Three weeks ago, on December 15th, 2010, Friends of the IAS Princeton invited the undisputed Lord of the Strings, Edward Witten, to give a talk about knots in general and about their relationship to the quantum theory in particular.

It is the discipline of mathematics in which the power of the physical intuition and the power of Witten's brain have been demonstrated most comprehensibly from the mathematicians' viewpoint - and the topic that has largely earned the Fields medal for Witten himself.

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On December 24th, 2010, your humble correspondent spent a few hours by untangling a knot on a closed circuit of Christmas tree light bulbs. Fortunately, I didn't know about this talk by Witten. If I had seen the talk before the Christmas, I would surely start to compute the knot polynomial.
And after dozens of hours, the knot polynomial would turn out to be equal to one, as I was able to experimentally demonstrate within a few hours. This story is an example of a context in which I am a better experimenter than a theorist. ;-)
We had recent discussions with Gordon (and maybe others) whether Penrose's popular texts on twistors may be read by undergraduates. Well, in this case, Edward Witten confirms my viewpoint that this stuff - like the Jones polynomials - could be taught to the high school students without losing much. There is a mechanical procedure to evaluate J_K and if it is distinct for two knots, they're inequivalent. In particular, J_K different from one means that the know cannot be untangled.
For adult people, however, a quantum definition could be simpler than the mechanical stuff that the high school kids could learn: the Jones polynomial is the expectation value of a Wilson loop evaluated in the fundamental representation for a SU(2) Chern-Simons theory defined on an S^3 (R^3 in which the single point at infinity can be surpassed).
How do the kids calculate J_K? For an unknot, a circle, J_K equals one. Then, pick your favorite numbers which can be anything but must be 2,3,5 for you to follow what Witten is going to say :-). If there is a relationship between three knots K,K',K'', then 2J_K+3J_K'+5J_K'' vanishes. This powerful identity is enough to calculate the J's.
How do the three related knots K,K',K'' differ? Well, they differ behind a circular shadow that has 4 external lines going from it. The 4 external lines can be connected into pairs in 3 simple ways, corresponding to K,K',K'': X-like connection with the Northeast-Southwest line on the top; two vertical connecting lines; X-like connection with the Northeast-Southwest line on the bottom. As I have mentioned, we postulate:

2JIt's rather easy to show that this rule is enough to calculate J; it's harder to prove that there can't be any contradiction - the worry is that one may calculate J in many ways which could contradict each other. Jones proved in the 1980s that there's no contradiction. Why it's so conceptually? Well, because you may calculate J_K from physics of Chern-Simons theory - which makes it clear that it doesn't depend on how you draw the knot. The polynomial is invariant under deformations. Khovanov homology is similar to the polynomial but it is tougher, so you need a very smart high school student. Witten eventually got to his more recent research, namely a clarification of a paper by Gukov, Schwarz, and Vafa. His thinking brought him to his recent paper about geometrizing of quantum mechanics (TRF). Again, the most important and natural insights about these objects can be obtained in quantum field theory and string theory. In the question period, the first question was from a friend who hates maths and who asked what it is good for. (The individual may still be a friend of the IAS buildings, one who plans to transform them into brothels.) Witten said that quantum computers could find the knot polynomials to fight decoherence - in quantum Hall systems. Topological insulators are another example close even to the Khovanov homology. Not sure whether it's applied enough for the "friend". :-) The second question was about the relevance of the polynomials for tangles in biology; Witten hasn't heard of it. The third question was where have the monopoles gone; Witten said that Alan Guth overdid his job and in trying to protect the mankind from too many monopoles from the inflation, he threw the baby out with the bath water and diluted them exponentially so not even Sheldon Cooper could discover them. ;-) However, Witten has only worked on theories that are inconsistent with the absence of monopoles, as he puts it. The fourth question is how Witten chooses his problems; it's the hardest job, Witten says. He looks in the sweet spot - not too easy, not too hard. Semi-jokingly, he says that it turns out that he does nothing most of the time as a result. The fifth question led Witten to explain that the knots are his passion. Another question made him mention some work by Gaiotto and others that is seemingly so remote from the knots that he couldn't say anything useful. The seventh question led Witten to describe a paper by Albert Schwartz that only has a few citations and Witten claims to be the only person aside from the author who paid attention to the work. That's quite wrong! ;-) I did, too, and in fact, I checked that Nima had a copy of one of the papers a few days ago (I was rechecking that I made Nima pay attention as well). In 1998, I spent quite some time with fun discussions with Albert Schwartz at Rutgers. One more question was about some invariants that are not appropriate for that audience - and for TRF, either. ;-) So Witten chose to be silent and so will I. One more question made Witten praise string theory that has made many insights possible. String theory is on the right track but more accurate answers may be obtained by younger physicists only. Two more questions shared a unified simple answer: "All theories have their critics." To get an idea about the intelligence of these particular critics, the newest question that the critics of string theory discuss on their most well-known blog is whether Edward Witten is an extraterrestrial alien sent from Mars (and what about your humble correspondent?). The critic-in-chief is somewhat skeptical but he believes that he, the critic, was sent by the people from Venus. Via Clifford Johnson_{K}+ 3J_{K'}+ 5J_{K''}= 0.

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