### The QG-TQFT Blues

#### Posted by John Baez

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

- Professor Elvis Zap, The Quantum Gravity Topological Quantum Field Theory Blues.

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 $n$-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 BluesI’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 $n$-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.

## 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.