(bright music) - Welcome back to Conversations at the Perimeter. I'm Colin and I'm here with Lauren, and we are thrilled to share our conversation with Francois David. Francois is a mathematical physicist, which means he tackles really hard problems of physics, like quantum gravity, using a mathematical toolkit, and I have to admit that's a toolkit that I didn't have a lot of experience with growing up.
So I was a little apprehensive going into this conversation, but thankfully, Francois is a very gifted teacher. - Francois was actually one of my teachers when I first came to Perimeter as a grad student in the Perimeter Scholars International master's program, and he's been coming to teach in this program from France for many, many years and he has an amazing reputation among the students.
I'm now actually an instructor in that program myself, and so I've been able to interact with Francois, both as one of my teachers and now as a colleague. - So what's it like for you to put Francois in the hot seat now, where you ask all the hard questions and he has to answer them? - Honestly, it was a really different experience because, back when I was a student, I was usually too nervous to put my hand up in class and ask questions.
He even mentions during this conversation that he remembers I always had a lot of questions, but I know that I would usually stay after class to ask those around just a smaller group of students, and so this was really different, that I got to ask questions and share the conversation with so many others. - And for me, that apprehension I had off the bat, it melted away so quickly when I realized just how much he loves physics and how infectious his love is for it.
I'm excited for other people to get that sense of the joy of physics and math from Francois, so let's step inside the Perimeter. - Thank you so much, Francois, for joining us for a conversation today, and it's great to have you here at PI all the way from France. Would you mind telling us a little bit about what you do as a mathematical physicist and what it means to work in that field? - Well, first, thank you very much for this invitation to this kind of interview.
That's my first experience in this, almost my first. Okay, about my experience as a mathematical physicist, but I must say that I don't really know exactly what is mathematical physics, because it depends a bit on the country, on the culture, or the person. So I am partly a theoretical physicist and partly a mathematical physicist or both. And mathematical physics is a field of research. There is no real border, but interface between mathematics and theoretical physics.
Mathematical physicists are more involved in using recent and sophisticated mathematical techniques and ideas because mathematics are way much than just techniques of calculations. They are concept, ideas.
So mathematical physicists are more interested in the structure of physical theory and understanding how that works, what one can tell out of the mathematics that governs the physical theory, and understand, often on simple models, not always, but they take a simple model, not often directly related to some real physical systems.
It may be, but they're often idealized in order to keep track just of the important physical feature they want to understand and working out, as deeply as possible, the math and the theory and see what comes out. Are those theoretical models consistent, for instance? That's very important. Can we compute exactly and prove properties of this model, or are we just able to use what are called phenomenological model?
So one makes assumptions, some approximation, and then one relies on calculation and also physical intuition, and often it works, but sometimes it doesn't work. You really have to work hard and do hard math and some deep, and sometimes unexpected results come out. So that's mathematical physics. - Francois, you used the word consistent there to describe the research. Does consistent mean that an idea is true, or that it's true enough for now, and is inconsistency an enemy of science?
- In my mind, consistency is a mathematical consistency. It's related to another concept, very important for some physicists, not all of them, but it's a mathematical beauty of a theory. So it's something which was very important for Paul Dirac, one of the creator and inventor of quantum mechanics, who considered that a theory had to be true if it was beautiful.
This led him, for instance, to discover the Dirac equation, though often, beauty is associated to mathematical consistency in the mind of mathematician and in the mind of many theoretical physicists. There is something which is more than just mere beauty because some very simple object can be very beautiful. Consistency means that, often in theoretical physics, one needs to start with some assumption. There is space, there is time.
For instance, one important assumption is there is no difference between the future and the past, which seems a bit, of course, contradictory with our daily experience, but that's the deep principle of, nowadays, theoretical physics. So one makes assumption, let's say what physical problem or physical system is described by one makes some assumption.
One assume the rules, for instance, the rules of classical mechanics or the rules of the law, other than the rules of the law of quantum mechanics, the law of hydrodynamics, the law of classical physics, Newton Law, et cetera, and one see, whether building out of that, one doesn't run up into some mathematical inconsistency. Sometimes it's easy to see that there should be some inconsistency in some direction, so don't look in this direction.
