Cumrun Vafa: String Theory | Lex Fridman Podcast #204

The following is a conversation with Kamaran Vafa, a theoretical physicist at Harvard specializing in string theory. He is the winner of the 2017 Breakthrough Prize in Fundamental Physics, which is the most lucrative academic prize in the world. Quick mention of our sponsors Headspace, Jordan, Harmony's show, Squarespace and Allform. Check them out in the description to support this podcast. As a side note, let me say that string theory is a theory of quantum gravity that unifies quantum mechanics and general relativity. It says that quarks, electrons and all other particles are made up of much tinier strings of vibrating energy. They vibrate in 10 or more dimensions depending on the flavor of the theory. Different vibrating patterns result in different particles from its origins for a long time string theory was seen and as too good not to be true, but has recently fallen out of favor in the physics community, partly because over the past 40 years it has not been able to make any novel predictions that could then be validated through experiment. Nevertheless, to this day it remains one of our best candidates for a theory of everything or a theory that unifies the laws of physics. Let me mention that a similar story happened with neural networks in the field of artificial intelligence, where it fell out of favor after decades of promise and research, but found success again in the past decade as part of the deep learning revolution. So I think it pays to keep an open mind, since we don't know which of the ideas in physics may be brought back decades later and be found to solve the biggest mysteries in theoretical physics. String theory still has that promise. This is the Lex Friedman podcast and here is my conversation with Kamran Van what is the difference between mathematics and physics?

Well, that's a difficult question because in many ways math and physics are unified in many ways, so to distinguish them is not an easy task. I would say that perhaps the goals of math and physics are different. Math does not care to describe reality. Physics does. That's the major difference. But a lot of the thoughts, processes and so on, which goes to understanding the nature and reality are the same things that mathematicians do. So in many ways they are similar. Mathematicians care about deductive reasoning, and physicists or physics in general, we care less about that. We care more about interconnection of ideas, about how ideas support each other, or if there's a puzzle discord between ideas, that's more interesting for us. And part of the reason is that we have learned in physics that the ideas are not sequential. And if we think that there's one idea which is more important and we start with there and go to the next idea and next one, and deduce things from that, like mathematicians do. We have learned that the third or fourth thing we deduce from that principle turns out later on to be the actual principle. And from a different perspective. Starting from there leads to new ideas which the original one didn't lead to. And that's the beginning of a new revolution in science. So this kind of thing we have seen again and again in the history of science. We have learned to not like deductive reasoning, because that gives us a bad starting point to think that we actually have. The original thought process should be viewed as the primary thought. And all these are deductions, like the way mathematicians sometimes does so in physics. We have learned to be skeptical of that way of thinking. We have to be a bit open to the possibility that what we thought is a deduction of a hypothesis. Actually, the reason that's true is the opposite. And so we reverse the order. And so this switching back and forth between ideas makes us more fluid about deductive fashion. Of course, it sometimes gives a wrong impression. Like, physicists don't care about rigor, they just say random things. They are willing to say things that are not backed by logical reasoning. That's not true at all. So despite this fluidity in seeing which one is a primary thought, we are very careful about trying to understand what we have really understood in terms of relationship between ideas. So that's an important ingredient. And in fact, solid math being behind physics, is, I think, one of the attractive features of a physical law. So we look for beautiful math underpinning it.

Can we dig into that process of starting from one place and then ending up at the fourth step and realizing all along that the place you started at was wrong? So does that happen when there's a discrepancy between what the math says and what the physical world shows? Is that how you then can go back and do the revolutionary idea for different starting place altogether?

Perhaps I'll give an example to see how it goes. And in fact, the historical example is Newton's work on classical mechanics. So Newton formulated the laws of mechanics. The force F equals to MA and his other laws. And they look very simple, elegant, and so forth. Later, when we studied more examples of mechanics and other similar things, physicists came up with the idea that the notion of potential is interesting. Potential was an abstract idea which kind of came. You could take its gradient and relate it to the force. So you don't really need a priori, but it's solved, helped some thoughts, and then later euler and Lagrange reformulated Newtonian mechanics in a totally different way in the following fashion. They said, if you want to know where a particle at this point and at this time, how does it get to this point at the later time is the following. You take all possible paths connecting this particle going from the initial point to the final point, and you compute the action. And what is an action? Action is the integral over time of the kinetic term of the particle minus its potential. So you take this integral and each path will give you some quantity. And the path it actually takes, the physical path, is the one which minimizes this integral or this action. Now this sounded like a backwards step from Newton's. Newton's formula seemed very simple. F equals to ma and you can write F is minus the gradient of the potential. So why would anybody start formulating such a simple thing in terms of this complicated looking principle? You have to study the space of all paths and all things and find the minimum and then you get the same equation. So what's the point? So Euler and Lagrange's formulation of Newton, which is a, which was kind of recasting in this language, is just a consequence of Newton's law. F equals MA gives you the same fact that this path is a minimum action. Now, what we learned later last century, was that when we deal with quantum mechanics, Newton's law is only an average. Correct. And the particle going from one to the other doesn't take exactly one path. It takes all the paths with the amplitude which is proportional to the exponential of the action times an imaginary number I. And so this fact turned out to be the reformulation of quantum mechanics. We should start there as the basis of the new law, which is quantum mechanics. And Newton is only an approximation and the average. Correct.

