What Is Quantum Field Theory and Why Is It Incomplete?

Strogatz said he was enjoying himself. This is really fascinating. I didn't get to study quantum field theory. I got to take a quantum mechanics class. That was great. I enjoyed that a lot, but never did it again. I am just in the position of many of our audience here, just looking at the wonders that you are describing, and quantum field theory is one of them.

There is an aspect of the Standard Model that makes it hard or impossible to do on a computer. I can add a Hollywood slogan. The slogan is "things can happen in the mirror that can't happen in our world." Chien-Shiung Wu discovered parity violation in the 1950's. When you look at something happening in front of you, or in a mirror, you can tell if it's happening in the real world or in the mirror. The laws of physics show that what is reflected in a mirror is different from what is actually happening. This theory says that it is difficult or impossible to recreate that aspect.

The lattice wouldn't have a problem with the parity. I'm pretty sure it's a subtle thing.

I can tell you a little bit about why the particles in our world are made of quarks and electrons. There are two particles. They are called left and right handed. It is related to how their spin changes as they move. Left-handed particles feel a different force than right-handed particles. This is the reason for the violation.

It is difficult to write down mathematical theories that are consistent and have the property that left-handed particles and right-handed particles are different. You have to go through some loopholes. Anomalies is a type of cancellation in quantum field theory. When space is continuous, you only see these loopholes when spaces or requirements are present. The lattice doesn't know anything about this. The lattice doesn't know anything about these anomalies.

You can't write down an inconsistent theory on the lattice The lattice has to cover its ass in order to make sure that the theory you get is consistent. It does that by not allowing theories where left- and right-handed particles feel differently.

Strogatz thinks he gets the taste of it. There are some anomalies that are required to see in the case of the weak force that can be seen with the help of a topology. Something about the continuum is important.

I think you said it better than me. It's all about the structure of the universe. It is correct. It's true.

Strogatz: Alright. I think it's good. I wanted to talk about what quantum field theory has done for mathematics, because that is one of the great success stories. For physicists who care about the universe, that may not be a primary concern, but for people in mathematics, we are very grateful and perplexed at the great contributions that have been made by thinking about mathematical objects. Tell us a little about the beginning of that story.

This is a wonderful thing that comes out of quantum field theory. There is no irony here. We are using mathematical techniques that mathematicians are suspicious of because they don't think that they are rigorous. We can leapfrog mathematicians and almost beat them at their own game in certain circumstances, where we can turn around and give them results that they are interested in, in their own area of specialty.

I would like to give you some idea about how this works. There are ideas to do with geometry that this has been most useful in. The other ones are not the only ones. It is the one that we have made the most progress in thinking about as a physicist. Geometry has always been close to physicists' hearts. Einstein believes that space and time are geometric objects. Geometric space is what mathematicians call a manifolds. The surface of a soccer ball is something you can think about. There is a hole in the middle of a doughnuts. There are a few holes in the center of a pretzel. The big step is to push it to some higher dimensions and imagine a higher-dimensional object with higher-dimensional holes.

There are a lot of questions mathematicians are asking us about objects like this, such as what kind of holes they can have, the structures they can have on them, and so on. Physicists tend to have some extra intuition.

This secret weapon of quantum field theory is also present. There are two secret weapons in our possession. We have a disregard for rigor. The two combine well. When we put a particle on a space, we will ask how it responds to the space. Something interesting happens when the particles or quantum particles spread over the space. It has the ability to know about the global nature of the space because of its quantum nature. It can sense all of the space at once and figure out where the holes are and where the valleys are. Things like getting stuck in holes can be done by quantum particles. Tell us something about the space's structure.

The area called symplectic geometry was changed in the early 1990s by something called mirror symmetry, which is one of the major successes of applying quantum field theory. The four-dimensional quantum field theory was solved by Nathan and Edward Witten. Physicists will come up with new ideas from quantum field theory but are unable to prove them often because of the lack of rigor. When mathematicians come along, they usually take the ideas and prove them in their own way, introducing new ideas.

