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December 29, 2007

The QG-TQFT Blues

Posted by John Baez

Here at last is the music video we’ve all been waiting for!

Elvis Zap, also known as Scott Carter, is a quantum topologist from way down south. He’s one of the guys who first got me interested in possible applications of higher-dimensional knot theory to quantum gravity. That eventually led me to nn-categories, and I’ve been on a downhill slide ever since. I know the blues he’s singin’ about.

Lyrics follow… and more.

The Quantum Gravity
Topological Quantum Field Theory Blues

I’ve been calculating
I said I’ve been calculating
calculating all night long
Got a quasi-triangular Hopf algebra
and I wrote down the coproduct wrong.

I’ve been integrating
integrating the whole day through
I said I’ve been integrating
integrating the whole day through
Got a Chern-Simons functional integral
and its convergent, too.

I’ve been writing down knot diagrams
converting them to braids
Using the Alexander isotopy
you know I’m not afraid I’ve been
assigning modules
to each of these six strings
been doin’ it for weeks now
and I still don’t understand a thing.

I’ve got them old Quantum Gravity
Topological Quantum Field Theory Blues
I’ve got them old Quantum Gravity
Topological Quantum Field Theory Blues
And without NSF funding I think that you would, too.

My remark about a “downhill slide” was sort of a joke, but not completely. I’m very happy about working on nn-categories, but I suffered from the quantum gravity blues for many years.

Every year around this time John Brockman asks a bunch of people a question and makes a book of their replies. This year’s question is:

WHAT HAVE YOU CHANGED YOUR MIND ABOUT? WHY?

Here’s my answer. Other answers can be found at The World Question Center.

Should I be thinking about quantum gravity?

One of the big problems in physics - perhaps the biggest! - is figuring out how our two current best theories fit together. On the one hand we have the Standard Model, which tries to explain all the forces except gravity, and takes quantum mechanics into account. On the other hand we have General Relativity, which tries to explain gravity, and does not take quantum mechanics into account. Both theories seem to be more or less on the right track - but until we somehow fit them together, or completely discard one or both, our picture of the world will be deeply schizophrenic.

It seems plausible that as a step in the right direction we should figure out a theory of gravity that takes quantum mechanics into account, but reduces to General Relativity when we ignore quantum effects (which should be small in many situations). This is what people mean by “quantum gravity” - the quest for such a theory.

The most popular approach to quantum gravity is string theory. Despite decades of hard work by many very smart people, it’s far from clear that this theory is successful. It’s made no predictions that have been confirmed by experiment. In fact, it’s made few predictions that we have any hope of testing anytime soon! Finding certain sorts of particles at the big new particle accelerator near Geneva would count as partial confirmation, but string theory says very little about the details of what we should expect. In fact, thanks to the vast “landscape” of string theory models that researchers are uncovering, it keeps getting harder to squeeze specific predictions out of this theory.

When I was a postdoc, back in the 1980s, I decided I wanted to work on quantum gravity. The appeal of this big puzzle seemed irresistible. String theory was very popular back then, but I was skeptical of it. I became excited when I learned of an alternative approach pioneered by Ashtekar, Rovelli and Smolin, called loop quantum gravity.

Loop quantum gravity was less ambitious than string theory. Instead of a “theory of everything”, it only sought to be a theory of something: namely, a theory of quantum gravity.

So, I jumped aboard this train, and for about a decade I was very happy with the progress we were making. A beautiful picture emerged, in which spacetime resembles a random “foam” at very short distance scales, following the laws of quantum mechanics.

We can write down lots of theories of this general sort. However, we have never yet found one for which we can show that General Relativity emerges as a good approximation at large distance scales - the quantum soap suds approximating a smooth surface when viewed from afar, as it were.

I helped my colleagues Dan Christensen and Greg Egan do a lot of computer simulations to study this problem. Most of our results went completely against what everyone had expected. But worse, the more work we did, the more I realized I didn’t know what questions we should be asking! It’s hard to know what to compute to check that a quantum foam is doing its best to mimic General Relativity.

Around this time, string theorists took note of loop quantum gravity people and other critics - in part thanks to Peter Woit’s blog, his book “Not Even Wrong”, and Lee Smolin’s book “The Trouble with Physics”. String theorists weren’t used to criticism like this. A kind of “string-loop war” began. There was a lot of pressure for physicists to take sides for one theory or the other. Tempers ran high.

Jaron Lanier put it this way: “One gets the impression that some physicists have gone for so long without any experimental data that might resolve the quantum-gravity debates that they are going a little crazy.” But even more depressing was that as this debate raged on, cosmologists were making wonderful discoveries left and right, getting precise data about dark energy, dark matter and inflation. None of this data could resolve the string-loop war! Why? Because neither of the contending theories could make predictions about the numbers the cosmologists were measuring! Both theories were too flexible.

I realized I didn’t have enough confidence in either theory to engage in these heated debates. I also realized that there were other questions to work on: questions where I could actually tell when I was on the right track, questions where researchers cooperate more and fight less. So, I eventually decided to quit working on quantum gravity.

It was very painful to do this, since quantum gravity had been my holy grail for decades. After you’ve convinced yourself that some problem is the one you want to spend your life working on, it’s hard to change your mind. But when I finally did, it was tremendously liberating.

I wouldn’t urge anyone else to quit working on quantum gravity. Someday, someone is going to make real progress. When this happens, I may even rejoin the subject. But for now, I’m thinking about other things. And, I’m making more real progress understanding the universe than I ever did before.