When you say amplitude, you mean probability.

Yes. The amplitude means if you sum up all these paths with the exponential I times the action. If you sum this up, you get the number complex number. You square the norm of this complex number gives you a probability to go from one to the other.

Is there ways in which mathematics can lead us astray when we use it as a tool to understand the physical world?

Yes. I would say that mathematics can lead us astray as much as all physical ideas can lead us astray. True. If you get stuck in something, then you can easily fool yourself that just the thought process, we have to free ourselves of that. Sometimes math does that role. Like say, oh, this is such a beautiful math, I definitely want to use it somewhere. And so you just get carried away. And you just get maybe carried too far away. So that is certainly true. But I wouldn't say it's more dangerous than old physical ideas. To me, new math ideas is as much potential to lead us astray as old physical ideas which could be long held principles of physics. So I'm just saying that we should keep an open mind about the role the math plays. Not to be antagonistic towards it and not to over welcoming it. We should just be open to possibilities.

What about looking at a particular characteristics of both physical ideas and mathematical ideas, which is beauty? Do you think beauty leads us astray? Meaning, and you offline, showed me a really nice puzzle that illustrates this idea a little bit. Now maybe you can speak to that. Or another example where beauty makes it tempting for us to assume that the law and the theory we found is actually one that perfectly describes reality.

I think that beauty does not lead us astray because I feel that beauty is a requirement for principles of physics.

So beauty is fundamental in the universe.

I think beauty is fundamental, at least that's the way many of us view it.

It's not emergent, it's not immersion.

I think Hardy is the mathematician who said that there's no permanent place for ugly mathematics. And so I think the same is true in physics, that if we find a principle which looks ugly, we are not going to be. That's not the end stage. So therefore beauty is going to lead us somewhere. Now it doesn't mean beauty is enough. It doesn't mean if you just have beauty, if I just look at something is beautiful, then I'm fine. No, that's not the case. Beauty is certainly a criteria that every good physical theory should pass. That's at least the view we have. Why do we have this view? That's a good question. It is partly, you could say, based on experience of science over centuries. Partly is philosophical view of what reality is or should be. And in principle it could have been ugly and we might have had to deal with it, but we have gotten maybe confident through examples after examples in the history of science to look for beauty.

And our sense of beauty seems to incorporate a lot of things that are essential for us to solve some difficult problems, like symmetry. We find symmetry beautiful and the breaking of symmetry beautiful somehow. Symmetry is a fundamental part of how we conceive of beauty at all layers of reality, which is interesting in both the visual space where we look at art, we look at each other as human beings, the way we look at creatures in the biological space, the way we look at chemistry and Then into the physics world as the work you do. It's kind of interesting. It makes you wonder which one is the chicken or the egg. Is symmetry the chicken and our conception of beauty of the egg, or the other way around, or somehow the fact that the symmetry is part of reality, it somehow creates a brain that then is able to perceive it. Or maybe this is just because we. Maybe it's so obvious, it's almost trivial. That symmetry, of course, will be part of every kind of universe that's possible, and then are any kind of organism that's able to observe that universe is going to appreciate symmetry?

Well, these are good questions. We don't have a deep understanding of why we get attracted to symmetry. Why do laws of nature seem to have symmetries underlying them? And the reasoning or the examples of whether if there wasn't symmetry, we would have understood it or not. We could have said that, yeah, if there were things which didn't look that great, we could understand them. For example, we know that symmetries get broken. And we have appreciated nature in the broken symmetry phase as well. The world we live in has many things which do not look symmetric, but even those have underlying symmetry when you look at it more deeply. So we have gotten maybe spoiled perhaps by the appearance of symmetry all over the place, and we look for it. And I think this is perhaps related to the sense of aesthetics that scientists have. And we don't usually talk about it among scientists. In fact, it's kind of a philosophical view of why do we look for simplicity or beauty or so forth. And I think in a sense, scientists are a lot like philosophers Sometimes, I think especially modern science seems to shun philosophers and philosophical views, and I think at their peril. I think in my view, science owes a lot to philosophy. And in my view, many scientists, in fact probably all good scientists, are perhaps amateur philosophers. They may not state that they are philosophers or they may not like to be labeled philosophers, but in many ways what they do is like what is philosophical takes of things. Looking for simplicity or symmetry is an example of that, in my opinion. Or seeing patterns, you see, for example, another example of the symmetry is like how you come up with new ideas in science. You see, for example, an idea A is connected with an idea B. Okay, so you study this connection very deeply, and then you find the cousin of an idea A, let me call it A prime, and then you immediately look for B prime. If A is like B, and if there's an A prime, then you look for B prime. Why? Well, it completes the picture. Why? Well, it's philosophically appealing to have more balance in terms of that. And then you look for B prime and lo and behold, you find this other phenomena, which is a physical phenomenon which you call B prime. So this kind of thinking motivates asking questions and looking for things. And it has guided some scientists, I think, through many centuries, and I think it continues to do so today.