New ideas are being fed into quantum field theory. There has been a great development between mathematics and physics. We often ask the same questions, but using very different tools, and talking to each other has made more progress than we would have done.

Strogatz thinks that the intuitive picture that you gave is helpful in thinking about the idea of a quantum field being delocalized. If there is time in the theory, or if we are just doing geometry, you have this object that spreads over the whole of space and time. The fields are well suited to detect global features.

That is not a standard way of thinking in mathematics. We think of a point and a neighborhood of a point. Our friend is that. The physicists are used to thinking of global objects, but we are more focused on the surface of the object.

That is absolutely correct. The feedback into physics has been crucial. A lot of our thinking in quantum field theory is based on the idea that we should think globally. One of the more optimistic ways to build quantum computers is to use a program.

One of the most powerful ways to build a quantum computer is to use the idea of quantum field theory, where information isn't stored in a local point but in a global location. If you push it somewhere at a point, you don't destroy it because it isn't stored at one point. It is stored all at once. There is a wonderful interplay between mathematics and physics.

Strogatz said to shift gears one last time back away from mathematics towards physics. More of the constellation of theories that we call quantum field theory has been tested recently at CERN. Is this the location of the Large Hadron collider?

That is correct. It is in the Swiss city of Lausanne.

Strogatz (30:05) It is my understanding that physicists have been predicting something like 50, 60 years ago, but I am not sure what the correct word is. Disappointed, angry, confused. Some of the things they wanted to see in the experiments have not come to fruition. Being one is called supersymmetry. Let us know a little about that story. We are hoping to see more from those experiments. What should we think about not seeing more?

The people were hoping to see more. I don't know how we should feel. I can give you the story.

The world's largest particle physics lab was built. It was built with the expectation that it would be able to find the particle that makes up the universe. The Standard Model was the last part of the theory of mass causality. The Standard Model would lead us to what comes next, the next layer of reality that comes afterwards, and there were reasons to believe that once we completed the model. There are arguments that can be made that when you discover the Higgs, you should find the same energy scale and particles in the same area. There is a special particle in the universe. The Standard Model only has one particle that doesn't spin. The electron and photon are the only particles that spin. There is only one particle that doesn't spin. It is the simplest particle in the standard model.

There are theories that say that a particle that doesn't spin should have a heavy mass. Heavy means are pushed up to the highest level possible. Good arguments can be found in these ones. Materials described by quantum field theory can be used in many other situations. If a particle doesn't spin, it's called a scalar particle It has a small mass. It is mass light.

We were expecting there to be a reason why the mass of the particle is so large. We thought that there would be some extra particles that will show up once the Higgs was present. It could have been something called technicolor. There were a lot of theories. The operation of the machine, the experiments, and the sensitivity of the detectors have exceeded all expectations. People are doing an experiment.

There isn't anything else at the energy scale that we're exploring. That is a challenge. I don't know what it is. It is a puzzle to a lot of people. We were wrong about the expectation that we should find something new. We don't understand why we're wrong. We don't know what happened with those arguments. They still think I'm right. There is something we are missing about quantum field theory. It is good to be wrong in this area of science because you can finally be pushed in the correct direction. We don't know why we're wrong.

So much progress has been made from these paradoxes, from what feels like disappointment at the time. I don't want to say you could be washed up by the time this is figured out, but it's a scary prospect.

If washed up, it would be fine. I want to live.

Strogatz said he felt bad saying that.

We can think about some of the issues from the small to the large. The early universe is one of the great mysteries. When we didn't really have particles yet, you study as one of your own interests. We just had quantum fields.

After the Big bang, there was a time when inflation was called for. It was a time when the universe expanded very quickly. There were quantum fields when this happened. One of the most amazing stories in all of science is the fact that the quantum fields had fluctuations. They bounce up and down due to quantum jitters. The Heisenberg uncertainty principle states that a particle can't be in a specific place because it will have infinite momentum. It's the same for these fields. The quantum fields can't be zero or worth anything. They are always shuffling up and down.