Posted at December 29, 2007 3:46 AM UTC

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Re: The QG-TQFT Blues

It turns out that the problem of quantum gravity is more difficult then we thought.
It is not just the problem of getting rid of the QFT infinities, but it is more a problem of ambiguities, since there is no unique way of quantizing a classical theory. Related to this is the problem of the definition of quantization, i.e. whether to quantize only the fields on a manifold (e.g. string theory approach) or to quantize both the manifold and the fields(e.g. noncommutative geometry).
In the absence of the quantum gravity phenomena, the best we can do is to explore various possibilities, since this knowledge will be usefull when the right time comes. At least, a new mathematics will be generated.

Posted by: Aleksandar Mikovic on December 29, 2007 11:36 AM | Permalink | Reply to this

Re: The QG-TQFT Blues

whether to quantize only the fields on a manifold (e.g. string theory approach) or to quantize both the manifold and the fields(e.g. noncommutative geometry).

I am not sure if the dichotomy referred to here is real.

I find it remarkable that Alain Connes’s noncommutative-geometry (really: spectral geometry) based approach to physics beyond the standard model, the spectral action approach, is the same kind of generalized sigma-model approach that string theory is based on:

in Connes’s model physical spacetime is realized as the target of a sigma-model describing a supersymmetric 1-particle – technically: a spectral triple.

in string theory physical spacetime is realized as the target of a sigma-model describing a supersymmetric 2-particle – technically: a 2-spectral triple.

While obvious, this is rarely amplified the way it should be.

Yan Soibelman is privately circulating notes on a big project fleshing out the 2-spectral triple perspective on 2-dimensional CFT, building on his work with Kontsevich.

Maybe when that is out, things will become clearer.

Posted by: Urs Schreiber on December 29, 2007 2:25 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

And, I’m making more real progress understanding the universe than I ever did before.

My impression is that the biggest danger in quantum gravity research is to try to do precisely what current criticism is trying to make people to do: to make direct contact with experiment. And more generally, to handle the investigation in the way physicists are used to handling it.

Progress is made here when re-thinking the basics of physical theories. Once we actually know what a quantum field theory really is in the first place, we can come back to seeing how quantum gravity might fit into the picture.

There is just no point in trying to be quicker than one can be.

Posted by: Urs Schreiber on December 29, 2007 8:18 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

There is just no point in trying to be quicker than one can be.

I agree with that. Some people think they know pretty well what sort of form a theory of fundamental physics should take, and that it’s just a matter of choosing one that fits the data. In my quantum gravity days I was trying to be radical, describing physics using higher-dimensional generalizations of Feynman diagrams. But, when it came to choosing a good theory from this class, I was just shooting in the dark, without even knowing what counted as hitting the target.

I now more clearly realize that a vast amount of work needs to be done to understand existing theories of physics: string theories, quantum field theories, classical field theories, even classical mechanics. Among other things, we need to fit these theories in a clear framework that’s not limited by the meager mathematical tools that happened to be around when people first dreamt them up. Clearly these theories are based on some very good idea. But, what exactly are these good ideas? We need to analyze this.

As begin doing this, we quickly find there’s a lot more ‘room’ for theories than people had realized. For example, what sort of spacetime can a particle, string or membrane move around in? It doesn’t need to be manifold: it can be a smooth nn-category, or an nn-category in any topos, or a spectral triple, or a categorified spectral triple, etcetera… Most of these possibilities, which may sound bizarre at first, are actually implicit in theories people already consider. So, we need to think about them. Simultaneously we need to think about questions like why the complex numbers show up in quantum mechanics, and so on.

Viewed from outside this may seem like a slow business. It probably won’t lead to testable predictions anytime soon! Sometimes it’s almost indistinguishable from pure mathematics. But the great thing about pure mathematics is that it reaches insights that stand the test of time and become essential for all new work in physics. And so, I’m happier now than I was back in the days of throwing darts in a darkened room and hoping to hit the bull’s-eye.

Posted by: John Baez on December 29, 2007 8:51 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

Over the holidays I have also been reading Peter Pesic’s
collection of articles in Beyond Geometry . These are Riemann, Poincare, Clifford, Klein, and others discussing the nature of geometry following Lobochevsky. The questions there are the same questions that are being asked here. I wonder if we are any closer to the answer. Clearly, both our physical and mathematical understanding has advanced, but the questions remain, “What is reality and how do we model it mathematically?” One thing interesting in these essays was that the questions were perceived as questions of philosophy, in particular Kant was talked about quite a bit.

The QG-TQFT blues were written on a Friday morning after a long week in Manahattan, Kansas. It was preceded by a week at the geometry center. Steve Sawin was among those at both conferences. K. State was a spring conference that was hosted by Crane and Yetter. Those of you (JB) who were there will remember the walk through the Kansa prairie. I had to rush ahead out of earshot of two physicists talking because the landscape was startlingly beautiful and they were discussing Chern-Simon’s actions. In my opinion, that discussion should have occurred elsewhere. Anyway, I was in the shower on the last day of the conference and the lyrics came to me as soon as the title was finished.

About a month ago, I was doing some calculations in a Frobenuis algebra and kept getting the wrong answer. Sure enough, I wrote down the wrong co-product. Arrgh!

Posted by: Scott Carter on December 31, 2007 12:54 AM | Permalink | Reply to this

Re: The QG-TQFT Blues

I was lamenting back here the fact that 130 years ago a leading English language philosophy journal could host Helmholtz giving his modified Kantian view of geometry.

In the twentieth century geometry became taken to be divisible (by Reichenbach and associates, and hence Anglo-American philosophy) into a physical geometry and a mathematical geometry, the latter the mere working out of logic consequences of axiomatic systems, where it became increasing hard to define which were ‘geometric’.

The philosophy of geometry has become, by and large, a sub-branch of the philosophy of general relativity, itself a branch of philosophy of physics. I take Cartier’s Mad Day’s Work to be clear proof that a philosophy of geometry from a mathematician’s perspective is possible.

Posted by: David Corfield on December 31, 2007 9:54 AM | Permalink | Reply to this

Re: The QG-TQFT Blues


“And, I’m making more real progress understanding
the universe than I ever did before.”