And I think if you look at the long arc of history, I suspect that the things that would be remembered is the philosophical flavor of the ideas of physics and chemistry and computer science and mathematics. Like, I think the actual details will be shown to be incomplete or maybe wrong, but the philosophical intuitions will carry through much longer. There's a sense in which, if it's true, that we haven't figured out most of how things work currently, that it'll all be shown as wrong and silly. It'll almost be a historical artifact. But the human spirit, whatever, like the longing to understand the way we perceive the world, the way we conceive of it, of our place in the world, those ideas will carry on.

I completely agree. In fact, I believe that almost. Well, I believe that none of the principles or laws of physics we know today are exactly correct. All of them are approximations to something. They are better than the previous versions that we had, but none of them are exactly correct and none of them are going to stand forever. So I agree that that's the process we are heading, we are improving. And yes, indeed, the thought process, and that philosophical take is common. So when we look at older scientists, or maybe even all the way back to Greek philosophers and the things that. The way they thought and so on, almost everything they said about nature was incorrect. But the way they thought about it and many things that they were thinking is still valid today. For example, they thought about symmetry breaking. They were trying to explain the following. This is a beautiful example. I think they had figured out that the Earth is round. And they said, okay, Earth is round. They have seen the length of the shadow of a meter stick. And they had seen that if you go from the equator upwards north, they find that depending on how far away you are, the length of the shadow changes. And from that they had even measured the radius of the Earth to good accuracy.

That's brilliant, by the way, the fact.

That they did that, very brilliant, very brilliant. So these Greek philosophers, very smart. And so they had. Taking it to the next step, they ask, okay, so the Earth is round, why doesn't it move? They thought it doesn't move. They were Looking around, nothing seemed to move. So they said, okay, we have to have a good explanation. It wasn't enough for them to be there. So they really want to deeply understand that fact. And they come up with a symmetry argument. And the symmetry argument was, oh, if the Earth is spherical, it must be at the center of the universe for sure. So they said, the Earth is at the center of the universe.

That makes sense.

And they said, you know, if the Earth is going to move, which direction does it pick? Any direction it picks. It breaks that spherical symmetry because you have to pick a direction. And that's not good because it's not symmetrical anymore. So therefore, the Earth decides to sit put because it would break the symmetry. So they had the incorrect science. They thought, earth doesn't move, but they had this beautiful idea that symmetry might explain it. But they were even smarter than that. Aristotle didn't agree with this argument. He said, why do you think symmetry prevents it from moving? Because the preferred position not. So he gave an example. He said, suppose you are a person and we put you at the center of a circle and we spread food around you on a circle around you, Loaves of bread, let's say. And we say, okay, stay at the center of the circle forever. Are you going to do that just because of the symmetric point? No, you are going to get hungry. You're going to move towards one of those loaves of bread, despite the fact that it breaks the symmetry. So from this way, he tried to argue, being at the symmetric point may not be the preferred thing to do. And this idea of spontaneous symmetry breaking is something we just used today to describe many physical phenomena. So spontaneous symmetry breaking is the feature that we now use. But this idea was there thousands of years ago, but applied incorrectly to the physical world. But now we are using it. So these ideas are coming back in different forms. So I agree very much that the thought process is more important and these ideas are more interesting than the actual applications that people may find today.

Did they use the language of symmetry and the symmetry breaking and spontaneous symmetry? That's really interesting. Yes, because I could see a conception of the universe that kind of tends towards perfect symmetry and is stuck there like that. Not stuck there, but achieves that optimal and stays there. The idea that you would spontaneously break out of symmetry, like have these perturbations jump out of symmetry and back. That's a really difficult idea to load into your head. Where does that come from? And then the idea that you may not be at the center of the universe, that is a really Tough idea, right?

