Dijkgraaf on quantum physics' gifts to mathematics

LHC: Collisions will begin mid June, despite the ULO in the tube (Unidentified Lying Object). The speed is disappointing to me.

One year ago, the director of the Institute for Advanced Study in Princeton Robbert Dijkgraaf gave a public lecture at the Perimeter Institute.



The 1-hour video was posted one month ago so it's about time for the TRF readers to have the opportunity to watch it. The words written below are meant to be moderately disorganized notes – which will not be proofread – rather than a full-fledged TRF blog post nominated for the Pulitzer Prize.




The title was "The Unreasonable Effectiveness of Quantum Physics in Modern Mathematics" and Robbert covered lots of cute mathematical legends, numerology, and things that many people find exciting, as well some more technical ways in which string theory allows you to answer very difficult problems in pure mathematics.




All communist viewers will also be happy to learn what is the right, communist way to move, around 39:50.

At the beginning, Dijkgraaf is introduced and some of his jobs and prizes are mentioned. He is also the chieftain of the international InterAcademy Council that has, among other things, recommended sexual pervert Rajendra Pachauri to establish some rules in the IPCC that would introduce some reliability and impartialness to the work of the institution. The recommendations were ignored.

Incidentally, Pachauri wanted to escape the prison a few days ago but they didn't allow him to attend the water summit.

Back to mathematics and physics. He describes them as his two lovers – who nevertheless like each other. A picture shows a military failure of applied mathematics. Dijkgraaf immediately admits that his title is a parody to Wigner's "opposite" 1960 quote about the unreasonable effectiveness. Wigner began by a story in which a shoe shop owner is told that the normal distribution may tell him how many shoes he has. He is stunned by the \(\pi\) entering the denominator. What do circles have to do with shoes? And the entrepreneur's question is a deep one.

Dijkgraaf cites Galileo and Feynman as scientists who were impressed by this effectiveness hiding in mathematics. Dijkgraaf mentions that Feynman otherwise "didn't love mathematics too much". I am not sure it is fair. He mostly didn't like the formal framework in which most mathematicians around us operate. But that's something else than not being a fan of mathematics, or not being good at it. He was very good at many aspects of mathematics – just think about the evaluation of the path integrals and loop diagrams and all the tricks you need to do so effectively.

Well, Feynman has said that if all mathematics abruptly disappeared, physics would be returned back exactly by one week (the week in which God wrote the world, a mathematician unfamiliar with cosmology and evolution added). This quote only says that Feynman considered the qualitative ideas primary and mathematics to be the necessary toolkit that gives them muscles. "One week" is surely exaggerated but I do agree that much of the mathematics that is actually helpful in physics may be reconstructed rather quickly if all the right physical ideas are preserved.

Dijkgraaf shows an Android tablet from the Babylonian Empire with the number 343768681 on it. Four thousand years ago, a Babylonian engineered figured out that this number was equal to\[

18541^2 = 13500^2 + 12709^2

\] It is not clear to me whether the Babylonians actually knew the Pythagorean theorem. One could discuss whether the importance of the Pythagorean theorem should be "expected" given the existence of integer solutions to the quadratic "Fermat" equations above. I am not sure.

In Greece, they loved the Platonic polyhedra and invented (bogus) isomorphisms of these solids to the elements, planets, and other things. The fifth element was the quitessence (the heaven is made of it) – which was mapped to the dodecahedron. Dutch politicians complained about a statue of the dodecahedron: Is it appropriate to hang a soccer ball there? A few years ago, the dodecahedral topology of the Universe was considered – the possibility was killed by further research.

Kepler himself destroyed the isomorphism between planets and the Platonic polyhedra by seeing that the orbits were imperfect – elliptical. I am not sure whether this is the right argument, either ;-), but the newer conclusion seems more right, of course. Kepler recognized that he "brought a cart of manure to science".

Dijkgraaf mysteriously celebrates the = sign in the equations. The principle behind equations is called Clinton's principle because "it depends what the meaning of 'is' is". Funny but if this joke has some serious underlying message, I have missed it.

He also shows a triangle above a circle for a split second and claims that people who remember the arrangement think with the right hemisphere (they prefer pictures) and those who prefer to agree with the wrong arrangement you tell them verbally think with the left one (they prefer texts), or something like that. It's a fascinating correlation that may be right but I am still not sure whether I believe it's so simple.