May 2008 bring everyone in the field more real progress towards understanding the universe than ever before, with more cooperation and less conflict, together with growing enthusiasm and creativity as everyone follows his/her own most satisfying and productive path toward that goal.

Happy New Year!

Posted by: Charlie C on December 31, 2007 12:02 PM | Permalink | Reply to this

Re: The QG-TQFT Blues


Why should gravity be quantized? You want a mathematically consistent theory that subsumes General Relativity and the Standard Model. This does not force you to take linear combinations of entire spacetimes.

In fact is there any compelling reason why a fundamental theory of physics should involve the real numbers at all, or even why there should be a notion of addition?

Posted by: Dirk Vertigan on December 31, 2007 5:32 AM | Permalink | Reply to this

Re: The QG-TQFT Blues

Dirk Vertigan wrote:

Why should gravity be quantized? You want a mathematically consistent theory that subsumes General Relativity and the Standard Model. This does not force you to take linear combinations of entire spacetimes.

Indeed. Combining quantum field theory with classical gravity is well-known to be problematic — certainly no easier than quantum gravity. So, most people find it more plausible that gravity obeys the principles of quantum theory, like all the other forces. But, there could easily be lots of options that have not been explored yet. One of these could be the right one. Most of them aren’t, obviously.

In fact is there any compelling reason why a fundamental theory of physics should involve the real numbers at all, or even why there should be a notion of addition?

It’s hard to tell ahead of time how radically we must rewrite physics before getting to the next really good theory. For a nice introduction, see:

Penrose’s spin networks, and the material we are just beginning to discuss in the geometric representation theory seminar, show how significant parts of quantum mechanics can be developed using purely combinatorial techniques, not referring to the real or complex numbers. Isham is currently working on doing physics in an arbitrary topos — a generalization of set theory.

Will such radical ideas be necessary to reconcile quantum theory and general relativity? Only time will tell. Luckily, it’s not my problem any more.

Posted by: John Baez on December 31, 2007 7:13 AM | Permalink | Reply to this

Re: The QG-TQFT Blues

significant parts of quantum mechanics can be developed using purely combinatorial techniques, not referring to the real or complex numbers.

And maybe the most remarkable aspect of this statement is – that it does not involve the words “quantum gravity”.

I perceive it as another big problem of the quantum gravity research we have seen and are seeing that it is based too much on prejudices.

Also a very daring and mind-bogglingly visionary prejudice is a prejudice.

What does look promising, though, is to go the way you just described: work out, starting from the basics instead of from fancy stuff like, say, the resolution of the big bang-singularity, what deep things are still to be said about our apparently familiar theories.

Posted by: Urs Schreiber on December 31, 2007 11:54 AM | Permalink | Reply to this

Re: The QG-TQFT Blues

…not referring to the real or complex numbers.

A possibly silly question I keep meaning to ask, so why not here on the last day of the year: if groupoids come to take the place of (non-negative) reals, do you expect to have a notion of distance between them? Is there a sense in which adding to a groupoid an object with a huge number of automorphisms is not changing it much?

Also how might you carry over the idea of multisets and ‘sets with a negative number of elements’ to groupoids?

Loeb, D., Sets with a negative number of elements, Advances in Mathematics 91 (1992), pp. 64–74.

Posted by: David Corfield on December 31, 2007 4:21 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

David wrote:

A possibly silly question I keep meaning to ask, so why not here on the last day of the year…

Indeed! We need to keep coming up with crazy new ideas, and New Year’s Even is a great time for it!

… if groupoids come to take the place of (non-negative) reals, do you expect to have a notion of distance between them?

I’ve never thought about that. Naively, the distance between real numbers xx and yy is |x−y||x - y|, so to generalize it we’d need to understand ‘absolute value’ and ‘subtraction’.

The concept of ‘absolute value’ or ‘size’ is something we now understand for groupoids and even some categories — it’s the groupoid cardinality, or Euler–Leinster characteristic. I’ve been trying to take as much advantage of this new discovery as possible… before everyone else catches on. The early bird gets the worm.

Subtraction is the hard part, except insofar as a pushout of groupoids generalizes a ‘non-disjoint union’ of sets

X∪Y≅X+Y−X∩Y X \cup Y \cong X + Y - X \cap Y

and gets a bit of subtraction going on, along with addition.

The big challenge is getting full-fledged ‘negative groupoids’, and integrating them thoroughly with everything else we know. We know the sphere spectrum is like the ∞\infty-categorified integers. And, we know the sphere spectrum arises by applying a systematic process invented by Graeme Segal

[symmetricmonoidalcategories]→[spectra][symmetric monoidal categories] \to [spectra]

to the symmetric monoidal category of finite sets! So, for a long time Jim Dolan has been advocating applying this same process to the symmetric monoidal category of finite groupoids, or finite nn-groupoids, or tame ω\omega-groupoids (‘tame spaces’). This might give the ‘∞\infty-categorified real numbers’.

But, the only homotopy theorist we’ve consulted about this idea said it was worthless, because the symmetric monoidal category of all tame spaces was too big and unmanageable.

I think they were just being a bit timid. After a few glasses of champagne, one gets bolder.

Happy New Year!

Posted by: John Baez on December 31, 2007 9:25 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

Now where can we find a bolder homotopy theorist?

So your candidates

the symmetric monoidal category of finite groupoids, or finite nn-groupoids, or tame ω\omega-groupoids (‘tame spaces’),

should all give generalized cohomology theories. Can these be so very different from the ‘A theory’ you mention in week 199?

…another example comes from taking the category of finite CW complexes, with disjoint union as the “tensor product” and the obvious braiding. This gives a generalized cohomology theory called “A-theory”, due to Waldhausen.

Anyone care to give us a brief intro to A-theory?