So symmetry, sometimes an explanation of being at the symmetric point is sometimes a simple explanation of many things. Like if you have a ball, circular ball, then the bottom of it is the lowest point. So if you put a pebble or something, it will slide down and go there at the bottom, and stays there and at the symmetric point, because the preferred point, the lowest energy point. But if that same symmetric circular ball that you had had a bump on the bottom, the bottom might not be at the center. It might be on a circle on the table, in which case the pebble would not end up at the center. It would be the lower energy point, symmetrical, but it breaks the symmetry once it picks a point on that circle. So we can have symmetry reasoning for where things end up, or symmetry breakings, like this example would suggest.

We talked about beauty. I find geometry to be beautiful. You have a few examples that are geometric in nature in your book. How can geometry in ancient times or today be used to understand reality? And maybe how do you think about geometry as a distinct tool in mathematics and physics?

Yes, geometry is my favorite part of math as well. And Greeks were enamored by geometry. They tried to describe physical reality using geometry and principles of geometry and symmetry. Platonic solids, the five solids they had discovered had these beautiful solids. They thought it must be good for some reality. There must be explaining something. They attached one to air, one to fire, and so forth. They tried to give physical reality to symmetric objects. These symmetric objects are symmetries of rotation and discrete symmetry groups. We call today of rotation group in three dimensions. Now we know. Now we kind of laugh at the way they were trying to connect that symmetry to the laws of the realities of physics. But actually, it turns out in modern days, we use symmetries in not too far away exactly in these kind of thoughts processes in the following way, in the context of string theory, which is the field I study, we have these extra dimensions. And these extra dimensions are compact, tiny spaces typically, but they have different shapes and sizes. We have learned that if these extra shapes and sizes have symmetries which are related to the same rotation symmetries that the Greek we're talking about. If they enjoy those discrete symmetries, and if you take that symmetry and quotient the space by it, in other words, identify points under these symmetries, you get properties of that space at the singular points which force emanates from them. What forces? Forces like the ones we have seen in nature today. Like electric forces, like strong forces, like weak forces. So these same principles that were Driving them to connect geometry and symmetries to nature is driving today's physics. Now much more modern ideas, but nevertheless, the symmetries connecting geometry to physics. In fact, often, sometimes we ask the following question. Suppose I want to get this particular physical reality. I want to have these particles with these forces and so on. What do I do? It turns out that you can geometrically design this space. To give you that, you say, oh, I put the sphere here. I would do this. I will shrink them. So if you have two spheres touching each other and shrinking to zero size, that gives you strong forces. If you have one of them, it gives you the weak forces. If you have this, you get that. And if you want to unify forces, do the other thing. So these geometrical translation of physics is one of my favorite things that we have discovered in modern physics in the context of string theory.

The sad thing is when you go into multiple dimensions and we'll talk about it, is we start to lose our capacity to visually intuit the world we're discussing. And then we go into the realm of mathematics and we lose that. Unfortunately, our brains are such that we're limited. But before we go into that mysterious, beautiful world, let's take a small step back. And you also, in your book, have this kind of through the space of puzzles, through the space of ideas, have a brief history of physics, of physical ideas. Now, we talked about Newtonian mechanics, leading all through different Lagrangian, Hamiltonian mechanics. Can you describe some of the key ideas in the history of physics, maybe lingering on each, from electromagnetism to relativity to quantum mechanics, and to today, as we'll talk about with quantum gravity. And sure.