This relationship between "text" and "images" is mentioned because of the interplay linking "algebra" and "geometry" or "microphysics" and "macrophysics". These are two "senses of beauty" that co-exist in mathematics. Do you find the macroworld or the microworld more beautiful? Algebra or geometry?

Dyson said in 1972 that the marriage of mathematics and physics ended in divorce but a black box was just opened and people – like the speaker – could suddenly write down the simple Lagrangian of the "Standaardmodel". If you ask why it's spelled "Standaardmodel", it's bbecaause thespeaaker is "Robbert Dijkgraaf". People may write the Lagrangian; or 1 page of the Lagrangian of components; or print a T-shirt with boxes indicating the elementary particles, "now with Higgs". (I wouldn't quite agree that the latter contains "everything" one needs without lots of extra clarifications.)

He's introducing extra dimensions, especially the 4th one in the spacetime. A 4D cube is rotating. A biologist looked at the rotating GIF for 2 years but he still couldn't see the fourth dimension. A movie is composed of images. Stack them on a pile and you get a spacetime. Now, why are all electrons exactly identical? A factory produces perfect copies without a little scratch.

Dijkgraaf promotes Wheeler's idea of the electron-positron trajectory going back and forth in the spacetime. That's cute but given the ability of the electron to be created (e.g. from a quark, an antiquark, and a neutrino), it can't be taken too seriously. The "single path of one electron through spacetime" is a possible explanation why electrons are identical. But quantum field theory or string theory simply have another explanation. The electrons are created from quantum fields that obey the same laws everywhere, so they are identical, much like the factory picture suggested.

Wheeler made a call to Feynman, "there is just one electron in the Universe", on Saturday night. He says that "these zigzag trajectories" of one electron are now called Feynman diagrams. OK, I disagree that it's "quite the same". Feynman diagrams have vertices and the whole point of Wheeler's "efficient" idea was to get rid of all vertices and repeated propagators! Wheeler's telephone call could have been conceivably helpful but I don't really think that Wheeler was anywhere close to be considered a co-father of the Feynman diagrams because of that idea. Yes, in Feynman diagrams, a positron is an electron going backwards in time (and the Feynman-Wheeler work on retarded and advanced propagators was needed to get the full picture of all components in Feynman diagrams), but this is really just a different way of describing Dirac's picture of a positron as a hole in the sea of electrons. The "hole" is what makes the positron present when the electron is not, and vice versa, which is why the arrow of time is reverted.

They predicted that instead of formulae, articles would be full of Feynman diagrams. Right. The usual story about "why are there Feynman diagrams on your van? They are actually meaningful in physics, they are called Feynman diagrams." – "I know, I am Mrs Feynman."

Virtual particles. Everything can happen as long as it happens quickly enough before it's detected. He thinks that it's what the Dutch society is all about – but that's only because he doesn't know the Czechs. Vacuum energy one-loop graphs. Closed loops of Wheeler.

Knots are maps of \(S^1\) to \(\RR^3\). Mathematicians have tried to classify them for centuries. Only one "letter" in the database of knots was found around 1920. But Chern-Simons theory clarified all of them. The knots are the "Feynman diagrams". Fields medal winners have to be below 40, wait for the next life.

Enumerative geometry. Counting of curves in the quintic (a compact 6D space). Rational curves, 2875. Conics... All these numbers may be nicely counted by string theory's mirror symmetry. This was viewed as a method to solve "hard recreational problems" but the status of this enumerative geometry has changed thanks to the mirror symmetry. When showing the curves in the quintic, Dijkgraaf suddenly shows the fifth-order equations – much more mathematics than elsewhere in the talk. These equations are labeled Gromov-Witten theory, and Gromov and Witten are just the tip of the Seiberg.

The story how the physicists outsmarted mathematicians in counting those 317 million cubic curves in the quintic. Mathematicians needed a massive collaboration with supercomputers and there was a mistake that physicists advised them to correct. But physicists already knew the numbers up to the 10th order and higher. Mathematicians have finally realized that their whole lives have been meaningless – or, to put it optimistically, they were worth one week of a string physicist's life.

