Director’s Innovation Speaker Series: How Looking at Genetics and Networks Led to Solving a Quantum Gravity Problem
JOSHUA GORDON: Hello everyone. I am Joshua Gordon, Director of the National Institute of Mental Health. It is my pleasure to welcome you here for today for our Directors Innovation Speaker Series. It is my pleasure to welcome, in particular, Dr. Sylvester Jim Gates of Brown University. I will introduce him in just a little bit. Before I do, I want to go over some of the ground rules for today.
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It is my time to introduce Jim. Jim Gates is a theoretical physicist. We are really excited to have him here today because as you know the main point of the Directors Innovation Series is to spur innovation. I think what he is going to talk about today is an example of innovation that we can take to heart for our own purposes here at NIMH.
Jim was for a long time at the University of Maryland in the Department of Physics there, where he was the University System Regence Professor and the John S Toll Professor of Physics and the Director of the String and Particle Theory Center. In 2017, he retired from the University of Maryland and moved to Brown University, where he is the Theoretical Physics Center Director as well as the Ford Foundation Professor of Physics. He is also a faculty fellow at the Watson Institute for International Studies and Public Affairs. Jim explained to me in our discussion beforehand that he really enjoys working at this intersection between science and society. I think you will get some of that as well today.
Jim is best known for his work on super symmetry, super gravity, and super string theory. I guess he is just all-around a super scientist. He actually received two BS degrees and a PHD from the Massachusetts Institute of Technology where his doctoral thesis was the Institution’s first on the subject of super symmetry. In 1984, he coauthored, Super Space, the first comprehensive book on the topic. He is the past president of the National Society of Black Physicists and an NSBP fellow as well as a fellow of the American Physical Society and the American Association for the Advancement of Science and the Institute of Physics in the UK.
In 2021, Professor Gates was the recipient of the 2021 Andrew Gemant Award, which is one of many awards that he has received over his lifetime. He has also been elected as a fellow of the South African Institute of Physics and was elected to the Mathematical Sciences Research Institute Board of Trustees. He has been president of the American Physical Society. He is also an elected member of the American Academy of Arts and Sciences, the American Philosophical Society, and the National Academy of Sciences. In fact, when he was named in 2013 to that body, we was the first African-American theoretical physicist to be recognized and admitted to the National Academy.
President Obama, himself, awarded Professor Gates the National Medal of Science at the White House in 2013.
In addition to his academic work, he has served his country on the Board of Scientific Advisors to the President during the Obama administration and has had numerous other policy roles over time.
I am really excited that Jim is here today to talk to us about something that is actually quite more related than we might expect from a theoretical physicist to our work here, at the NIMH: how looking at genetics and networks led to solving a quantum gravity problem. I am sure you appreciate genetics and networks. Those are two things that we care deeply about at NIMH. I am really looking forward to this talk. Jim, thanks for coming.
SYLVESTER JAMES GATES: Josh, thanks for the invitation. I am very interested in asking you a question. There was a slide up a moment ago and under my name it said Vice President of Research at the Institute for Clinical and Economic Review. I am not quite sure what that is about, but if that is accurate, you have got the wrong guy. I am going to let you folks worry about that.
JOSHUA GORDON: We will worry about fixing that up. I think that is left over from the last Innovation Speaker series, which was really interesting. You might want to listen to it on the web. For now, I just want to let you know your camera has not started. You can go ahead and start that if you like to show us your face and if you would like to share your slides, as well.
SYLVESTER JAMES GATES: Thank you. I always like to tell people that – don’t be afraid when you see a strange image on the screen. It is just me. Let’s get started today and have some fun in physics because my students will tell you that I always find physics great fun.
I am someone who got to have their childhood dream come true. I was four years old when I started thinking about becoming a scientist. Here I am at age 71. I have had a 40 plus long career doing science, as well as a 51 long consecutive career of teaching mathematics or science to university students.
Because of that latter set of activities, I think that young people fire the imagination when you are an old person trying to do something innovative. Today’s talk will be an example of what I get from working with young people, sometimes even high school students. So, let’s get on with the talk.
Can someone confirm that we have full-screen mode?
JOSHUA GORDON: I can see your slides. It is in full-screen mode.