So I mentioned the classical mechanics and the Euler Lagrange formulation. One of the next important milestones for physics were the discoveries of laws, of electrician magnetism. So Maxwell put the discoveries all together in the context of what we call the Maxwell's equations. And he noticed that when he put these discoveries that Faraday's and others had made about electric and magnetic phenomena in terms of mathematical equations, it didn't quite work. There was a mathematical inconsistency. Now, one could have had two attitudes. One could say, okay, who cares about math? I'm doing nature, electric force, magnetic force, math I don't care about. But it bothered him. It was inconsistent. The equations you were writing, the two equations he had written down did not agree with each other. And this bothered him. But he figured out if you add this, jiggle this equation by adding one little term there, it works at Least it's consistent. What is the motivation for that term? He said, I don't know. Have we seen it in experiments? No. Why did you add it? Well, because of mathematical consistency. So he said, okay, math forced him to do this term. He added this term, which we now today call the Maxwell term. And once he added that term, his equations were nice. Differential equations, mathematically consistent, beautiful. But he also found the new physical phenomena. He found that because of that term, he could now get electric and magnetic waves moving through space at a speed that he could calculate. So he calculated the speed of the wave, and lo and behold, he found it's the same as the speed of light, which puzzled him because he didn't think light had anything to do with electricity and magnetism. But then he was courageous enough to say, well, maybe light is nothing but these electric and magnetic fields moving around. And he wasn't alive to see the verification of that prediction, and indeed was true. So this mathematical inconsistency, which we could say this mathematical beauty, drove him to this physical, very important connection between light and electric magnetic phenomena, which was later confirmed. So then physics progresses and it comes to Einstein. Einstein looks at Maxwell's equation, says, beautiful. These are nice equations, except we get one speed, light. Who measures this light speed? And he asked the question, are you moving? Are you not moving? If you move, the speed of light changes. But Maxwell's equation has no hint of different speeds of light. It doesn't say, oh, only if you're not moving, you get the speed. It's just you always get this speed. So Einstein was very puzzled, and he. He was daring enough to say, well, you know, maybe everybody get the same speed for light. And that motivated his theory of special relativity. And this is an interesting example because the idea was motivated from physics, from Maxwell's equations, from the fact that people tried to measure the properties of ether, which was supposed to be the medium in which the light travels through. And the idea was that only in that. In that medium, the speed is speed of, if you're at rest with respect to the ether, the speed, the speed of light. And if you're moving, the speed changes. And people did not discover it. Michelson and Morley's experiment showed there is no ether. So then Einstein was courageous enough to say, you know, light is the same speed for everybody, regardless of whether you're moving or not. And the interesting thing is about special theory of relativity is that the math underpinning it, it's very simple. It's linear algebra. Nothing terribly deep. You can teach it at A high school level, if not earlier. Okay. Does that mean Einstein's special relativity is boring? Not at all. So this is an example where simple math, you know, linear algebra, leads to deep physics. Einstein's theory of special relativity, motivated by this inconsistency at Maxwell equation, would suggest for the speed of light, depending on who observes it.

What's the most daring idea there? That the speed of light could be the same everywhere.

That's the basic. That's the guts of it. That's the core of Einstein's theory. That statement underlies the whole thing. Speed of light is the same for everybody. It's hard to swallow, and it doesn't sound right. It sounds completely wrong on the face of it. And it took Einstein to make this daring statement. It would be laughing in some sense. How could anybody make this possibly ridiculous claim and it turned out to be true?

How does that make you feel? Because it still sounds ridiculous.

It sounds ridiculous until you learn that our intuition is at fault about the way we conceive of space and time. The way we think about space and time is wrong because we think about the nature of time as absolute. And part of it is because we live in a situation where we don't go with very high speeds. There are speeds that are small compared to the speed of light. And therefore the phenomena we observe does not distinguish the relativity of time. The time also depends on who measures it. There's no absolute time. When you say it's noon today now, it depends on who's measuring it. And not everybody would agree with that statement. And to see that, you will have to have fast observer moving close to the speed of light. So this shows that our intuition is at fault. And a lot of the discoveries in physics precisely is getting rid of the wrong old intuition. And it is funny because we get rid of it, but it always lingers in us in some form. Like even when I'm describing it, I feel like a little bit like, isn't it funny as you're just feeling the same way? It is. Yes, it is. But we kind of replace it by an intuition. And actually, there's a very beautiful example of this, how physicists do this, try to replace their intuition. And I think this is one of my favorite examples about how physicists develop intuition. It goes to the work of Galileo. So, you know, again, let's go back to Greek philosophers or maybe Aristotle in this case. Now, again, let's make a criticism. He thought that objects, the heavier objects, fall faster than the lighter objects.

Makes sense.

It kind of makes sense. And people say about 50 and so on, but that's because of the air resistance. But you might think, like, if you have a heavy stone and a light pebble, the heavy one will fall first. If you don't do any experiments, that's the first gut reaction, I would say. Everybody would say that's the natural thing. Galileo did not believe this, and he kind of did the experiment. Famously, it said he went on the top of Pisa Tower and he dropped these heavy and light stones, and they fell at the same time. When he dropped at that same time from the same height. Okay, good. So he said, I'm done. You know, I've showed that the heavier and lighter objects fall at the same time. I did the experiment. Scientists at that time did not accept it. Why was that? Because at that time, science was not just experimental. The experiment was not enough. They didn't think that they have to soil their hands in doing experiments to get to the reality. They said, why is it the case? So Galileo had to come up with an explanation of why heavier and lighter objects fought the same rate. This is the way he convinced them, using symmetry. He said, suppose you have three bricks, the same shape, the same size, same mass, everything, and we hold these three bricks at the same height and drop them. Which one will fall to the ground first? Everybody said, of course, we know it. Symmetry tells you they're all the same shape, same size, same height. Of course they fall at the same time. Yeah, we know that. Next. Next is trivial. He said, okay, what if we move these bricks around with the same height? Does it change the time they hit the ground? They said, if it's the same height again by the symmetry principle, because the height translation, horizontal translation of symmetry. No, it doesn't matter. They all fall the same rate. Good. Does it matter how close I bring them together? No, it doesn't. Okay, suppose I make the two bricks touch and then let them go. Do they fall at the same rate? Yes, they do. But then he said, well, the two bricks that touched are twice more mass than this other brick. And you just agreed that they fall at the same rate? They say, yeah, yeah, we just agreed. That's right. That's strange. Yes. So he deconfused them by this symmetry reasoning. So this way of repackaging some intuition, a different intuition. When the intuitions clash, then you. Then you side and you replace the intuition.

