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Edward Witten: connecting quantum mechanics and geometry

The postulates of quantum mechanics seem to be truly immutable and fundamental for the working of this Universe - and multiverse, if the latter exists.

However, it's also pretty important that events take place in time. Special relativity completes time into spacetime and general relativity allows this spacetime to get curved.

String theory teaches us many details about the unification and ways how various mathematical possibilities are realized in physics - defined in the broader sense.

Aside from unification, some of the physicists have dreamed about a tighter form of union between quantum mechanics and geometry - some kind of duality that shows that one of the pillars implies the other (at least when it is generalized in an appropriate way).




For example, millions of readers of The Elegant Universe by Brian Greene remember the following paragraph from the last chapter that is dedicated to this very question:

Currently, no one knows how to do this. But many string theorists foresee a reformulation of how quantum principles are incorporated into our theoretical description of the universe as the next major upheaval in our understanding.

For example, as Cumrun Vafa has said, "I think that a reformulation of quantum mechanics which will resolve many of its puzzles is just around the corner. I think many share the view that the recently uncovered dualities point toward a new, more geometrical framework for quantum mechanics, in which space, time, and quantum properties will be inseparably joined together."

And according to Edward Witten, "I believe the logical status of quantum mechanics is going to change in a manner that is similar to the way that the logical status of gravity changed when Einstein discovered the equivalence principle. This process is far from complete with quantum mechanics, but I think that people will one day look back on our epoch as the period when it began."

[From 1998 interviews/chats of Brian Greene with Vafa and Witten.]
Those of you who have talked about some deeper issues with Cumrun Vafa and/or Edward Witten know that they feel that these visions were or are important for the future of physics but they would often be modest about this esoteric knowledge.

Edward Witten has dedicated a part of his luxurious time to these fantasies, too. Although both quotes above are relevant, it actually turns out that Vafa's quote more accurately describes Witten's new preprint that has just arrived:
A new look at the path integral of quantum mechanics (PDF)
The preprint depends on topological string theory whose research has had two epicenters - the vicinity of Cumrun Vafa's office at Harvard and the neighborhood of Edward Witten's office at the IAS in Princeton.

The 78-page preprint also depends on complex numbers that are fundamental in physics as well as some new analytical continuation that generalizes the Wick rotation, the most frequently used type of analytic continuation in physics.

Witten's derivations are presented - and probably have to be presented - in Feynman's path integral formalism. It's the approach to quantum mechanics that arguably preserves the spacetime geometry in as manifest a form as it can - which is helpful for Witten's purposes. It is no coincidence that Feynnman's pioneering paper about path integrals in non-relativistic quantum mechanics was titled Space-time approach to non-relativistic quantum mechanics: note that Adobe's PDF format hasn't changed much since 1948 (just kidding).

At any rate, Witten simply takes a path integral and complexifies its phase space coordinates p and q. Visually, the dimension of the phase space gets doubled. In such an extended space, there are many ways how to deform the contours. In the past, people have deformed contours in many ways but they just converted one quantum way to look at a quantum theory to another, equally quantum way to look at a quantum theory (recall Wick's rotation, for example).

However, Witten's special new contours do more than that: they pretty much "explain" where quantum mechanics comes from. Just to be sure, it's not the kind of "emergence" that many of you hope - that would conclude that quantum mechanics doesn't exist and everything is classical: indeed, Witten is no crackpot who would believe that quantum mechanics is fundamentally wrong. It is something that makes quantum mechanics even more abstract and dependent upon advanced mathematics - but you have been warned that this is the only reasonable type of progress that could be expected.

By this procedure, the path integral of a non-relativistic quantum mechanical theory is converted to a two-dimensional quantum field theory. This model turns out to be a topologically twisted A-model and the main actors in this model that do the interesting maths are actually not point-like particles: the main players are exotic types of A-branes known as coisotropic branes. Recall that A-branes are middle-dimensional branes - generalizing "real" contours in the complex plane to many dimensions. The naive type of A-branes are those on the Lagrange cycles (recall 3D cycles in 6D Calabi-Yau manifolds) but here a special type is used, instead. The Floer theory of complexified spaces becomes helpful here.