SYLVESTER JAMES GATES: Thank you. Ok, let’s start our fun, “How Looking at Genetics and Networks Led to Solving a Quantum Gravity Problem”.
I am going to have to take you through a course on the Standard Model 101 before we get to the problem. So, we are going to spend part of our time just making sure that we are all on the same page. Let’s first start with graphs, networks, and polytopes in science.
I hope everyone is familiar with this lady, Rosalind Franklin. She is the first human being to conduct an experiment that actually determined the structure of the DNA, which is the base of all of our genetics. Of course, Watson and Crick actually did a theoretical analysis. In fact, they surreptitiously used her data. I like to point out that science is interesting and it has room for everyone to make great discoveries.
The structure of DNA, as we know, is the double helix. There is a whole lot of effort these days going into genetics, both for forensic genetics, for medicinal and treating of diseases, genetic diseases. So, it is a blooming part of our structure of our country’s science, increasingly so for our economic structures.
The whole story is told in a series of books. These are just some that I have picked out at random that people can go and learn about the double helix and how it was discovered and what have you.
There are other networks in nature besides the double helix. Here, we have the unit cell of a diamond. It consists of three tetrahedrons in the unit cell and 18 carbon atoms. Of course, in addition to looking at crystal – I mean molecules like diamond, we can look actually at more complicated substances. So, there is a class of substances called faujasites. You can see a picture of a faujasite crystal here. It turns out that you – we are going to come back to that crystal in a moment. You can also see things that are like networks in genomics.
A friend of my, Rick Lenski, lent me this slide from a presentation that he made in Nature in 2016. This actually shows the development of genomes. In fact, Rick and his collaborators ask a very interesting question, which has been asked for a long time, which is if you take say two genetic systems and you give them exactly the same starting point, do they always evolve to the same endpoint? The answer that they found was no. This is actually showing the pathways of some of these generations as they check for this result.
That faujasite crystal I showed you earlier has a unit cell. It is composed of silicon and aluminum atoms arranged in a truncated octahedron. Here, for someone like me, is the way I see an octahedron. It is a mathematical set of equations. I just pulled this one from Wikipedia, where one can find this particular structure described in terms of the number of vertices, which is 24, the number of edges, which is 36, and the number of faces. Interesting enough, this structure has hexagonal and square faces.
Now, we are going to run through particle physics. So, I am going to start with chemistry. I bet many of you have seen this. Actually, probably many of you have not seen this. This is Mendeleev’s Table of Elements as he first discovered it. As you can see, there are holes in this. One of the power that mathematics can base upon – a table, by the way, is an element of mathematics, graphs, tables, what have you. When you have a mathematical setting, you can actually see things that are invisible to others. For example, those holes here were significant because that meant there were more elements than had been discovered. If we go to the modern table of elements, you can see all of those holes have been filled in.
This also illustrates something about science. Science is not static. It is dynamic. It is always changing. When people talk about truth in science, the question is when are you asking the question because otherwise we will give you different answers based on our observations of nature.
Here is one version of the modern Table of Elements. This is a schema that shows the particles of the Standard Model. On the schema, you see the symbol, E. That stands for the electron, which is the most familiar of the elementary particles. In fact, it was actually first hypothesized by a Scottish electrochemist named GJ Stoney. He actually first hypothesized this and then he actually found evidence for it and he gave it a name. So, that is the electron.
But there are lots of things in nature that are like the electron. There is something else that my cursor is point to. This is a new particle. It is a particle that in all ways is exactly like an electron, but 200 times as heavy.
About five decades ago, we discovered there is another copy of the electron in nature. It is called the Tau particle.
So, you don’t get atoms by just taking electrons. You have to have protons and neutrons. Inside protons and neutrons are even smaller particles. We call them quarks. You can see them listed in the column.
Now, these two columns are what I call the parts list for how you build a universe. If you just had those, you couldn’t build our universe because you have to bind these objects together in definite patterns. That is where forces come in. So, the electromagnetic force, which is actually the force most responsible for chemical structure, is carried by a different particle called the photon.