Look in the problems where inconsistency doesn't appear. And sometimes the inconsistency appears in a surprising way. And of course, if you run into a mathematical inconsistency, it means that you are to think more. Either one of our assumptions was wrong, or it might be a paradox, but not a real inconsistency if you work out enough. Science and knowledge progress by making errors. If everything was clearly understandable and consistent from the very beginning, it wouldn't be interesting.
- And could I also say maybe that, if in physics we often tend to start with assumptions and, as you said, sometimes those assumptions might lead to inconsistencies and sometimes not, would a goal of mathematical physics be to provide more structure to those assumptions so that there may be, at some point, no longer assumptions? - Yeah, this happens, too.
Sometimes you start, from assumptions, you work or after some other researchers come out from different field or different ideas, or even some mathematicians come out also, and when discovered that those assumption were were correct, it was not coming from some naturalness or intuition that things should be that way. It comes out that they had to be this way. And that's a difference between often, one start by, oh, things should work this way or that way.
And then you may have different theory, which start from different point of view. After working often very hard by a team of very different people, one comes out of that that, in fact, oh, things had to be that way, this way, and not that way, or sometimes, oh, things had to be this way and your two approach were seemingly contradictory, but consistent. One time, this happens in the early days of quantum mechanics, very often, where people were starting from some kind of wild assumptions.
- I often hear mathematicians talk about the sense of beauty in mathematics, and that's a beauty that, personally, I haven't been able to experience because I grew up a little bit afraid of math. Can you describe the sense of beauty that you see in mathematics? - I'm not a mathematician, so I won't speak as a mathematician, although I know some mathematics.
I was educated in mathematics since the French high school, and the university system is more focused on mathematics than in other countries. Also, I married a mathematician and two of my daughters are mathematicians.
My impression is that mathematicians see beauty in simplicity of structure, but consistencies of structure, objects can be mathematical, theories can be complicated, but there is some underlying structure which enables you to come out to theorems by abstract reasoning, not just heavy and technical calculation. Although they are also very important, they also both in theoretical physics, science in general, or in mathematics, you see simplicity after a lot of hard work.
It's a bit like digging an archeological dig. You find some beautiful archaeology, but you had to work, work, and once you find something, you say, "Oh, but I should have looked in this direction," come to the results very easily, but of course, you just know because you worked hard. So that's my feeling of what a mathematician feel about beauty. So one of my daughter is a mathematician.
She's doing algebra, geometry, a number of theories, and she said, "I prefer math to physics because in math, "we are dealing with objects we have created ourselves "and so we know it's consistent, while in physics, "there is some external world and we start from that." We want to understand the universe, we want to understand how a cell works or how the solar system works or why there are chemical reactions, and that's something which is given to us or which is there for us to understand.
That's probably one reason why I prefer to be a physicist than a pure mathematician. So probably my brain prefers to be a mathematician. That's why I'm a mathematical physicist, but my curiosity or my intuition prefers to have surprises coming from where we live. Especially here, you have a group of very good people working with the foundations of physics and the foundation of some philosopher, too.
They will be able to tell more, but it's unclear whether the mathematics are part of the real world or something completely outside. That's a view of many mathematicians, that mathematics exists by themself. This is more considered, mathematics as a tool. There is a debate that goes back to the great philosophers about what are mathematics and physics, since they are intertwined since they were created or discovered.
- From what you say, I mean, you're giving us a nice description that mathematics involves some beautiful structures that we can create, and physics is about describing these really interesting phenomena in our world, so maybe mathematical physics is working from both of those ends to give some structure to the universe, and oh, maybe that's not correct, but- - No, I think that's a good view, but I'm not an historian of science, but many of the mathematical object were created
from the real worlds and then evolved on their own, and some structure of the real worlds have been discovered through mathematics. - And is that why we need mathematical physics, so that we make sure that those two ends are talking to each other? - The interface has been there. It has been important, depending on the historical period in science and also on the countries, but the interface has to be.
Otherwise, there won't be good physics without mathematics, of course, because I think Galileo stated, one of the first to state, that mathematics is a language, physics. Also, a lot of mathematicians now, not all of them, but of course, it depends, get inspiration from physics, and the ideas which, somehow, a bit clumsy ideas, created by theoretical physicists, common mathematics, challenge things, and then come back to physics as a neat tool and with new ideas provided mathematicians.