That's brilliant. In some of these more difficult physical ideas, physics Ideas, in the 20th century and the 21st century, it starts becoming more and more difficult to then replace. The intuition, you know, what does the world look like for an object traveling close to the speed of light? You start to think about like the edges of supermassive black holes and you start to think like, what, what's that look like? Or I've been into gravitational waves recently. It's like when the fabric of space time is being morphed by gravity. Like, what's that actually feel like? If I'm writing a gravitational wave, what's that feel like? I mean, I think some of those are more sort of hippie, not useful intuitions to have. But if you're an actual physicist or whatever the particular discipline is, I wonder if it's possible to meditate, to sort of escape through thinking, prolonged thinking and meditation on a world like live in a visualized world that's not like our own in order to understand a phenomena deeply. So like replace the intuition like through rigorous meditation on the idea in order to conceive of it. I mean, if we talk about multiple dimensions, I wonder if there's a way to escape the three dimensional world in our mind in order to then start to reason about it. The more I talk to topologists, the more they seem to not operate at all in the visual space. They really trust the mathematics, which is really annoying to me because topology and differential geometry feels like it has a lot of potential for beautiful pictures.

Yes, I think they do, actually. I would not be able to do my research if I don't have an intuitive feel about geometry. And we'll get to it, as you mentioned before, that how, for example, in string theory you deal with these extra dimensions. And I'll be very happy to describe how we do it, because without intuition we will not get anywhere. And I, I don't think you can just rely on formalism. I don't, I don't think any physicist just relies on formalism. That's not physics, that's not understanding. So we have to intuit it and that's crucial. And there are steps of doing it and we learned it might not be trivial, but we learn how to do it. Similar to this Galileo picture. I just told you, you have to build these gradually, but to connect the bricks, you have to connect the bricks exactly. You have to connect the bricks literally. Yeah. So going back to your question about the path of the history of the science, so I was saying about the electricity, magnetism and the special relativity were simple idea led to special relativity. But then he went further thinking about acceleration in the context of relativity and he came up with general relativity where he Talked about the fabric of space, time being curved and so forth, and matter affecting the curvature of the space on time. So this gradually became of connection between geometry and physics. Namely, he replaced Newton's gravitational force with a very geometrical beautiful picture. It's much more elegant than Newton's, but much more complicated mathematically. So when we say simpler, we mean in some form it's simpler, but not in pragmatic terms of equation solving. The equations are much harder to solve in Einstein's theory, and in fact so much harder that Einstein himself couldn't solve many of the cases. He thought, for example, you couldn't solve the equation for a spherical symmetric matter, like if you had a symmetric sun. He didn't think you can actually solve his equations for that. And a year after he said that it was solved by Schwarzschild. So it was that hard that he didn't think it's going to be that easy. So yeah, the formalism is hard. But the contrast between the special relativity and general relativity is very interesting because one of them has almost trivial mathematics and the other one has super complicated math. Both are physically amazingly important. And so we have learned that the physics may or may not require complicated math. We should not shy from using complicated math like Einstein did. Einstein wouldn't say, I'm not going to touch this math because it's too much tensors or curvature. And I don't like the four dimensional spacetime because I can't see four dimension. He wasn't doing that. He was willing to abstract from that because physics drove him in that direction. But his motivation was physics. Physics pushed him just like Newton pushed to develop calculus because physics pushed him that he didn't have the tools. So he had to develop the tools to answer his physics questions. So his motivation was physics again. So to me, those are examples which show that math and physics have these symbiotic relationship which kind of reinforce each other. Here I'm giving you examples of both of them. Namely, Newton's work led to development of mathematics, calculus. And in the case of Einstein, he didn't develop Riemannian geometry, just used them. So it goes both ways. And in the context of modern physics, we see that again and again. It goes both ways.

Let me ask a ridiculous question. You know, you talk about your favorite soccer player to bar. I'll ask the same question about Einstein's ideas. Which is, which one do you think is the biggest leap of genius? Is it the E equals MC squared? Is it Brownian motion? Is it special relativity? Is it general relativity? Which of the famous set of papers he's written in 1905. And in general, his work was the biggest leap of genius, in my opinion.

It's special relativity. The idea that speed of light is the same for everybody is the beginning of everything he did.