There are four different forces that are fundamental in nature. There is the gravitational force. There is the weak nuclear force. There is the electromagnetic force. Finally, there is the strong nuclear force. Each of these forces has carriers. So, for example, the strong nuclear force has eight carriers. These are the things that keep the quarks glued together inside of protons. The weak nuclear force actually has three carriers, the so called W- boson, the W+ boson, and the Z boson. The gravitational force – well, you see the gravitational force is actually not part of the Standard Model. If you are trying to combine gravity with the Standard Model, we get mathematical nonsense. That is one reason why we know that our understanding of nature is incomplete and we have to keep striving for more complete structure.
There are, in fact, eight gluons. I neglected to mention that. In 2012, we found another one of these particles. This is a so-called Higgs Boson. It is also a carrier of the weak nuclear force.
Often, once these tables that have the structure that you can see here – it looks nice and symmetrical and so you say, oh, my goodness, it is so symmetrical and we probably have all the particles. Well, if you look at them a slightly different way, you can see that it is not so symmetrical. In fact, all of the particles that carry forces are in this upper righthand quadrant. These things do not obey the Pauli exclusion principle. Whereas things like the electron at in this lower lefthand quadrant. Everything in that quadrant obeys the Pauli exclusion principle. The object in the upper row carry forces. The object on the lower row are subject to forces. So, nature has this funny asymmetry, even though when you first look at it you think we are at the end of the story.
We know these sorts of things by doing observations. Observation, as Einstein stated – observations and Einstein stated that we know this because Galileo drummed this into our head – observations are the basis of science. If you went back 15 years ago and you measured a magnetic property of the electron called the g-value, you could measure it. You would get this number. Or you can use the mathematics of the Standard Model to calculate it and you get this number. You will notice that the last two columns don’t agree. Well, that is because in science there is no such thing as absolute certainty. We, in fact, are aware that we are not perfect, our instrumentation is not perfect. Therefore, whenever we measure something we have to be cognizant of how likely is it that we are making an error. We do this by using what we call the Delta in the two value. As you can see, the Delta exactly if you ignore everything before these last two digits, you get zero. That means agreement between theory and observation.
However, about three years ago, the experiment was done again. In fact, this experiment is done almost continuous because we keep improving our technology. It is just like when Mendeleev first found the Table of Elements there were some missing. With increased technology, we were able to find all of them. So, now you can see currently we are up to over a billion digits in our understanding of this magnetic property of the electron. There is nothing else in science so precise.
We have a theory, which tells us what this value – how this value is calculated, and we have observation that tells us the calculated values are in agreement with nature, again, except for these last two digits. These last two digits are not because of calculational error. They are what are called systemic errors. That is in the devices that we use, in the electronics that we use. So, we always have to account for that.
How do these particles get together and describe our universe? I have a cartoon here for you. In this cartoon, you see two white dots. Those are supposed to be electrons. So, you imagine two electrons. Since electrons have the same charge, they repel each other. So, how do they repel each other? The answer is they send a message carrier. That is what you see with that wiggly line in the middle of the cartoon. That represents a photon telling one electron there is another electron close by, you ought to be repelled from me.
Now, this is classical physics. If you do quantum – and the word quantum is a buzzword these days. We hear so much about quantum, not just in science, but in society and government and business. So, what is quantum?
Well, this picture is all that classical physics allows for the repulsion of two electron. However, if you study quantum mechanics, this picture can contribute. Again, we have two electrons as the white dots moving along. In this picture, we are doing something different. The cursor is showing the path of one of the electrons, which emits a photon that it later reabsorbs. In between that emission and absorption, it sends a second photon to the second electron to say you should be repelled. So, that is what quantum mechanics says. It opens up new pathways for the forces to act.
Here is another such diagram. The one we just went through is called a vertex correction. This one is called vacuum polarization. This one we are going to find very important.
So, here we have two electrons. One gets to this point where my cursor is. It emits a photon. That photon disintegrates into an electron and its antiparticle. An electron in negatively charged. Its positron antiparticle is positively charged. So, they have opposite charges. They start to attract each other. When they get back together, they create a second photon, which then tells the second electron to be repelled.
Now, this picture of quantum mechanics – these sorts of pictures come to us from the physicist Richard Feynman. These, in fact, are Feynman diagrams I am showing you. If you take a course in relativistic quantum field theory, you actually learn how to turn these pictures into mathematical equations. Those mathematical equations make predictions about the physical particles. It is those predictions that the observations are checking.