There are many examples that one can think, but a few in the last decades. - So mathematics, you said, is a tool that we can use to make progress in big problems in physics. So what are some of the big problems in physics that you are trying to tackle using mathematical techniques? - I've been very much interested.
In fact, I realized all along, my career, not only this question, but about random geometry, let's say starting from geometrical objects, and see what's the role of randomness, and one of my interests in that comes from quantum gravity, so quantum physics and gravitation. Theory of gravitation has been born with Kaplan, Newton, all the great mind in the 19th century.
Then Einstein discovered that, in order to make habitation compatible with the theory of relativity that he discovered in order to understand the behavior between light and matter, no habitation, he discovered that, in fact, spacetime orders that you shouldn't consider space and time as two separate notion or entities, but they have to be taken as a part of spacetime.
Einstein discovered that, in order to formulate the consistency of gravity, the spacetime itself as a internal structure, it has a metric and it can be a geometrical object. In fact, it is a curved object. All of spacetime, so both space is curved. Usually, you often form this fact. You said that you have flat space, you put the body in it, like the sun, and it curves the space.
And then therefore, it's like a ball, and you can have a marbles way to explain empirically why the planets orbit around the sun. The theory of general relativity of Einstein says that, also, time is curved, and that's something which is more difficult, too, that it's space and time which are curved, not only space. Productivity tells us that, in fact, time is associated to space, so times has to be considered as a separate time at different points in space.
When you start to compare what's happening, when you go to a different place, you let run time and then you come back at the same place, you discover that space behaved in a different way that you could have expected if time was something uniform, like in Newton theory of time, especially when there is a gravitational field.
If you have a black hole and you are far from the black hole, or if you go close to the black hole and come back or close to the sun and then come back, then time has very differently approach a black hole. You come back, then the clocks are desynchronized. There was a very nice example of that in a movie, this "Interstellar." This is checked in laboratories, not going near black holes, but just having two atomic clocks.
As you raise one of the atomic clocks by a few meters, drop it back on the table where it started from, and you can see such effects, tiny effects, but they are measurable and I agree with the theory. Now come quantum mechanics, great discovery of last century. Einstein also played a role, but less central, compared to relativity. And in quantum mechanics, some very special kind of randomness, rather than randomness, one choose it. The role of chance is very important.
There is some indeterminacy. You are never sure of what the results of a measurement will be, but this randomness, in some senses, uncertainty is governed by mathematical role which are very, very precise, so it's not randomness just because we don't know exactly what's going on.
When you are interested in, for instance, the theory of quantization of gravity, one of the great problems nowadays of present physics, you have to treat spacetime as a curved object, a curved spacetime, but with some randomness coming from the quantum nature of the universe.
And we know that, for consistency, this idea of consistency, the beauty of the theory, the geometry of spacetime, the curvature of spacetime, has to be treated as a random object, but an object with randomness agreeing with the law of quantum mechanics, if, indeed, gravitation is consistent with quantum mechanics, and we don't really know if they are consistent.
We hope that it's consistent, we are trying to make a consistent theory of quantum gravity, but maybe we'll come up into an inconsistency, which means that we will have to build a new theory of nature, which will be post-quantum and post-gravitational. - So quantum gravity, it's essentially the quest to reconcile two theories, quantum mechanics and general relativity, and to come up with a bridge between the two?
- We need to have a consistent physical theory, which leads us to a complete understanding of quantum mechanics and a complete understanding of gravity. We have to build such a theory. Some physicists think that it's not necessary, that we can still live with those two theories, but the vast majority thinks that, for just this reason of consistency and beauty, in the sense of logical consistency, there has to be such a theory. It depends with whom you talk, though.
There are several direction of research, and it's a very active subject, in part, well represented here in the Perimeter, of course, and there are many different ideas. Some are mathematically well-developed, some are less and more rely on intuition or some toy model.
The two main ones are string theory, and the other one is based on still treating the geometry of spacetime, how four-dimensional spacetime as some basic data and quantizing it according to the law of quantum mechanics, while string theory is wider and more speculative. - A lot of your contributions are specifically to two-dimensional quantum gravity, and we had a really good question sent in from Tebra in Bangladesh- - Ah, okay, yes. - So maybe we can listen to his question.