Dijkgraaf explains how the calculation proceeds in string theory. String theory uses muscular Feynman diagrams on steroids. The quintic is used for the hidden dimensions. Nice animations of the Calabi-Yaus. How do the strings move on that spacetime? Like everything, the strings are summing over the histories. The counting of the curves comes from coefficients in a formula resulting from the path integral (a quantum mechanical amplitude).

Classification of Calabi-Yaus. A plot in the \(h^{1,1}\)-\(h^{1,2}\) plane that made the mirror symmetry obvious, too.

Once shown the "spirit of the truth" that results from string theory, mathematicians were able to make the proof of the formulae that make the enumeration simple. These proofs still include a nearly infinitesimally smart portion of the wisdom of string theory, I would say: it is just a recreational projection of what really follows from the string picture.

Symmetry. Greeks loved it but they failed to invent group theory which we use to study symmetries. An explanation of \(SU(3)\) transformations in QCD and why the "color" is a good label. Communism governs the global \(SU(3)\) transformations of the quarks. Much of the world is free, however, and quarks enjoy the capitalist gauge symmetry. Each of them may rotate differently.

Flying carpets called D-branes. A stack of D-branes is like a set of plastic transparencies. \(U(N)\) gauge theories live on them. Those are important to understand black holes. The degrees of freedom living on the horizon may be represented by open strings. He thinks that this quantum gravity research hasn't yet produced formulae that may stun mathematicians.

I like the mathematicians' chauvinist quotes. Reality is the first approximation to mathematics. The difference between physics and mathematics is that physics describes the laws that God ordered the Universe to obey while mathematics describes the laws that even God has to obey.

So mathematicians aren't bound by the reality but again and again, putting mathematical questions in a physical context turned out to be useful. Every context seems to enrich a mathematical concept. He mentions people who think that it's better and more intuitive to "define" derivatives at school using the speed or slopes than by the \(\epsilon\)-\(\delta\) gymnastics.

Dijkgraaf talks about the "complementarity" between the heuristic approach of physics and the rigorous attitude of mathematics and emphasizes that the mathematical proofs didn't convert the whole physics reasoning to a rigorous language. It's a complementarity, like the brain hemispheres, and so on. Well, I don't think so. I think that this reaction of the mathematicians only shows that they were satisfied with returning to their recreational problems and just get "inspired" by the wisdom they saw in string theory. But if they could convert the framework of physics "entirely" to the rigorous language which is almost certainly possible – after all, we deal with CFTs and they admit a rather perfect axiomatization – they would have much more powerful tools. So it's really mathematicians' laziness and low ambitions, not some "symmetric complementarity", that decided about the mathematicians' reaction.

Mathematics is a temptation for a physicist but he must often resist.

Plato's cave: we're prisoners who only watch the shadows on the cave. But the true and perfect mathematical figures are outside the cave. No shadow or picture of the polyhedron actually has the perfect symmetry that it has in our mind. Dijkgraaf describes the holographic principle as the upside down Plato's cave: the shadow is the real thing.

Mathematics and physics fell in love again. They should marry. Applause at 52:10.

The first question: are the interactions between mathematics and physics recent or old? Dijkgraaf describes the ups and downs. The interactions were intense in the 17th century etc. But it becomes more remarkable with quantum mechanics because in the previous centuries, we haven't been trained to think quantum mechanically and the mathematically rich description still works. Nature just love equations (not just beetles, as Darwin thought).

Second question. A girl asks whether he can prove the graviton to exist thanks to the strange particles in string theory. I don't know how to understand the question, and neither did Robbert. He began to talk about the experimental discovery of the graviton. I think that she wanted a theoretical answer and he could have given one. But he did at least say that the existence of gravitons is analogous to that of photons and it isn't a difficult mathematical problem.

The third question came from the host. What sparked your interest? At age of 11, he got a book on DNA. So Robbert cried in the kitchen that there was a new world order conspiracy that had prevented him from knowing about DNA. When he was 15-16, he decided that there could have been a conspiracy trying to prevent him from knowing atoms, strings, and Calabi-Yau manifolds, so he began to do his own research, too. ;-) Encyclopedias and teachers were helpful.