Posted by: David Corfield on January 1, 2008 1:05 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

David wrote:

Now where can we find a bolder homotopy theorist?

There’s no shortage of ‘em, so I should try another one.

Indeed, the one I talked to is very bold in his own way — he’s just the sort of guy who likes spectra that give generalized cohomology theories where you can compute stuff. Building a generalized cohomology theory from the symmetric monoidal category of all tame spaces seemed useless to him because he didn’t see how you could compute stuff with it. But, that was never the point here. The point was to understand stuff.

Can these be so very different from the ‘A theory’ you mention in week 199?

…another example comes from taking the category of finite CW complexes, with disjoint union as the “tensor product” and the obvious braiding. This gives a generalized cohomology theory called “A-theory”, due to Waldhausen.

Very astute, David! Here’s the difference: a finite CW complex is the sort of thing that has a well-defined Euler characteristic, taking values in ℤ\mathbb{Z}. A tame space is the sort of thing that has a well-defined homotopy cardinality, taking values in ℝ +\mathbb{R}^+.

As you know, Euler characteristic and homotopy cardinality are like Jekyll and Hyde: they’re morally the same, but it’s hard to prove, because you never see them both side by side, and they act very different. Euler characteristic arises when we take finite sets and do our feeble best to throw in additive inverses, while homotopy cardinality arises when we take finite sets and do our feeble best to throw in multiplicative inverses.

Anyone care to give us a brief intro to A-theory?

I wish someone would. Barring that, we can read starting on page 20 of this review article:

I should warn you that the founder of A-theory — Friedhelm Waldhausen — didn’t quite use Segal’s original machine

[symmetricmonoidalcategories]→[spectra][symmetric monoidal categories] \to [spectra]

to turn [finiteCWcomplexes][finite CW complexes] into the spectrum for AA-theory. Instead, he used a subtle generalization of this idea, which starts with a ‘category with cofibrations and weak equivalences’. This is why Rosenberg refers to Waldhausen’s original paper on A-theory as a ‘100-page technical tour de force.’ But, I suspect the ‘moral essence’ of A-theory could be grasped without delving into these technicalities, if only someone would explain it to us.

Here’s part of the moral essence. Finite CW complexes are a generalization of finite sets. So, we have a symmetric monoidal functor

[finitesets]→[finiteCWcomplexes][finite sets] \to [finite CW complexes]

and this gives a map of spectra from the sphere spectrum to the A-theory spectrum. So, there’s a map from the stable homotopy groups of any space to its A-theory groups!

And, if I’m not mixed up, the nnth A-theory group of a space ‘splits’ as a direct sum of its nnth stable homotopy group and some mysterious extra stuff. This extra stuff is what I’d really like to understand.

Posted by: John Baez on January 1, 2008 7:53 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

What other symmetric monoidal functors are around? I guess there’s

[finitesets]→[finitegroupoids]. [finite sets] \to [finite groupoids].

And then various fundamental nn-groupoid functors from finite CW complexes.

What are you going to call your generalized cohomology theory for tame spaces?

Posted by: David Corfield on January 1, 2008 11:34 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

Right — there’s a whole network of symmoncats of ‘spaces’, where the tensor product is disjoint union, and all these give spectra and thus generalized cohomology theories!

David wrote:

What are you going to call your generalized cohomology theory for tame spaces?

Well, let’s see: there’s already A-theory, and…

… and K-theory, and L-theory, and M-theory, and…

I’m not sure how many letters are already used! How about T-theory?

Posted by: John Baez on January 1, 2008 11:49 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

…and some mysterious extra stuff.

Hmm, the algebraic K-theory of the sphere spectrum is equivalent (p. 2) to the algebraic K-theory of EE, the small rig of category of finite sets and these are equivalent to the A-theory of a point, ℤ×|BGL(E)| +\mathbb{Z} \times |B G L (E)|^+.

As we’re still in festive mood, any thoughts on

[symmetric monoidal 2-categories] →\to [2-spectra]?

Posted by: David Corfield on January 2, 2008 5:10 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

David wrote:

Hmm, the algebraic K-theory of the sphere spectrum is equivalent (p. 2) to the algebraic K-theory of EE, the small rig category of finite sets and these are equivalent to the A-theory of a point, ℤ×|BGL(E)| +\mathbb{Z} \times |BGL (E)|^+.

I find the dense jargon here makes it a little hard to separate out the stuff I understand from the stuff I don’t!

I very much understand how to turn a category into a space: a space built of simplices, where nn-simplices are composable nn-tuples of morphisms in our category. This space is called the nerve of our category.

And, I pretty much understand that when our category is symmetric monoidal — equipped with a tensor product that’s commutative up to coherent natural isomorphism — then its nerve gets a product that’s commutative up to coherent homotopy. This sort of space is called an E ∞E_\infty space: a space that’s a commutative monoid up to coherent homotopy.

And, I pretty much understand that we can take an E ∞E_\infty space and throw in ‘inverses’ to get a space that’s an abelian group up to coherent homotopy. Such a space is called an infinite loop space. It’s a special case of a ‘spectrum’.

So, we’ve got a process going from from symmetric monoidal categories to spectra.

And, I pretty much understand certain examples of this. For example, the classic example that started the whole game! The symmetric monoidal category of finite sets (with disjoint union as tensor product) gives us the sphere spectrum.

I think I understand this stuff and you do too.

Now for the stuff that I understand less well.

First of all, unless I’m going crazy, I think this process of going from symmetric monoidal categories to spectra is sometimes called ‘KK-theory’. Unless I’m going crazy, I think people say sentences like “the KK-theory of this symmetric monoidal category is this spectrum.”

For example: “the KK-theory of the symmetric monoidal category of finite sets is the sphere spectrum.”