- Hi, Francois, this is Tebra. I'm a theoretical physicist based in Bangladesh. Of course, you and I know each other, so this is for other people, other listeners. Anyway, I have a question for you.
Recently, there have been some buzz in the physics circle about your work in two-dimensional gravity and how that has helped breakthroughs in recent years, so I was just wondering if you could explain in general terms what your contribution was to the field of two-dimensional gravity and how that contributed to recent breakthroughs in two-dimensional gravity. Thank you for listening and thank you for your answer. - Thank you, Tebra.
I've been specifically interested and worked and got some interesting results in a subfield of quantum gravity called two-dimensional gravity. It's both a toy model and a very interesting model for some physical application. It's a model which is very much simplified, a core model where you can study one aspect of the physics. - But the idea would be that, by working with this toy, we can still gain some insights that will still help us to understand the more complicated system?
- Yes, and so an example of a toy model, which is a very useful example for studying quantum gravity is to consider that spacetime, instead of having three dimension one time, or as in string theory, nine or 10 dimensions of space and one dimension of time, or maybe nine dimension of space and two direction of times, would consider a very simplifying model of spacetime, where you have one direction of space, so space is just a line, and one direction of time,
so spacetime is just a sheet of paper. So it's a very simple model, and you lose many aspects of habitation theory. In particular, you lose a very important aspect of your operation. You lose the law of attraction, Newton's Law, for some technical reason. So you have no habitation anymore, but you have geometry because a sheet of paper can be curved. If it's a rubber sheet, it has curvature, so you keep one of the basic point, that spacetime is curved.
So you can quantize it and you can study the quantum effects. In particular, that's the simple case where you can build a consistent quantum model of gravity, and you can build a theory on simple axioms and compute things and go to the end of your calculation and get insights about what quantum gravity could be, or some aspects of quantum gravity could be or could not be.
So working with a two-dimensional model or either one-plus-one-dimensional model, spacetime, rather than two four-dimensional, three-plus-one-dimensional spacetime is very important and is very interesting. And I've been working, I think, since the 80s, by some period on those models. My contribution in this idea, I've been twofold. I've been one of the first to implement the idea that, instead of taking a continual spacetime, you can approximate it by a discrete object.
Typically, you can see that you can build a surface out of taking triangles, flat triangles, but gluing them, and if you glue them in a proper way, you can build polyhedra. So you can build curved surfaces or curved spacetime out of discrete objects and realize the quantum nest of a quantum spacetime by looking at the common matrix of this construction you can make by building what's called triangulation.
If you glue a triangle, you build a triangulation of a surface or you build a discretized surface or a discrete surface, and treating this object at quantum means look at the status, see that's a surface. That's a typical, average size, average shape, average curvature, or such an object, and it seems they're naive and simple ideas, but it was motivated by the fact that this procedures is now to work already in quantum physics without gravitation.
When this idea was introduced, it was in the 80s. Theoretical physicists had introduced what they called lattice gauge theory, discretized theory of the strong interaction, for instance, but on a discrete spacetime by extension and energy, we put high in it. Other theoreticians and some mathematician, too, started to look at can you make this idea working for very simple, one-plus-one theory of quantum spacetime?
And it turns out that you can work and make calculation in these toy models using mathematical theory, which came out from something completely different, which is called the theory of random matrices, which comes from the study of quantum systems, which are very complicated dynamics.
So not toy models, but very, very complicated models, and looking for whether they still exhibit some universal feature, which are there because the system are very, very complicated instead of being very, very simple. - The idea of a toy model, is it akin to building a toy car with just a wooden rectangle and four round wheels, making sure it rolls, and then eventually, gradually adding more and more features until you've got a sports car?
- If we didn't have the toy model to think about, it would have been very difficult to find in the very complicated system. So that's one aspect of the toy model, but then I could say that there are other kind of toy models, which is exemplified by this idea of random matrices. Want to explain, but think about the matrices are just a table of numbers, like an Excel spreadsheet, where you can add them. You know that you can add the cells up in a spreadsheet, but you can also multiply them.