The Higgs boson, as I mentioned, was discovered in 2012. It answers a puzzle. Let me start this cartoon over here. The Higgs boson in this picture is this dot that is rolling down this energy surface. The three dots on the bottom of this represent the Z particle, the photon, and the W particles. When the Higgs particle rolls down, the Z particle over here and the W particle become massive. The photon stays massless. It turns out this is the only mathematical way we know how to combine the theories of special relativity of Einstein with mass in equations that are very similar to Maxwell’s equations.
Here is the Higgs boson at the bottom of this energy surface, having caused the Ws to gain mass, the Zs to gain mass, but leaving the photon massless.
These particles spin, at least most of them do. All of the particles in the Standard Model have a rate of spin, which is characterized by an integer we call j. So, the Higgs boson, which we are going to meet in a moment, actually has j equals zero. It doesn’t spin. This has very important implications because when we look at the equations for how it moves those equations have a lot of similarity to the clump that you can see moving along on this slinky. Mathematically, the equations are almost identical. There are some complications, but they are certainly in the same family.
On the other hand, electrons spin. This quantity, h bar, is a rate by which we measure spins. So, for the electron, this number, j, is one half. Quarks also spin. You can see I have two quarks here spinning, a left-handed, right-handed census here. That is an important property to know about elementary particles. Most of them have a spin.
Now, the electron spin is not like ordinary spin. The spin of the electron also is why it acts like a little bar magnet. That is the property of g minus two that we were measuring. If the electron is placed at the center of the earth, which is what these two cartoons represent – there is a dot at the center that the electron is – and it is moving upward toward the north pole, then the direction of its north magnetic pole can only be pointed along a latitude of 35.23 degrees north or pointed the direction of - the north pole of the bar magnet property of the electron can be pointed at 35.23 degrees in the south latitude. No magnet that we know has this property. If you take an ordinary magnet and move it, the direction in which the north pole points is not constrained. But this is what quantum mechanics teaches us.
So, it turns out that electromagnetic energy – rather light is actually two forms of energy. There is an electrical wave and a magnetic wave. I have some cartoons to show them. The green line represents the electrical wave. It is moving up and down in the animation that I am showing you. On the other hand, the magnetic wave is shown in purple. It is moving left to right. So, they are at right angles to each other, one going up and down and one going side to side, left to right. When you get the two of them that is what light is.
So, it turns out that if you look at that picture, you can see it was going straight up and straight down. However, whenever you have something oscillating straight up and straight down – by the way, this purple line really should be a green line. I apologize for that error. Whenever you have something going up and down, you can decompose it into something that winds around a clockface in an anticlockwise sense or something that winds around a clockface in a clockwise sense.
Now, notice when they are directly pulling apart this is when this disappears. That is because it is like a tug of war with two teams that are equal. When they pull at angles to each other, this center line grows. That is the up and down oscillation that we saw for the electrical field in my previous cartoon.
Here, we are taking a physics instrument called a quarter wave plate. To the left of the plate, we are taking the up and down wave and we are just pulling out the piece of it that is moving in an anticlockwise direction. On the bottom slide, we are going to do the opposite, again, using a quarter wave plate, but oriented in the other sense. Things like the photon have a j of equal one. That is their rate for the photon.
Maxwell’s Equations. Well, you know, we physicists love equations. This is the language of nature, apparently. A lot of other scientists use this also. These are the equations that basically underlie the Standard Model I have just taken you through. You have to generalize these equations so that you find something called the Dirac equation to describe the quarks, the electron, and all of the particles that were in that lower left-hand corner. You have to generalize these equations to what are called Yang-Mills equations to describe all of the particles that were in that upper right-hand quadrant. But that has happened in our history. That is how we study the mathematics of the Standard Model.
However, since we know some math, we can ask questions about things that are not in the Standard Model. In particular, that rate of spin could it be something other than zero, one-half, or one? Well, mathematically, the answer is yes.
This is a picture of what the spin vector would look like if you had a spin three-halves particle in the center of the earth. Again, a little dot representing the particle. We are watching the directions in which its spin vectors can point. They also can only point at certain latitudes if the object is moving upward. But here you can see there are more options.