Interestingly enough, Witten can do a "similar" thing not just for a 1-dimensional field theory i.e. non-relativistic quantum mechanics - which produced the 2-dimensional field theory above. He can also perform an "analogous" exercise for 3-dimensional theories, namely his favorite Chern-Simons theories. Their path integral may now be extracted from N=4 supersymmetric gauge theory with the same gauge group (and with a boundary) in 4 dimensions.

Unfortunately, there is not enough room in this footnote to complete the article. It will be continued later. ;-)



P.S.: When you open the PDF file, be ready for a distraction on the very first line. Several years ago, I argued it was an extremely bad idea to change the identifiers of preprints on the arXiv, especially the removal of the "discipline" such as hep-th from the identifier. I argued that people who matter wouldn't get used to the new system which is inferior, anyway.

Needless to say, the first line of Witten's paper says hep-th/yymm.nnnn. The author hopes that this sequence of characters gets translated to the right numerical identifier. Well, it didn't. ;-) All the templates how to write similar things have changed and frankly speaking, I don't even remember what exactly Witten should have written instead of hep-th/yymm.nnnn.



Andy Strominger and a black hole talk

Andy Strominger gave a very amusing talk about the harmonic oscillators of the 21st century (watch, 67 minutes) which are the black holes - both the simplest and the most complex objects we know in the Universe.

When Andy was a student, his advisor Roman Jackiw would insist that Andy and others had to convert everything to a harmonic oscillator. And you know, field theory showed that this thing can be done to all elementary particles (and also to the Hydrogen atom and other things!). Andy is now what Roman used to be, and he insists that the students transform everything to a black hole. :-)

Strominger starts with the history - Schwarzschild in the World War I - and how it took 50 years for the experts to agree about the right interpretation of his solution. Various event took place, Penrose understood the causal structure, and Wheeler gave it the name "black hole" in 1967. In the 1970s, the analogies of thermodynamics and black holes began to emerge. The thermodynamic-statistical relationship is the relationship between simple and complex.

He eventually gets to the breakthroughs of himself and Cumrun Vafa in the mid 1990s, mentioning the role of string theory as well as the relative independence of the calculations on string theory. Any sufficiently consistent picture of quantum gravity has to achieve the same result. Of course, this can't include LQG or any picture that is obviously far from being consistent, but it is an OK statement for various AdS/CFT-like descriptions of situations that are not "quite stringy", or that are at least not "manifestly stringy".

Some comments about observed Kerr black holes that are near extremal. It's been discuss on TRF, too. Andy also shows the NASA animation of that hole that I uploaded to YouTube - and for mostly irrational reasons, it became the most viewed video I uploaded with 300,000+ visits. (The anthems and Smetana and Mládek songs are not too far, however.)

Strominger then shows how the 2D conformal symmetry may be identified even in the geometry of black holes that don't "obviously" look like black holes with this symmetry. Discussed of various AdS/CFT applications - nuclear matter, superconductors, graphene, non-Fermi liquids etc. - are said to reduce to black holes, giving us a new superior method to study these physical systems.

Navier-Stokes equations of hydrodynamics (with turbulence) are a largely unsolved mathematical problem that may be reduced to a black hole exercise, too.

According to Andy, string theory acts as an "engine" running behind all of this - while it doesn't force the condensed matter and other physicists to believe that it's the right theory of particle physics. However, Andy does spend some time with the bogus claims about (or against) string theory and reviews some basic facts and history here. Strings became a theory of quantum gravity in the 1970s and remained the only known consistent solution as of today.

The other, non-gravitational forces were added in the 1980s. Of course, Andy has played an important role in these developments, too. Meanwhile, string theory has grown into a large and cohesive framework that contains pretty much all the good ideas for beyond-the-Standard-Model issues in theoretical physics. It should no longer be called just string theory but it is: nevertheless, all important physical phenomena are different facets of the same theoretical structure.