More complicated, but the mathematician and physicist know very well what it means. And so order comes out of complexity, or to mention a word of a famous physicist, E. W. Anderson, the sum is more than the parts. It appears, for deep mathematical reasons, then if you take a very complicated object made out of simple objects, instead of it becoming just a mess, it becomes something which exhibit very simple feature.
Some universal behavior comes out of complexity, and the property of the sum of the subject is not just emerging from the properties of the small parts. It's come out from the rule. This is also an idea which is important, for instance, in quantum gravity. Many suspect that, in fact, the fact that we have a smooth, neat spacetime with a bit of curvature, explain gravity.
It may come out from something at quantum scales and below at some post-quantum scales, which is completely different and maybe random, both the idea of taking toy models to understand the real systems and taking complicated system to understand what's going on for large systems. There are two trends in common, not in completely not incompatible ideas, which are very important in modern and theoretical physics, because then you can make toy models of very complex system and study them.
That's the idea of those fundamentals, but they are simple, complex models. - And I wanna go back to a word you said a little while ago, which is this word universal. You said sometimes in these systems, you can end up finding something that's actually universal, so can you tell us what that means? - Universality means that, out of very different systems, exhibits the same behavior, although, in some sense, this behavior is universal.
This concept, which is now one of the very important concept in theoretical physics, come out, not from high-energy physics, not from gravity. It comes out from condensed matter. It led to the discovery or the creation of a theory which is called the theory of homogenization group, but forget about the group. You have some different physical system, completely different, which in fact, exhibit, in some regime, exactly the same behavior.
If you study the behavior of ice and water, water can be a liquid, can be solid, and it can be a gas. Usually, it's one or the other, but there is a very special point when you have water at a very specific temperature and a very specific pressure. You reach what is called a critical point, where water is neither a liquid or a gas, it's both. At this point, there are huge fluctuations of pressure and density. These behaviors occurs for water, but it occurs also for other gases.
In fact, it's better studied in other gases or other liquids. Usually, you have this function. You heat water, and at some point, it boils. It's very simple, suddenly, vapor starts to happen, so it's called the first-order transition, but if you increase the pressure, there is a point where the transition becomes smaller and smaller. At some point, it disappear. It turns out that you have system of magnet.
I don't know if, in high school, you might have done the experiment that you take a magnet. So the magnet has some magnetic property. And if you heat a magnet, you put it under a Bunsen flame, at some point, the magnet stops being a magnet. It's just a dull piece of metal. So there is a critical temperature where a magnet stops being a magnet, and it turns out that the property of this magnet are the same or very similar to the property of water. That's very strange.
This has not been understood for many years, and in the beginning of the 70s and the 60s and end of the 70s, physicists working in condensed matter understood why this occurs, but they understood, thanks to one of the great high-energy physicists of that time, Ken Wilson, who started being interested in what's called critical phenomena. He built out of high ideas, which came from high-energy physics, the concept of randomization transformation and what's called now randomization group.
The idea is that, if you start from a system, for instance, which is described at microscopic scales by a collection of atoms, atoms can behave as small magnets, very little magnets, in fact. That's the origin of magnetism. You have atoms, you have electrons turning around, and the electrons have a magnetic moment. In the addition, they create magnetic moments because they go around the nuclei of the atom, et cetera.
Okay, anyway, so that's the origin of magnetism, but if you start from the magnet described just by its microscopic structure at the atomic scale and you start to look at what are the properties of this magnet, if you go at larger and larger scales, so changing the scales or making some averaging, the magnetic property of a magnet, instead of looking at whatever magnet you see at the property, at the scale of an atom, you see a cube, 10-by-10-by-10 atoms,
and you see what are the properties of this magnet. - Like zooming out on a picture? - No, it's exactly like zooming out, but zooming out being defined in a proper mathematical way. - (laughs) Right. - And if you do that, it was discovered by Ken Wilson and explained, and the other physicist working in that field, that this posed view sometimes converges in substance.
You zoom out, you zoom out, you zoom out, and when you have zoomed, you find something which is the same kind of object, wherever you were looking, at a magnet or at a fluid, where you could say, "Okay, this tiny region "of space can be either a liquid or a gas." So if you want, you would take the molecule of your water, and either they are very closely packed and they are connected by hydrogen bonds or there, they can wander around so they form a liquid. So it's exactly the same thing.