The beginning is the beginning. Once you embrace that weirdness, all the weirdness, all the.

I would say that's. Even though he says the most beautiful moment for him, he says that is when he realized that if you fall in an elevator, you don't know if you're falling or whether you're. Whether you're in the falling elevator or whether you're next to the Earth gravitational field. That to him was his aha moment, which inertial mass and gravitational mass being identical geometrically and so forth as part of the theory, not because of some funny coincidence that's for him, But I feel from outside at least, it feels like the speed of light being the same is the really aha moment.

The general relativity to you is not like the conception of space time.

In a sense, the conception of space time already was part of special relativity when you talk about length contraction. So general relativity takes that to the next step, but beginning of it was already space. Length contracts, time dilates. So once you talk about those, then, yeah, you can dilate more or less different places than its curvature. So you don't have a choice. So it kind of started just with that same simple thought. Speed of light is the same for all.

Where does quantum mechanics come into view?

Exactly. So this is the next step. So Einstein develops general relativity and he's beginning to develop the foundation of quantum mechanics at the same time the photoelectric effects and others. And so quantum mechanics overtakes, in fact, Einstein in many ways because he doesn't like the probabilistic interpretation of quantum mechanics and the formulas that's emerging. But physicists march on and try to, for example, combine Einstein's theory of relativity with quantum mechanics. So Dirac takes special relativity, tries to see how is it compatible with quantum mechanics.

Can we pause and briefly say what is quantum mechanics?

Oh, yes, sure. So, quantum mechanics. So I discussed briefly when I talked about the connection between Newtonian mechanics and the Euler Lagrange reformulation of the Newtonian mechanics and interpretation of this Euler Lagrange formalism in terms of the paths that the particle take. So when we say a particle goes from here to here, we usually think it classically follows a specific trajectory. But actually in quantum mechanics, it follows every trajectory with different probabilities. And so there's this fuzziness. Now, most probable it's the path that you actually see, and the deviation from that is very, very unlikely and probabilistically very minuscule. So in everyday experiment, we don't see anything deviated from what we expect. But quantum mechanics tells us that things are more fuzzy. Things are not as precise as the line you draw if things are a bit like cloud. So if you go to microscopic scales, like atomic scales, and lower, these phenomena become more pronounced, you can see it much better. The electron is not at the point, but the cloud spread out around the nucleus. And so this fuzziness, this probabilistic aspect of reality is what quantum mechanics describes.

Can I briefly pause on that idea? Do you think this is. Quantum mechanics is just a really damn good approximation, a tool for predicting reality, or does it actually describe reality? Do you think reality is fuzzy at that level?

Well, I think that reality is fuzzy at that level, but I don't think quantum mechanics is necessarily the end of the story. So quantum mechanics is certainly an improvement over classical physics. That much we know by experiments and so forth. Whether I'm happy with quantum mechanics, whether I view quantum mechanics, for example, the thought, the measurement description of quantum mechanics, am I happy with it? Am I thinking that's the end stage or not? I don't. I don't think we're at the end of that story. And many physicists may or may not view this way. Some do, some don't. But I think that it's the best we have right now, that's for sure. It's the best approximation for reality we know today. And so far, we don't know what it is, the next thing that improves it or replaces it, and so on. But as I mentioned before, I don't believe any of the laws of physics we know today are from exactly correct. It doesn't bother me. I'm not like dogmatic. Say I have figured out this is the law of nature. I know everything. No, no. The beauty about science is that we are not dogmatic and we are willing to. In fact, we are encouraged to be skeptical of what we ourselves do.

So you were talking about Dirac.