I really dislike this use of the term ‘KK-theory’, since that term means a bunch of other things too, and these other things show up in the very same context. For example, I think the KK-theory of the symmetric monoidal category of vector spaces is called… the KK-theory spectrum!

I think this is actually the origin of this annoying extension of the term ‘KK-theory’, but I still don’t like it.

And, it gets worse. If you carefully read the first page of that paper by Baas & Co., you’ll see what I mean. I think they’re using the term ‘KK-theory’ in at least 3 and possibly 4 different ways in two paragraphs here! That’s when I get the feeling I’m going crazy. I keep feeling I’m making some mistake or other.

Somewhere on page 2 of the paper you mentioned, Baas & Co. say this:

The sphere spectrum SS is the algebraic KK-theory of the small rig category of finite sets EE.

Now, that sounds a lot like something I just said: “the KK-theory of the symmetric monoidal category of finite sets is the sphere spectrum.” The only differences are the words “algebraic” and “small rig”.

Of course the category of finite sets is equipped not only with disjoint union, but also cartesian product, which distributes over disjoint union. So, it’s a ‘rig category’. You can say ‘small rig category’ if you’re nervous about size issues — sets versus proper classes. I tend to gloss over that, at least while blogging! So, the process I’ve just sketched turns small rig categories into special spectra called ‘ring spectra’.

So, you might think we already understand what Baas & Co. just said.

But no!

They in fact are discussing a different construction, called the ‘algebraic KK-theory spectrum of a rig category’. This is a construction from a previous paper of theirs, which generalizes something called the ‘algebraic KK-theory spectrum of a ring’. And that, in turn, is a special case of the KK-theory construction I just outlined!

Namely: if take the symmetric monoidal category of finitely generated projective modules of a ring RR, and turn it into a spectrum, we get the ‘algebraic KK-theory spectrum’ of RR. This idea was invented by Quillen.

I hope you’re confused now.

Anyway, I have a feeling that buried amid all this confusing (but ultimately very beautiful) stuff there is a tiny little fact about the AA-theory of a point. But, I’m too exhausted to find and understand that fact.

As we’re still in festive mood…

I might have been when I started writing this, but now I’m not. Too many kinds of KK-theory.

Posted by: John Baez on January 3, 2008 3:03 AM | Permalink | Reply to this

Re: The QG-TQFT Blues

Hey, those are some really cool quantum gravity blues! Great stuff.

As an addendum, lol, I happened to pause the video at an inopportune moment and I shockingly discovered that Willem Dafoe (a.k.a the Green Goblin) has now penetrated the TQFT community!

pic

pic

Posted by: Bruce Bartlett on December 31, 2007 3:33 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

Bruce, you would have made an excellent staff member of the Separated at Birth? department of the (now defunct) Spy Magazine.

Posted by: Todd Trimble on January 1, 2008 11:59 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

About ten years ago, I found a short story, “Like John Baez on Acid.” JB replied that any publicity is good publicity. Dafoe also had the starring role in “The Last Temptation…” That is a role that *I* really don’t want! But Elvis’s next music post will be related ;-)

Thanks for the belly laugh Bruce!

Posted by: Scott Carter on January 2, 2008 1:44 AM | Permalink | Reply to this

Re: The QG-TQFT Blues

All publicity is good publicity… if you behave honorably. I’m not sure why that story is called ‘Like John Baez on Acid’. Maybe because it’s a bit like a more rambling cousin of This Week’s Finds? It’s not very psychedelic.

Best passage:

I told him the story about how snapshots of my program to visualize a four dimensional database of fossilized pollen appeared in Banchoff’s “Beyond Three Dimensions” book before the program was written, and he told me the story of how he was officemates with Bill Thurston (Field’s Medal winning and in the opinion of some the greatest living mathematician), was assigned the same thesis problem as Bill (on foliations), and then suffered a severe crisis of self-esteem when Bill rewrote the entire subject in a month. I met Thurston briefly last summer at the Geometry Center, and had a similar thing happen to me: I described my research on fractals to him, and within ten minutes he had not only caught up and superceded all of the work I had done for the last couple of years (this is not as impressive as it may sound. Most mathematics has the quality of being obvious and easy to understand in retrospect. Plus, I later found out that he had worked on a similar problem involving reptiles.), but he then went on to reformulate the problem and completely change the way I looked at the field. I’ve known many intelligent people in my life – people who knew more than I did, people who were quicker learners than I am, people who were more creative than I am — but I have never met anyone besides Bill Thurston who deserved the word genius. True, the word has degenerated into a cliche these days, but it is the only word that captures the primordial, magical quality of intellectual power operating on a completely higher level. It is very, very scary to talk to Bill for extended periods of time, because you quickly develop the belief (justifiable or not) that none of your ideas, none of your intellectual accomplishments, none of your mental foundations are worth anything — that he could have done them all in a day if he bothered. I don’t want to call Thurston inhuman, for that would be an insult to both humanity and one of its finest jewels, but no science fiction writer I am aware of has ever created an alien as an alien as he.

Posted by: John Baez on January 2, 2008 7:02 AM | Permalink | Reply to this

Re: The QG-TQFT Blues

This is a thought-provoking story. It reminds me of something I read about René Thom in the early days of the IHES. Thom might have been at the height of his powers, but the trouble was that Grothendieck was too, and Thom found that very difficult to handle. There Grothendieck was, sweeping almost all of pure mathematics before him in his massive programme, and despite Thom having already won a Fields medal for his work in topology, he apparently felt suffocated. There’s an interview with him where he says something like “Grothendieck’s technical superiority was crushing”. So he got out of the traditional fields of pure mathematics and invented something completely new, catastrophe theory.