You take very different system, sometimes complicated objects, so the dynamics can be complicated, can be simple in your toy model. It can be complicated in your model. You zoom out, you zoom out, you zoom out, and if you go zoom out enough, sometimes you find the same object. So in this sense, simplicity or beauty is emerging by zooming out what's going on in the complicated system. So this is the idea of universality, which is very important in physics.
When you normalize, you average and see what has a property. This creates some kind of norm, and renormalization means that you normal the scales and you change the scale. You renormalize, and you change against the scale. You will renormalize, et cetera, et cetera. So you have this idea of toy models and this idea of normalization, so that the simple phenomenon come out of very complicated object, and irrespective of the detail of what's going on the small scales.
- And it seems, Francois, like some of these tools, like renormalization group or random matrix theory, they've allowed you to study quite different problems. You've talked just now about some problems in condensed matter. You were telling us about quantum gravity. Would you mind maybe telling us the story of your career and maybe the different problems that you've looked at along the way?
- Yes, in fact, I realize that this concept of universality and normalization group has been one of the guiding line of my research. Those tools were created when I was in high school, so I learned them when I started. I was a graduate student, and I've been trying to improve them and apply them. So I started in high-energy physics and theory, and then I started being interested in whether I could apply those idea to condensed matter.
And then when I was a post-doc in Princeton, I came in contact with a researcher working in quantum gravity, this idea of discretizing spacetime, and so I applied it to quantum gravity.
So I started to study this idea to work in quantum gravity, so I studied mission model, a bit of higher dimension, but this doesn't work so well, and then I came in contact with another field of theoretical physics, which is biophysics, in fact, and one very specific subject, which is the study of membranes who have two-dimensional themes in three dimensions, because when I was in touch with young physicists, visiting (indistinct), and one got a position and they were working in that field.
And this idea of universality is very important because, by discussing, we discovered that, in fact, some models of quantum gravity in two dimension and some models of membranes were very similar. They had some difference, in particular as a whole of bending in two-dimensional gravity. Bending is not important. Well, it's very important in a physical membrane. So I've been working in this concept, studying the physics of what's called fluid membranes and then crystalline membranes.
This was a very exciting field and it's still important, but then a few years later, there was some great progress in the theory of quantum gravity and in string theory, made by a group of theoreticians, especially Russian ones, this Russian school with Migdal, Polyakov, and we made progress in the two-dimensional quantum gravity, so I came back to that field.
And I was there, more interested in not discretizing spacetime, but taking continuum theory of two-dimensional gravity, a theory which was, well, created and invented by Polyakov, which is called Liouville theory. Liouville is a famous French mathematician from the 20th century. He was mostly a number theorist, but some of his equation were important in quantum gravity. So our model was neutral gravity, which is connected to string theory.
It was developed by this Russian school, and that tends to be known as the Liouville theory, but there are other theories up to Newton's quantum gravity, like Kiev's Titan Boom model and some other one, but one is the Liouville theory, and so I've been working on that. After that, I came back to quantum metric theory for several years and was interested in that, in particular for quantum cows, because quantum metric theory has application to quantum cows, and then I came back to quantum gravity.
- The first time we spoke, you used the term journey to describe your career, and you said that theoretical physics requires all sorts of different minds, so what kind of mind do you bring to the journey of theoretical physics? - I would say there are different kind of minds in theoretical physics. I'm not completely sure which mind I am. Some likes to wander around. I'm still a bit stubborn, so I come back to old problems. When I'm stuck, sometimes I look elsewhere, but I always come back.
I have some problem in my mind that I have them since 20 years. I'm just waiting for the good idea, if any, or if someone had a good idea to solve it, they are still there. - Some of these problems that you've described to us are incredibly challenging. Some of them are so difficult that they may not see a solution in our lifetimes, possibly ever. Francois, given the hugeness of these challenges, what keeps you going? - Well, I think that's curiosity.
As long as I've not understood something, I like to think about it. I feel disappointed. I feel the failure of not having made progress in a field. If someone else made the progress, that's fine. I said, "Okay, I was not smart enough. "I didn't have the idea." There is change in research. Sometimes you just have a good idea at a good time and sometimes you had it too early, and you couldn't make out something of it.