Andy makes fun of different attitudes to string theory: it is the theory that's solved everything, and we only need to go to Stockholm. However, you run into the non-uniqueness etc., and then you end up watching TV at home (which is sometimes not bad, by the way haha).

An easier way to watch TV is to say that it is a theory of nothing etc. Andy sensibly chooses the middle route and says string theory is something. ;-) Something has been solved, something is waiting, and you don't have to watch TV. :-)

Strominger prepared grades for string theory, with four As, two Bs, no Cs, three Ds, and two Fs. The most important is the A from "not being ruled out as a theory of the real world" - because every other theory has been. Not a surprising fact given the world's being such a complicated and surprising place.

String theory gets an F from "unambiguous testable predictions". Many people in the audience began to stupidly laugh when someone asks how it can be ruled out if it has no "unambiguous predictions". [Laughter.] Of course, the answer is: "Easy. You can rule a theory out if it made wrong predictions." Except that string theory hasn't.

The absence of chiral fermions is quoted as a lethal blow for (extended) supergravity. These people just don't understand that many tests have already occurred. If this is a Harvard colloquium, and the background at the end makes it very clear that it is one, it is extremely bad, indeed. Even ignoring politics, I would be afraid to walk there just because of this moronic laughter that looked like an endorsement of that dumb question. People could have falsified string theory but they have not. There can be another prediction in the future that will rule out string theory - or de facto prove it.

If you care, string theory gets an:

A for not being ruled out,
F for unambiguous testable predictions,
D for an LHC signal,
B for solving black hole puzzles,
A for inspiration to maths,
B for inspiration to the rest of physics,
A for unification (surely "a" solution to a previous superhard problem),
D for uniqueness,
F for solutions to the cosmological constant problem,
D for understanding of the Big Bang or the birth of the Cosmos,
A for solving Pauli's renormalizability problem of GR.

I think Andy is right that people would agree with the grades; they would disagree with whether it is a passing or failing report card. However, as Strominger emphasizes, string theory is the only student in the class. ;-) If you flunk her, you have to shut the school down.

To summarize, the road to understand the Universe inevitably has many twists and turns. There are interesting places to go.

In the question period, Melissa Franklin says that she actually feels warm for string theory. A question leads Strominger to lead to comments about getting energy out of rotating black holes. However, we don't expect to learn about microscopic physics from macroscopic physics.

Another question is an incoherent rant on the Titanic. I didn't understand what the question was but it was clearly some hostile rant. Andy points out that his talk was about traditional physics - so even when talking about the stringy physics, the actual insights (about the conformal symmetry etc.) did follow from the same diffeomorphisms we learned from Einstein in 1915.

Another question was too silent but it was something about some collisions. Again, Andy wanted to understand the macroscopic physics. One more question was more about experimental astrophysics, and an astrophysicist gave an answer instead of Andy. Another question led Andy to say that the accretion disk may approach the horizon for extreme black holes.

Hat tip: John C.

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reader Brian G Valentine said...

Few today are familiar with the work of Ottón Martin Nikodym (of the famed Radon-Nikodym theorem) in his construct of non-relativistic quantum mechanics as a purely algebraic formalism.

One aspect left indeterminate in this and Witten's construct is completeness: is every problem that can be posed in non-relativistic QM solvable in the space(s) in which the problem was posed?


reader Plato said...

Navier-Stokes equations of hydrodynamics (with turbulence) are a largely unsolved mathematical problem that may be reduced to a black hole exercise, too.

I would think the phenomenological correlation in early universe perspective being pushed back, we have created the opportunity to see further by LHC commensurable views of a space between "the colliding particles" and their determinations to surmise they come out of a "particulate location?"

How do you geometrically correlate that "in the geometries" with the help of Navier Stokes?

The Viscosity?:)

You had to "see the dynamics" in operation?


reader Pmer said...

can you go into Witten's paper in more detail?

And what's the big deal about Khovanov homology? Does one need to know about knots to evaluate a path integral or someting?