It’s maybe healthy to imagine meeting unbelievably superior mathematicians, even if you don’t meet them much in real life. Ron Graham described a fantasy of his involving an alien culture so mathematically advanced that a child might think of a proof of the twin prime conjecture in a short moment of distraction. Maybe that culture exists, and maybe one day we’ll be in contact with it. What would we do if we found another culture that had done all the mathematics that we’d ever done and a hundred times more? If every time we came up with a mathematical suggestion we thought was exciting and new, our alien friends informed us that they’d sorted it out long ago?

I gathered together various quotations on this topic (including, inevitably, one from Doron Zeilberger) and used them in a talk a couple of years ago; see especially the first and last few slides.

Posted by: Tom Leinster on January 2, 2008 4:35 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

Tom wrote:

What would we do if we found another culture that had done all the mathematics that we’d ever done and a hundred times more? If every time we came up with a mathematical suggestion we thought was exciting and new, our alien friends informed us that they’d sorted it out long ago?

Most people are in a vaguely similiar situation throughout school, if you replace ‘alien friends’ by ‘teachers’. And we all know what they wind up saying: things like “I was never all that good at math” and “I hate math.”

Of course it all depends on how insufferably superior our alien friends acted.

Personally I might take advantage of these folks — if they’d let me — by getting the answers to a lot of questions that are bugging me. In fact, I might enjoy having a pal who could easily answer any question I happened to come up with. But, it wouldn’t be so fun if they kept rolling their eyes, twiddling their thumbs and sighing with boredom.

It’s an interesting issue. It reminds me of the beginning of Geoffrey Dixon’s book, which I discussed in week59:

As you crack your eyes one morning your reason is assaulted by a strange sight. Over your head, humming quietly, there floats a monitor, an ethereal otherworldly screen on which is written a curious message. “I am the Screen of ultimate Truth. I am bulging with information and ask nothing better than to be allowed to impart it.”

It goes on, and the screen becomes more testy after a while.

Posted by: John Baez on January 2, 2008 5:48 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

Urs wrote:

My impression is that the biggest danger in quantum gravity research is to try to do precisely what current criticism is trying to make people to do: to make direct contact with experiment.

I have another thing to say…

What you write sounds like good advice for people like you and me — people who know some physics and enjoy elegant mathematics. But of course, there are also physicists who like to get their hands on lots of data and build rough-and-ready models to fit it, based on physical intuition more than mathematical elegance. I think there’s still a lot of room for this sort of work!

In particular, there’s a lot of data coming in about the power spectrum of the cosmological microwave background radiation, the dynamics of dark matter, and so on. With luck, we may even see the cosmological gravitational background radiation in our lifetimes. While none of this is quite the same as ‘quantum gravity’, it’s bound to be relevant eventually. And, understanding this stuff takes physicists with intuition and guts, willing to make up models, test them against the data, watch them crash and burn, and then try to fix them.

I admire such people a lot. But, I’m not one of them. I just enjoy pretty math too much.

Maybe the problem is trying to get the elegant mathematics to make contact with the experimental data prematurely. Right now this seems to lead to models that aren’t all that elegant and don’t quite make contact with the data.

Posted by: John Baez on December 31, 2007 10:21 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

While we’re exchanging New Year’s Eve greetings…

I just got an New Year’s email from Abhay Ashtekar, entitled ‘Your Blues’. I haven’t seen him for a long time. He said someone pointed him to this blog entry. And, he says there’s been a lot of progress on loop quantum cosmology in the last two years.

For anyone interested, he wrote a review article on the subject. It features quantized Friedman–Robertson–Walker models that exhibit a ‘quantum bounce’ at the Big Bang. These are ‘mini-superspace’ models, meaning they quantize only a highly symmetrical sector of general relativity. He said the field is not mature enough to make predictions in time for the Planck Mission, but people are starting to study perturbations around these mini-superspace models, which is what you’d need for that.

Posted by: John Baez on December 31, 2007 10:35 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

John mentioned, implicitly

[“loop quantum cosmology”]

I could never get reassured that it is beneficial to claim that it is known that loop quantum cosmology is indeed related to a holonomy variable quantization of some gauge theoretic formulation of gravity.

What LQC certainly is is the study of difference equations that approximate the Friedman-Robertson-Walker differential equations.

If that can be shown to be useful for matching data, that would be exciting.

What I could never quite follow are these claims that a holonomy quantization yields the kind of discretization that underlies “loop quantum cosmology”. My impression is that the discretization is always introduced by hand, when people pass to non-standard quantization prescriptions, introducing Bohr compactifications and the like.

In Ashtekar’s review this is mentioned on p. 9.

So, for what it’s worth, this is how I perceive the situation:

if “loop quantum cosmology” should one day indeed have experimental verification, it would be very interesting, but it would be interesting in the way in which MOND is interesting: it would be an ad hoc fit to the data, leaving us puzzled as to what it might really mean.

By the way: for a while I had the impression that I was the only one who didn’t think that the premises of loop quantum gravity really lead to discretization of spacetime. But a while ago apparently Thomas Thiemann got worried about that, too, and asked: Are the spectra of geometrical operators in Loop Quantum Gravity really discrete?

My last blogosperic remark on this was here, on Christine Dantas blog.

Posted by: Urs Schreiber on January 2, 2008 7:51 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

Thanks for the references!

Can you suggest any reading on any approaches to trying to get quantum theory (and similarly, other physics) to evolve/develop dynamically in some mathematical model of physics. In this view, the Big Bang would be `recent history’ and in the much larger entire universe, quantum theory would not necessarily be true, but it could evolve/develop by some dynamic processes and often stably remain true. Perhaps Black-Hole-to-Big-Bang transitions could be a crucial part of such a process,as they are in Smolin’s CNS.

Posted by: Dirk Vertigan on January 1, 2008 3:27 AM | Permalink | Reply to this

Re: The QG-TQFT Blues

You can now see everyone’s answers to the EDGE question of the year: “What have you changed your mind about? Why?”