- So Francois, we also got a question for you that was sent in from one of the students that you're currently teaching within your quantum field theory course within the Perimeter Scholars International program. Let's play the question from Anna Kanur. - You teach a course on quantum field theory, and one of the topics is ghosts. Without writing down any integrals, how would you explain what these ghosts are?
- Well, the denomination ghost has been given by the physicists who created this concept. It was a Russian, Faddeev and Popov, but I'm not completely sure anyway. Physics likes to find nice names when they have new objects. Sometimes the names are well-suited. Sometimes they are silly, but okay. So ghosts, in fact, are articles in a quantum theory with probability to be there is negative. If you think about probability, it's a very important tool of mathematics.
And the probability of an event, if you have some uncertainty on something happening, for example, if you play coins and it has a probability 1/2 to be heads and 1/2 to be tails. Okay, if the probability of some events is one, it means that it's certain, you are sure. If it's zero, it means that it never happens. So the probability are numbers which are in between zero and one, or 0% and 100%. You cannot have a probability of two.
The sum of the probability of all realization of an event has to be one, 'cause something happens. Whatever it is, you're sure it's going to happen. If you have a head and tails, 1/2 plus 1/2 is one. In quantum theory, whether it's change and uncertainty, you can calculate probabilities of something to be measured, and so some of the probabilities of all possible outcomes of the experiments or measurements has to be one.
In the inconsistent quantum theory, the sum of probability is one, it's called unitality, but it turns out that, in some quantum theory, you get probability two and probability minus one, but it's not a physical theory because you have a probability, for instance, to get a particle created, which is minus one. When you have a theory which such particles, they are called ghosts.
Sometimes when you make a theory and you get probability which are negative or greater than one, that's an example of an inconsistency. - I was gonna say, it sounds like something that must bother mathematicians. - It bothers mathematicians and it bothers physicists, too, (Lauren laughs) of course, because there are many theory of quantum gravity which have ghosts. The first theories of strong interaction are the ghosts.
Most of the ghost's particles, when you see them, it means there's a theory, you can put it aside and start with a better theory. In the lecture that I gave, it's a theory where you try to quantize the theory of strong interactions.
In this theory, well, you run into technical difficulties, and one way to deal with this difficulty and to solve the problem is to introduce a fiducial particle in the theory, which precisely has this property of having negative probability to be observed or larger than one probability to be observed. The fact that you have to consider those parts of those kind of ghostly particle when you make calculation comes out from the math.
So they have to be there, but when you work out more on the theory, you see that you can never observe them. They are virtual particles that are there in the quantum vacuum of the theory, or when you make calculation, you have two particles. You sew them on together in accelerator, you have a quantum theory that this type of what's going on when they interact, and you have a lot of virtual quantum process.
And then there is an outcome, some other particles, two, three, four, many, because you can create particles, come out. When you do the calculation, you see that you never see any of those ghostly particles. So those ghostly particles are there in your calculation. So in some sense, if you are mathematician, you see if it's in the calculation, it's something that exists, but you can never observe it. So in some sense, it's a feature of the calculation.
In some sense, they are like imaginary numbers in algebra. I never thought about this analogy, but I think it's a good analogy. Imaginary numbers are numbers a bit like real numbers, but the most important imaginary number is called i for imaginary, and i is a number so that its square is minus one.
So in some sense, you can say it's not a real number, but now when you are in high school, you learn about imaginary numbers because they are everywhere when you do calculation in math and in physics. And in fact, imaginary numbers were invented by, I think, Italian mathematician in the 15th century to solve a quadratic equation, algebraic equation that mathematician were solving since the Greek and the Egyptians and maybe the Babylonians.
Okay, and in order to find the solution of equation involving real numbers, they discovered that it was not quadratic equation, in fact, but it was question of degree theory. Anyway, so algebraic equation, they discover that it was very convenient to introduce this number where the square is minus one and consider it as a real number. Just make calculation and consider it at par with a number we're used to at that time.
And so now you discuss with a mathematician or with a physicist, or even with the engineering. Those are useful when you study electric currents. Well, they said, "Okay, well, i is a number, "as one or minus one." They treat it as just an ordinary number, although if you measure something, if you measure lengths, you measure an electric current, you are never going to find object where the length is minus i one meter or one inch.