The first few seconds are too long. The universe was expanded very quickly in the first few seconds of the Big-bang. The universe dragged the quantum fields apart after they were caught in the act. There were fluctuations there. One part of the fluctuation didn't know what the other part was doing because they were spread so far. The fluctuations are stretched across the entire universe.

The story is that we can see them now. We took a picture of them. The picture has a bad name. The radiation is called the Cosmic Microwave Background Radiation. The photograph is blue and red. There are ripples in the picture of the fireball that filled the universe 13 billion years ago. There were quantum fluctuations in the first few fractions of a second after the Big bang. We can calculate the quantum fluctuations. The fluctuations in the CMB can be measured. They concur. It is an amazing story that we can take a picture of.

There is a level of disappointment here too. The fluctuations that we see are just those from free fields. If we were able to see the fluctuations, it would be nice to know more about them. It would be great to hear about the interactions between the fields back in the very early universe. The non-Gaussianities that are there, if there are any at all, are smaller than the Planck satellite can detect.

There is hope that the non-Gaussianities might show up in the way that galaxies form, the statistical distribution of galaxies through the universe holds a memory of these fluctuations. It's amazing that you can trace the fluctuations for 14 billion years, from the very beginning of the universe to the present day.

That has given me a lot of information about the quantum fluctuations on the microwave background. I would always wonder. What do you mean by the free theory? Is there anything right? It is the vacuum itself.

The fields get excited as the universe grows. It is just a field that is bouncing up and down and is not interacting with any other fields. The points bounce up and down like springs. It is the most boring field you could think of.

We didn't have to postulate a particular quantum field at the start of the universe. It's just, that's what you mean.

The person says it's vanilla. It would have been nice to know if these interactions are happening or if the field has this property. It doesn't seem like we're there yet.

Maybe we should close with your hopes. If you had to pick one thing that you would like to see solved personally in the next few years, or for the future of research in quantum field theory, what would it be? If you were able to imagine.

The number is so many.

You can choose more.

There is something on the mathematical side. The fact that you can't discretize certain quantum field theories is something I would like to learn more about. Is there a way to get around the Theorem? Is it possible that we can somehow succeed in doing it?

The term "no-go" theorems is used in physics. You can't do this. They are often signposts about where you should look, because of the strict assumptions that come with a mathematical theorem. Maybe you can make progress on that by throwing out the assumption. Progress on the mathematical side is something I would like to see.

New hints of what lies beyond are some of the things that we have talked about on the experimental side. We are getting hints fairly frequently. It seems strange that the mass of the W boson on your side of the Atlantic is different from the mass on my side of the Atlantic. There are hints about either dark matter or dark matter. It's made of quantum fields. There's no question about that.

The dark energy is too strong a word, but there are suggestions from quantum field theory. That is much larger than we are actually seeing.

The same puzzle is there with the Higgs. There is a question about the light nature of the Higgs. There is dark energy as well. We don't understand why the universe is so small compared to what we are. It is a strange situation to be in. We have a theory. It is absolutely incredible. There are things we don't know.

Strogatz would like to thank David Tong for the wide-ranging and fascinating discussion. I would like to thank a lot of people for joining me today.

Thank you a lot.

Susan Valot, one of the producers of The Joy of Why, is the host of the Quanta Magazine SciencePodcast. Tell your friends about the show and follow where you listen. It helps people find things to listen to.

Steve Strogatz is the host of The Joy of Why. The selection of topics, guests, and other editorial decisions are not influenced by funding decisions made by the Simons Foundation. The Joy of Why is produced by two people. The editors are John Rennie and Thomas Lin. The theme music was written by a man. The artwork for the episodes is done by Michael Driver and Samuel Velasco. Steve is your host. Please email us at quanta@simons foundation.org if you have any questions. Thanks for taking the time to listen.