Well, that was math. What about back in the real world where we measure things? Many of you know about LIGO, the Laser Interferometer Gravitational-Wave Observatory. That is how we have measured waves of gravity. LIGO, at least when this picture was taken, consisted of three sites. There is a site in Johnston Parrish in Louisiana, a LIGO Hanford facility in Washington State, and Virgo, which is a French-Italian collaboration. So, when we look at gravity waves, these are the devices that tell us those stories.
But a gravity wave is a wave. So what does it actually do? I have a cartoon here to illustrate this for you. I want you to imagine looking down a long circular tunnel. I am going to show you what happens to that tunnel when a gravity wave comes from the back of it towards your eyes. You can see the tunnel deforms.
Now, this is the analog of what we saw with linear polarization for light. Do those right-handed and left-handed clock face rotations exist? The answer is yes. Those are much more interesting for gravity.
Here are the anticlockwise rotations. As you can see not only does it deform, but it does a beautiful ballet-like motion, the size of the tunnel that we are looking down. So, that is anticlockwise.
Here is clockwise. Now, the motions that I am telling you about here we know from mathematics. We have never actually measured them. But you see this is one thing that LIGO is going to do for us. This allows us to see these ballet-like motions that gravity waves cause for all material objects.
I just took you through the Standard Model. You are an expert now in understanding these small particles. But there is a boundary of our knowledge. One of them was talked about in a 2020 article because the Muon, which is that particle that was 200 times as massive as the electron that I pointed to, people also tried to measure its magnetic property. In April of 2021, it was reported that if you measure the property of the Muon, what you find is that the magnetic property does not agree with the calculations that I have described to you that come from the Standard Model. This is very, very exciting for people like me and my community because if it doesn’t agree with our mathematics that means we have to find out why.
More recently, there was this announcement that is of the same ilk. It was made on April 7th of this year. People were looking at the calculations of the mass of those W bosons in the context of the rest of the particles in the Standard Model. What they found according to this CDF collaboration at the Tevatron in Batavia, Illinois, is that there is a disagreement between the calculations and what is observed.
So, what are they actually talking about in both cases? Well, here, I have an electron. We went through this a moment ago. Mainly that the electron, in order to repel a second electron, has to send out photons. That is also true if you are going to measure its magnetic moment. So, this is a measurement of that classical magnetic moment of the electron. If you just do the Feynman graph associated with this, you find out the magnetic moment is the number two. Remember that two that I showed you? It was two point dot dot dot dot dot. Well, the two comes from this Feynman diagram.
Let’s now include the effects of quantum mechanics. Remember this diagram where we had our electron emitting a photon that it later absorbed? Well, this diagram also contributes to a measure of the magnetic property of the electron and so would this.
Now, the reason I am taking you through these is because I am only showing you electron-positron pairs here in the middle of this diagram. You see any particle and antiparticle could be in that loop. In particular, particles that we don’t know about might be in such a loop. If they are there, they will cause the measurement of a magnetic moment of the electron – I’m sorry of the new particle to be different from the calculations we are currently able to make.
This same kind of diagram actually applies to the question of the mass of the W particle. The fact that the W particle seems to be telling us that the mass is not what comes from the particles in the Standard Model as well as the g minus two or magnetic property of the new one seems to indicate the same thing, this makes it a very exciting time for physics. This is why colliders like the LHC and the Tevatron and all of those multibillion dollar devices are built, to explore our universe at these levels.
Well, we have gone through the Standard Model. We have gone through the boundary. I am now going to take you beyond the boundary.
Here is the Standard Model picture. Those are all of the particles in the Standard Model. I should probably mention, however, that the quarks actually come in three different colors. You have to sort of take the first – you have to take the first column and the second column and imagine three copies of those also. So life gets a little bit more complicated, but that is a pretty symmetrical looking diagram. It is not when you do this.
You see if I classify the particles by whether they are acted upon by the forces or if they are carriers of the forces, we get this diagram. Furthermore, I am looking at whether the spins are integers or half-integers. The integer spins are all in that right-hand column. The half-integers are in the left-hand column. As you can see, this is a highly unsymmetrical image. It reminds us once again of that Mendeleev diagram that had the holes in it. This suggests that nature may have more particles in store waiting for us to discover.