So ghost particles are similar, particles that you never observe, so in some sense, they do not exist, but if you introduce them and treat them in your calculations, they'll obey the same rule. For instance, i is maybe considered as a ghostly number, The first ghostly number ever- - (laughs) Okay. - To be considered. One shouldn't be too much afraid about ghosts.
- (laughs) Good, and Francois, you've been teaching here at Perimeter for more than 10 years, teaching students about ghosts and quantum field theory, and actually, I wanted to share that you taught me many years ago when I was a student in this program. - Yes, I still remember you. (Francois laughs) - (laughs) You remember. - Which means that you ask question. - I ask, oh, good, well, I'm still asking questions now.
(laughs) I wanted to tell you I still remember, there was one day after one of your lectures where a group of my classmates and I were talking, and one student came over and he said, "That lecture by Francois today was just perfect." He said, "There's no way that anyone could have been "in the room and not understand "everything that he wrote down," and I never heard him say that again about any other lectures, (laughs) so yours was definitely one of the best. - Okay, great, thank you.
- And we have one more question about your teaching, in fact, from another student from a few years ago that you taught. - Hey Francois, this is Farthi from PSI, 2019 Class. I was wondering, actually, when did you realize that you loved teaching? Would you mind telling more about your journey into becoming a teacher? - Good question, in fact, I realized I love teaching when I started teaching.
I don't know if it's a chance or an unfortunate fact to get researcher position in France at CNRS when I was a young scientist. From start, I didn't have any teaching duty. It's good to teach, but I had all my time for doing my research, and I know that most young scientists nowadays in France and everywhere, they have to teach. As long as they have to teach a reasonable amount of time, that's okay, but often, it's too much. So I had this great chance and I think this helped me.
So I was not especially looking for doing teaching, but I was offered first in France, whether I was already older, to give some lecture at a level of master or graduate school. I realized that I liked it. So I had the chance, in fact, to teach first in France in Ecole Normale with a group, it was for about more than 15 years, some course in application of quantum theory to structural mechanics. This has been a very good experience because the students were some of the best student in France.
Then I was offered this change. One of the greatest experience in my career to teach at PSI, which was really great. Well, first I discovered a new research institute, Perimeter Institute, which was still in the phase one building. I discovered entire different worlds of students coming from many, many different countries with different backgrounds.
This was different from teaching in France, where I had very, very good students, but somehow, more from the same mouth, very good mouth, but the mouth of French physics educational system in Paris. So this was complimentary. It was an international problem, where, in France, we mostly had French students. Well, now this has changed in the last year.
It's really European, but here, it was the first time I had student from Africa, South Africa, Far East, and this mixture and seeing how the students were interacting together, how the Perimeter was accommodating them, taking care of them, also having a decent proportion of women compared to men. Great things about this program. This was a discovery for me. - Francois, I'd actually like to read something that you wrote a couple of years ago.
It's from a book that Perimeter Institute put together to celebrate the 10th anniversary of the Perimeter Scholars International program, the PSI program, which you've been involved with since practically the beginning. You wrote, "Every year was memorable, "with a special remembrance for the adventures "and heroic first years in the old post office." The old post office, by the way, was Perimeter's first building, just a few blocks from where we are now.
You wrote, "The old post office building, "with its sofas and the billiard table "and the big coffee machine, "an evening spent preparing the next day's tutorials. "Long life to the PSI program "and to all the students who have benefited from it." Now I just thought that was a beautiful sentiment in the book, and now there are a lot of students after 10 years who have benefited from that PSI program. What keeps you coming back year after year to teach, and what do you get out of it nowadays?
- Well, I come because I'm very happy to come. I think it's a chance for me. I hope the students still enjoy it, but I consider it as both a privilege and this bring me happiness teaching in front, enjoying the students. Very interesting group, all the interacting with the other lecturer and teacher.
Well, last year and this year had been much disrupted by pandemics, and also, seeing this, that's an opportunity for me to visit the Perimeter as a scientific research institute, which is a great, new, vibrant place for doing theoretical physics. - Great, well, we're really glad to have you here and part of the teaching here and the research community. Thank you so much for sharing your time with us today. - Thanks. (bright music) - Thanks so much for listening.
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