the environments that are scale-invariant seems universal – following the same scaling laws with the same (often fractional) exponents, regardless of the "material" in which the behavior is implemented.

These approaches culminated in the 1983 BPZ paper written by the holy trinity of three Russian Alexanders: Belavin, Poljakov, and Zamolodčikov (the latter, while teaching a course on conformal field theories at Rutgers, told me that he liked this Western Slavic spelling of his name so I hope it's still OK). They derived lots of properties – in some sense all properties, especially the correlation functions – in all two-dimensional quantum field theories analogous to the Ising model. The bootstrap seems to completely work for two-dimensional CFTs (conformal field theories). The theories in this set are completely determined and classified by the consistency conditions and there are many discrete theories without any continuous deformations – similar to the Ising model (which is one of the "minimal models") – where all the correlation functions and their power laws may be calculated from the first principles.

This power of bootstrap may be "explained" by the infinite-dimensional conformal symmetry and that's also a reason why it was generally believed that none of these methods are powerful in dimensions \(D\geq 3\). However, a 2008 paper by Rattazzi, Ryčkov, and two more folks restarted an enterprise in these higher dimensions where one can at least derive some inequalities (for the dimensions of operators etc.) and this paper has led to hundreds of followups.

The theories where the bootstrap is powerful are still conformal – scale invariance is a necessary condition (and in some "seemingly complete" classes of theories in \(D\geq 3\), also a sufficient condition). But non-conformal theories may be described as some "deformations" of some conformal limits of theirs. The set of conformal field theories represents a useful collection of railway stations in the set of all quantum field theories.

Wolchover also says that people studied the inequalities in various parameter spaces of would-be theories that follow from the bootstrap methods. And the most interesting theories – or the theories that we actually know to exist – live not only on the "boundaries" of the consistent regions – it's cool to take the laws to the limit – but actually on the intersections of several boundaries, i.e. on the "kinks". For example, the three-dimensional Ising model lives on the vertex of a polyhedron in the picture above.

Because of the AdS/CFT correspondence, the bootstrap insights about the CFTs also have consequences for quantum gravitational theories in the AdS space. Theories of quantum gravity may be reasonably believed to worship the bootstrap paradigm in the full generality. In particular, I would add that it is rather clear that quantum gravity never allows you to strictly separate objects to elementary and composite ones. All "elementary" particles in quantum gravity may be interpreted as very tiny black holes – that are so tiny that the quantization conditions on their properties become hugely constraining. As you study increasingly heavy particle species, the density of their types becomes exponential and they basically fill the continuous sets of black holes solutions parameterized by several continuous parameters. Larger black holes can't be said to be "quite composite" and the tiny ones can't be said to be "quite elementary" because the two groups only differ quantitatively.

String theory has already taught us many additional reasons to be sure that the separation of particles into elementary and composite ones can't be done in the most general situation. For example, perturbative string theory considers strings and their excitations to be "elementary objects". However, S-duality – or going to the strong coupling – may exchange these fundamental strings with other objects such as D-strings, and the identification which particles are elementary and which particles are composite may get mixed up. Quite generally, the lightest and weakly interacting objects tend to be those that we liked to call "elementary". But if you study the theory at the generic coupling constant \(g_s \sim O(1)\), all objects are "comparably heavy" and "comparably strongly interacting", so there is some "brane democracy" in which objects of all types are equally elementary or equally composite and you just can't quite discriminate against anybody even if you wanted it badly.

There exist very particular exact structures linked to the bootstrap philosophy such as the "bootstrap equation". But the broader philosophy is basically the same as e.g. Cumrun Vafa's "Swampland Program". String theory – or, equivalently, the consistent theory of quantum gravity – forces you to respect many rules that aren't necessarily obvious from the beginning. And these rules are so constraining that it's possible to learn much – or everything – about the theories.

As I predicted decades ago, I still predict that the knowledge of the constraints that are imposed by consistency will be getting better. People will understand the "bootstrap equation", various inequality, as well as typical swampland-like inequalities such as the "weak gravity conjecture". And they will hopefully realize that many of those conditions are really conditions of the same kind, they may be deduced more logically and more systematically, and these insights may be used to build the consistent theory of quantum gravity in a non-perturbative, non-reductionist, bootstrap-or-swampland-like way.

The pure constructive or "reductionist" description of physics will be viewed as an artifact of expansions (e.g. perturbative expansions) around a point in the parameter space, around a certain environment but the true "theory of everything" is bound to be more general, more "spiritual", and more "impartial" than the specification of some fundamental degrees of freedom and their interactions.

"Bootstrap" appears in 35 TRF blog posts.

I recommend you Simmons-Duffin's TASI 2015 lectures on conformal bootstrap if you want to do these things seriously. Or watch hours of video with the same lectures.

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