Yes, I was talking about Dirac. Right. So Dirac was trying to now combine this Schrodinger's equations, which was described in the context of trying to talk about how these probabilistic waves of electrons move for the atom, which was good for speeds which were not too close to the speed of light, to what happens when you get to the nearest speed of light. So then you need relativity. So then Dirac tried to combine Einstein's relativity with quantum mechanics. So he tried to combine them and he wrote this beautiful equation, the Dirac equation, which, roughly speaking, take the square root of the Einstein's equation in order to connect it to Schrodinger's time evolution operator, which is first order in time derivative, to get rid of the naive thing that Einstein's equation would have given, which is second order. So you have to take a square root. Now, square root usually has a plus or minus sign when you take it. And when he did this, he originally didn't notice this, didn't pay attention to this plus or minus sign. But later physicists pointed out to Dirac, he says, look, there's also this minus sign. And if you use this minus sign, you get negative energy. In fact, it was very annoying that somebody else tells you this obvious mistake you make. Pauli, famous physicist, told Dirac, this is nonsense. You're going to get negative energy with your equation, which negative energy without any bottom, you can go all the way down to infinite energy. So it doesn't make any sense. Dirac thought about it, and then he remembered Pauli's exclusion principle. Just before him, Pauli had said, you know, there's this principle called the exclusion principle that two electrons cannot be on the same orbit. And so Dirac said, okay, you know what? All these negative energy states are filled orbits occupied. So According to you, Mr. Pauli, there's no place to go. So therefore they only have to go positive. Sounded like a big cheat. And then Pauli said, oh, you know what? We can change orbits from one orbit to another. What if I take one of these negative energy orbits and put it up there? Then it seems to be a new particle which has opposite properties to the electron. It has positive energy, but it has positive charge. What is that? Dirac was a bit worried. He said, maybe that's proton, because proton has plus charge. He wasn't sure. But then he said, oh, maybe it's Proton. But then they said, no, no, no, it has the same mass as the electron. Cannot be proton because Proton is heavier. Dirac was stuck. He says, well, then there may be another park we haven't seen. By that time, Dirac himself was getting a little bit worried about his own equation and his own crazy interpretation. Until a few years later, Anderson in photographic cosmic. In the photographic place that he had gotten from these cosmic rays, he. He discovered a particle which goes in the opposite direction that the electron goes when there's a magnetic field. And with the Same mass, exactly like what Dirac had predicted. And this was what we call now positron. And in fact, beginning with the work of Dirac, we know that every particle has an antiparticle. And so this idea that there's an antiparticle came from this simple math. You know, there's a plus and a minus from the Dirac's quote, unquote mistake. So again, trying to combine ideas, sometimes the math is smarter than the person who uses it to apply it. And you try to resist it, and then you kind of confront it by criticism, which is the way it should be. So physicist comes and says, no, no, that's wrong, and you correct it, and so on. So that is a development of the idea. There's particle, there's antiparticle, and so on. So this is the beginning of the development of quantum mechanics and the connection with relativity. But the thing was more challenging because we had to also describe how electric and magnetic fields worked. With quantum mechanics, this was much more complicated because it's not just one point. Electric and magnetic fields were everywhere. So you had to talk about fluctuating and a fuzziness of electrical field and magnetic fields everywhere. And the math for that was very difficult to deal with. And this led to a subject called quantum field theory. Fields like electric and magnetic fields had to be quantum, had to be described also in a wavy way. Feynman, in particular, was one of the pioneers, along with Schwinger's and others, to try to come up with a formalism to deal with fields like electric and magnetic fields interacting with electrons in a consistent quantum fashion. And they developed this beautiful theory, quantum electrodynamics. From that, and later on, that same formalism, quantum field theory, led to the discovery of other forces and other particles, all consistent with the idea of quantum mechanics. So that was how physics progressed. And so basically, we learned that all particles and all the forces are in some sense related to particle exchanges. And so, for example, electromagnetic forces are mediated by a particle we call photon and so forth. And same for other forces that they discovered strong forces and the weak forces. So we got the sense of what quantum field theory is.

Is that a big leap of an idea that particles are fluctuations in the field? Like the idea that everything is a field is the old Einstein. Light is a wave, both a particle and a wave kind of idea. Is that a huge leap in our understanding of conceiving the universe's fields?

I would say so. I would say that viewing the particles, this duality that Bohr mentioned between particles and Waves. That waves can behave sometimes like particles, sometimes like waves is one of the biggest leaps of imagination that quantum mechanics made physicists do. So I agree that that is quite remarkable.

Is duality fundamental to to the universe, or is it just because we don't understand it fully? Like we'll eventually collapse into a clean explanation that doesn't require duality like that a phenomena could be two things at once and both to be true. That seems weird.

So in fact, I was going to get to that when we get to string theory, but maybe I can comment on that now. Duality turns out to be running the show today. And the whole thing that we are doing in string theory, duality is the name of the game. So it's the most beautiful subject. And I want to talk about it.

Let's talk about it in the context of string. The.

Let's talk about the kind of string theory.

So do you want to take a next step into. Because we mentioned general relativity, we mentioned quantum mechanics, is there something to be said about quantum gravity?

Yes, that's exactly the right point to talk about. So, namely, we have talked about quantum fields, and I talked about electric forces, photon being the particle carrying those forces. So for gravity quantizing gravitational field, which is this curvature of space time, according to Einstein, you get another particle called graviton. So what about gravitons? Should be there, no problem. So then you start computing it. What do I mean by computing it? Well, you compute scattering of one graviton off another graviton, maybe with graviton, with an electron, and so on. See what you get. Feynman had already mastered this quantum electrodynamics. He said, no problem, let me do it. Even though these are such weak forces, the gravity is very weak. So therefore to see them, these quantum effects of gravitational waves was impossible. It's even impossible today. So Feynman just did it for fun. He usually had this much.