Nathan Seiberg on How Math Might Complete the Ultimate Physics Theory

Math and physics have a long history together. What are some of the most important ways they have influenced each other over the centuries?

From the time of the ancient Babylonians and Greeks, there hasn’t been a real distinction between math and physics. They studied similar questions. There has been a lot of cross-fertilization between what today we call math and physics. [Isaac] Newton is a great example. He was motivated by physics when he invented calculus. Over the 20th century, things were a bit more complicated. People specialized in math or in physics.

Physics usually offers very concrete questions and very concrete puzzles associated with reality and experiment. It’s also kind of grounded in reality. Math usually provides more generality, more powerful methods, and more rigor and precision. All of these elements are needed.

Do you think they’ll continue to be increasingly separate fields?

Given that they started as one field and lately diverged, but continue to influence each other, in the future I’d guess they’ll continue to influence each other to the point that there would be no clear separation between them. I think that there will be one deep, intellectual structure that encompasses math and physics.

Why has QFT, and physics in general, been such a provocative stimulus for math?

I think physicists and mathematicians are motivated by different questions. And different kinds of questions lead to different insights. There have been many examples where physicists came up with some ideas — which in most cases were not even rigorous — and mathematicians looked at them and said, “This is an equality between two different things; let’s try and prove it.” So the input from physics is another source of influence for the mathematicians. From this perspective, physics is like a machine that produces conjectures.

And the track record with these conjectures has been quite amazing, so mathematicians have learned to take physics in general and quantum field theory in particular very seriously. But what is perhaps surprising for them is that they still can’t make QFT rigorous; they still can’t figure out where these insights come from.

Let’s focus on the physics side for now, and that amazing track record. What are some of its biggest triumphs?

QFT is by far the most successful theory ever created by mankind to explain anything. There are many [predictions] that agree perfectly with experiments to unprecedented accuracy. We’re talking about accuracy of up to the order of 12 digits between theory and experiment. And there are literally trillions and trillions of experiments that match the theory. I don’t think historically there has ever been any theory as successful as quantum field theory. And it includes as special cases all the previous discoveries, like Newton’s theory, [James Clerk] Maxwell’s theory of electromagnetism, and of course quantum mechanics and Albert Einstein’s special relativity. All these things are special cases of this one coherent intellectual structure. It’s an amazing, spectacular achievement.

And yet we also think QFT is incomplete. What are its limitations?

The biggest challenge is to merge it with Einstein’s general theory of relativity. There are many ideas how to do this. String theory is the main one. There has been a lot of progress, but we’re still not at the end of the story.

You’ve referred to QFT as not yet “mature.” What do you mean by that?

I have my preferred maturity test for a scientific field. That is to look at textbooks and at courses at universities that teach the topic. When you look at a mature field, most of the textbooks are more or less the same. They follow the same logical sequence of ideas. Similarly, most of the courses are more or less the same. When you learn calculus, you first learn one topic, then another, and then the third. It is the same sequence in all institutions. For me, this is a sign of a mature field.

That’s not the case for QFT. There are several books with different perspectives from different points of view, with [ideas presented] in a different order. For me this means that we have not found the ultimate, streamlined version of presenting our understanding.

You’ve also mentioned that it’s a sign of incompleteness that QFT doesn’t have its own place in mathematics. What does that mean?

We cannot yet formulate QFT in a rigorous way that would make mathematicians perfectly happy. In special cases we can, but in general we cannot. In all the other theories in physics — in classical physics, in quantum mechanics — there is no such problem. Mathematicians have a rigorous description of it. They can prove theorems and make deep advances. That’s not yet the case in quantum field theory.

I should emphasize that we do not look for rigor for the sake of rigor. That’s not our goal. But I think that the fact that we don’t yet have a rigorous description of it, the fact that mathematicians are not yet comfortable with it, is a clear reflection of the fact that we don’t yet fully understand what we’re doing.

If we do have a rigorous description of QFT, it will give us new, deeper insights into the structure of the theory. It will give us new tools to perform calculations, and it will uncover new phenomena.

Are we even close to doing this?

Whatever approach we take, we get stuck somewhere. One approach that gets close to being rigorous is we imagine space as a lattice of points. Then we take the limit as the points approach each other and space becomes continuous. We describe space as a lattice, and as long as we’re on the lattice there is nothing non-rigorous about it. The challenge is to prove that the limit exists as the distance [between points on the lattice] becomes small and the number of points [on the lattice] becomes large. We assume this limit exists, but we cannot prove it.

So if we do it, will a rigorous understanding of quantum field theory actually merge it with general relativity? That is, will it provide a long-sought theory of quantum gravity?

It’s quite clear to me that there is one intellectual structure that includes everything. I think of quantum field theory as being the language of physics, simply because it already appears like the language of many different phenomena in many different fields. I expect it to encompass also quantum gravity. In fact, in special circumstances, quantum gravity is described by quantum field theory.

It might take a century or two, even three centuries, to get there. But I personally don’t think it will take that long. This is not to say that in 200-300 years science will be over. There will still be many interesting questions to address. But with a better understanding of quantum field theory, I think [discovery] will be a lot faster.

What could remain to be discovered after QFT is fully understood?

Most physicists aren’t trying to find a more fundamental description of nature. [Instead they say,] “Given the rules, and given what we know, can we explain known phenomena and find new phenomena, like new materials that exhibit special properties?” I think this will continue for a long time. Nature is very rich, and once we fully understand the rules of nature we’ll be able to use these rules to predict new phenomena. This is not less exciting than finding the fundamental rules of nature.

You mentioned that one indication the field of QFT is not complete is that it doesn’t yet have a canonical textbook. I mentioned this to another physicist recently, and he said a lot of people hope you’ll write it.

I tried, actually, but I stopped. Around 2000, I took one summer, and at the end of the summer I had many pages written, and I realized I hated what I’d written.

Honestly, my problem is that there are all these different ways of starting to write it, but I can’t find a preferred angle. I think it’s a reflection of the status of the field, a sign that it’s not yet mature enough. The fact that there isn’t a clear starting point, to me, is a sign that we haven’t yet found the ultimate way to think about it.