Besides me, here are some other people who have changed their minds about physics:

  • Carlo Rovelli no longer believes there’s nothing to add to the standard interpretation of quantum mechanics.
  • Lee Smolin now thinks that time is real, not just emergent.
  • Marcelo Gleiser no longer believes in the unification of forces and particles.
  • Max Tegmark no longer thinks we need to understand consciousness to understand physics.
Posted by: John Baez on January 2, 2008 12:48 AM | Permalink | Reply to this

Re: The QG-TQFT Blues

A healthy QG depression does in any case look more healthy than the insanity in which others have found shelter. It should not be allowed to ask on hep-th questions like Does God So Love the Multiverse?

I am not sure what worries me more: the religious elements becoming popular here, or the confusion which apparently motivates them.

Posted by: Urs Schreiber on January 3, 2008 1:25 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

What really upsets me about this particular paper is that Don Page is capable of good physics. If you look through his papers in reverse chronological order — currently numbering more than 120 — you’ll see a curious progression. The explicitly religious papers are first found this year, and are clearly linked to the Templeton Foundation. So, that organization is having some success in changing the relation of religion and physics.

On the other hand, we shouldn’t forget that Isaac Newton spent most of his later years on biblical chronology and theology. So, playing the devil’s advocate , one could say that concern with deep issues in physics and concern with religion are closely allied and only kept separate by a strenuous and unnatural sort of mental discipline.

However, regardless of whether it’s ‘natural’, I find this discipline to be immensely important.

Posted by: John Baez on January 3, 2008 4:38 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

one could say that concern with deep issues in physics and concern with religion are closely allied

What I am really missing in much of what is promoted as “religion” is that sense for the unspeakable.

So I find Templeton-foundation things – or at least what I have seen of it from the hep-th perspective on life – as much a pain religion-wise as physics-wise.

Even though also abused a lot by physicists back in the 60s, 70s, I find that a little of that Zen-attitude towards religion might be beneficial: if you find yourself talking about it directly, you are already missing it.

Posted by: Urs Schreiber on January 3, 2008 6:15 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

Urs wrote:

Even though also abused a lot by physicists back in the 60s, 70s, I find that a little of that Zen-attitude towards religion might be beneficial: if you find yourself talking about it directly, you are already missing it.

I agree utterly, which is why I’m gonna shut up now.

Posted by: John Baez on January 3, 2008 8:16 PM | Permalink | Reply to this

Jacob have I loved, but Esau have I hated

I knew this paper reminded me of something, but it took a while to remember what that something was: the rabbinical students in Surely You’re Joking, Mr. Feynman! (1985). They were concerned by the sparks coming from an elevator button, and asked Feynman whether electricity were fire, because if it were, then the use of electricity would be forbidden on the Sabbath. (I was told, incidentally, that this is why Brandeis uses only mechanical locks, instead of the card-swipe devices found at other universities nowadays.) Feynman said,

It really was a disappointment. Here they are, slowly coming to life, only to better interpret the Talmud. Imagine! In modern times like this, guys are studying to go into society and do something — to be a rabbi — and the only way they think that science might be interesting is because their ancient, provincial, medieval problems are being confounded slightly by some new phenomena.

I could rant at great length about all this, but I promised myself that I’d write more equation-type stuff instead of just reacting to everything which provokes my ire, so I’ll stop now.

Posted by: Blake Stacey on January 3, 2008 6:19 PM | Permalink | Reply to this

Re: Jacob have I loved, but Esau have I hated

Blake Stacey linked to

equation-type stuff

Following the link I saw that curious icon

That’s maybe an interesting idea. I’ll try to find some time later today to figure out if we might also want to include that icon here for research entires.

Posted by: Urs Schreiber on January 3, 2008 6:38 PM | Permalink | Reply to this

Blogging on Peer-Reviewed Research

The BPR3 people are still getting their organization underway, so a great many policy-and-procedure questions have yet to be asked, let alone answered. However, based on what has been said so far, I believe the intent is to stimulate and aggregate blogging on research which has already gone through peer review, rather than research which is still being worked out (and hasn’t yet been written up). This Week’s Finds would qualify, but perhaps not Supercategories.

What counts as peer review is still, to an extent, undecided.

Posted by: Blake Stacey on January 3, 2008 6:49 PM | Permalink | Reply to this

Re: Blogging on Peer-Reviewed Research

Yes, I understand that from the text on the icon. I guess entries like BF-Theory as a Higher Gauge Theory, The Principle of General Tovariance, Loday and Pirashvili on Lie 2-Algebras (secretly), to name just the last three discussions of peer-reviewed research which I posted, would qualify, for instance.

(Meaning: hey, I am not only talking about my ideas here! :-)

Posted by: Urs Schreiber on January 3, 2008 6:56 PM | Permalink | Reply to this

Re: Blogging on Peer-Reviewed Research

Gosh, would I suggest that you were? ;-)

The “BPR3” people have set up a new aggregation system at ResearchBlogging.org. It looks like a promising start, although their registration code is still a little buggy (I was able to register and log in, but in the process, it spat some raw SQL at me).

Posted by: Blake Stacey on January 21, 2008 7:50 PM | Permalink | Reply to this

Re: Jacob have I loved, but Esau have I hated

There is no objection to fire on the Sabbath, just to lighting one, e.g. striking a spark.
Solution: light the fire before the Sabbath starts and overnight bank it.
No wonder that culture produced a lot of logical thinkers who took wordings (e.g. definitions) seriously.
The Isreli solution to the elevator problem: On the Sabbath, the elevators are programmed to operaate continuously, stopping at each floor - no need to push a button.


My recently moved into house even has a Sabbath setting on the electric ovens.

Posted by: jim stasheff on January 4, 2008 1:28 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

John Baez:” On the other hand we have General Relativity, which tries to explain gravity, and do not take quantum mechanics into account. Both theories seem to be more or less on the right track - but until we somehow fit them together our picture of the world will be deeply schizophrenic…
So, I eventually decided to quit working on quantum gravity. Someday, someone is going to make real progress. When this happens, I may even rejoin the subject.”

One needs the mathematical tools suitable to do that. I mean you need vector derivative to formulate ED and covariant derivative for GR. These tools are outside of the classical analysis. I mean something like that, something that was unknown to the mathematicians. I have enough confidence to predict that you will get them (obviously flexible) through arxiv during January 2008.

Welcome back to Holy Grail.

Regards, Dany.

Posted by: Daniel Sepunaru on January 4, 2008 6:58 AM | Permalink | Reply to this

Re: The QG-TQFT Blues

With respect to Count Basie, that’s not “the blues I like to hear” !-) More seriously, I have a question about TQFT:

Back at the old String Coffee Table, Urs Schreiber discussed this paper by Aaron Lauda and Hendryk Pfeiffer which ends by describing a state sum construction of an open-closed TQFT with a finite set of D-branes via the groupoid algebra of a finite groupoid.

In Section 1 of this 3.5 page paper , Andrew Baker defines profinite groupoids as the inverse limit of (automorphism) finite groupoids.

My question is whether it would make any sense to try to translate the Lauda-Pfeiffer construction into the setting of profinite groupoids (categories)? What I am further aiming for is something like Edward Witten’s conjecture about constructing background independent open string field theory via an infinite set of D-branes. Thanks.

Posted by: Charlie Stromeyer Jr on January 21, 2008 10:25 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

In general I consider the blues recollections a feature of the alte kakers. Therefore, I beg your pardon for the inconsistency.

John Baez:” Most people are in a vaguely similar situation throughout school, if you replace ‘alien friends’ by ‘teachers’.”

I had remarkably good teachers, no one tried to demonstrate his superiority. Prof. M.I. Petrashen insisted that I should be accepted at Theor-Ph Department (exceptionally non-trivial then at Leningrad University since I am a Jew) just for one remark that her proof of some theorem was not clean enough. Later, my best teacher, Prof. V.N.Gribov required to call him “Volodia” just for one question that he was not able to answer (during Landau min preparation of the “Field Theory”; it was not his fall, L.D. was wrong). By the way, when I finished preparations, I asked him what his requirements are to pass. He said:” Very simple, you close the book and write it alone”. I didn’t say a word, turned 180 degrees and moved out. “Where you are going?” “I am not able to withstand that”. “I am joking, IF YOU WILL KNOW AND UNDERSTAND PHYSICAL OPTICS, IT WILL BE ENOUGH”.

Tom Leinster: ” It reminds me of something in the days of the IHES”.

At May 1980 I worked at IHES and was invited to dinner by L. Michel. It started with hand-cart filled with incredible collection of exotic wines, liquors, etc. I realized that it is unique opportunity for me to test them. It was just the matter of the natural curiosity. Shamefully, half hour later I was deadly drunk.

Next morning we (L.C.B. and L.P.H.) continued discussion of tensor products in QQM and S.L. Adler’s papers. I remember a large room full of light with huge window, sun and forest outside. I stand near the blackboard and L.C. (in the chair) desperately tries to explain me why the algebra of the tensor product must be quaternion and why S.L. Adler is wrong. Suddenly, the coin fall and I cried: “No, that is what I tried to find all the time”. Three months later I published paper entitled “Quantum Mechanics of Non-Abelian Waves” where was described how it is.

Regards, Dany.

Posted by: Daniel Sepunaru on January 29, 2008 1:50 PM | Permalink | Reply to this

Alte Kaker

Just one question: what’s an ‘alte kaker’? Is this a Yiddish expression meaning ‘old fogey’, ‘old geezer’ or something like that?

Okay, never mind — I answered my own question. But, I’d still like to know the etymology. What’s a ‘kaker’?

Posted by: John Baez on January 29, 2008 9:03 PM | Permalink | Reply to this

Re: Alte Kaker

What’s a ‘kaker’?

Answering this might get us into non-family-blog territory. ;-)

Posted by: Urs Schreiber on January 29, 2008 9:19 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

John Baez:” Just one question: what’s an ‘alte kaker’? Is this a Yiddish expression meaning ‘old fogey’, ‘old geezer’ or something like that?”

“alte kakers: old farts”; however, old geezers is better. It has roots in Russian, but I prefer not to enter into details.

Regards, Dany.

P.S. Do you like 0801.3395?

Posted by: Daniel Sepunaru on January 29, 2008 11:43 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

Google found lots of kaker
and at least one non-de-plume: Alte Kaker

jim

Posted by: jim stasheff on January 30, 2008 1:33 PM | Permalink | Reply to this

Re: The QG-TQFT Blues

John Baez:” Okay, never mind — I answered my own question”.

Your question about kakers somehow brings me to the world of sounds and music. First of all, I had in mind only myself and I believe that I am not yet. There is deep and mystery connection (for me) between math and music. The math paper should sound. I am not rocker and mine is definitely not rock service. I believe it is more close to A.Corelli or A.Vivaldi. A stupid typo (Eq. (26)) causes me pain.

Regards, Dany.

Posted by: Daniel Sepunaru on February 1, 2008 6:01 AM | Permalink | Reply to this

Re: The QG-TQFT Blues

CC Hennix has the homotopy maqam-blues: http://www.grimmuseum.com/blog-18/blog-17/index.html .

Surprisingly, in addition to there existing multiple topological blues, there are multiple arabic blues: http://m.youtube.com/watch?v=FtkLOQdipHQ . (Desert Blues musicians made bluesy music before they heard the western blues. Allegedly their (historical) music is what gave rise to the more commonly known blues.)

Posted by: Trent on March 30, 2015 7:58 PM | Permalink | Reply to this

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