Now, do we know what those particles are now? The answer is no. But people like me have been studying this subject called super symmetry. One of those possibilities is that the missing particles that we haven’t been able to measure because our technology is not good enough – maybe they one day will fill out this table so that it looks like this. Wow, a whole bunch of new particles showed up. This is like filling those holes in the Mendeleev table. Maybe this is what nature is going to do. We cannot guarantee that. The only way to find out is to continue working on experiments.
Now, all of those particles actually have some beautiful math. That is where I am going to take you because that gets me close to what my research is. These are pictures of the quarks, these red dots here. These are pictures of the antiquarks. You will notice that if I wanted to move from this quark to this quark, I have to move along that direction. If I want to move from this quark to that quark, I have to go from left to right. If I want to move from this quark to the quark at the bottom, I have to go move rightward, but also slanting down. So, there is some motions indicated by this figure.
Now, interestingly enough, if we look at the antiquarks, if I start with the quark here and I want to move to this quark, I have to move rightward and slanted up. What is interesting is the slant up here and the slant up here is actually the same, although that is not so obvious from this diagram. The left-right motion is clearly the same. The slant down motion, which I am indicating with my cursor here and here, those are the same. So, quarks, interestingly enough, if you put them along pictures like this, they kind of naturally generate a set of motions. I am showing you those motions in these arrows.
Mathematicians know these motions as group theory. In particular, if you showed this to a mathematician, they would tell you these are realizations of a group called SU3. Here, I am just coloring the motions to make sure you get the point that the different motions that we get by just thinking about what the quarks are have these very symmetrical patterns.
In fact, if you fill out an entire plane, what you find is a lattice. A lattice is a polytope. It is like looking at a crystal. So, there are these mathematical crystals that sit behind the physics of the Standard Model. Here is that lattice without any points taken out. I have just drawn a segment of it. We can put the quarks and the antiquarks in. There they are.
We can also ask are there other collections of particles that would fit this lattice? Here is another collection. This is called the eight representation of SU3. In nature, do we see particles that form this pattern? The answer is yes. The proton and neutron actually are of a set of family of eight objects that fit this pattern. There are other particles called pions. They also fit this pattern.
So, math sometimes tells you how to fill out particles that you haven’t seen. That is what the idea of supersymmetry is all about.
There is also this pattern I should show you. This is called the decuplet. Here it is with the particles filling it in.
So, now, the problem that I have been wrestling with and that I am going to tell you about in the remaining 14 or 15 or so minutes here. There is a problem – big data has a colloquial use these days. I am going to tell you about a big data problem that stymied my field for about 40 years.
Remember I told you gravity is not part of the Standard Model. The reason it is not part is because if we try to combine the mathematics of Einstein, who taught us how to think about gravity, with the equations of the Standard Model and you try to get predictions out, you get nonsense. This is an indication that you need to find something that will describe gravity and is consistent with the Standard Model. That is – we can call that the search for a quantum gravity.
Back in 1995, Edward Witten of Princeton University proposed something called M-Theory. Now, M-Theory is supposed to be this thing that will fit the bill of joining together all of the Standard Model with a theory of gravity that is consistent with Einstein’s theory of gravity. It turns out that M-Theory is actually an element of String Theory. On the right-hand side of this, I have shown you all of the parts of String Theory that are parts of this overarching theory called M-Theory.
What is the Einstein problem here? Well, back in ’74, Salam and Strathdee, two physicists, showed us how to write down the mathematics of that balanced table that I showed you that had left-right balance. The way to do that is to write something called a superfield. In 1978, three physicists, Eugene Cremmer, Bernard Julia, and I think it was Joel Scherk, wrote down a theory of gravity that matches up with Witten’s M-Theory. Notice that they did this before Witten came along.
The question is can you write down that kind of balanced table that includes this work? In fact, from 1978 to 2020, no one knew how to write that. This problem was solved in 2020 by my meeting two graduate students, Yangrui Hu and Hazel Mak. We found some very interesting way to achieve that. That is the problem we solved. I am going to tell you how we solved it because it may surprise you.
So, this is what Cremmer, Julia, and Scherk told us about this 11-dimensional gravity. It has three different pieces. I am showing you here. One of them is a fermion and two of them are bosons in the same way that we talked about the Standard Model. But if you actually count – you can see two is not equal to three. Remember, our tables balanced before. Clearly, this is not balanced. There must be something missing.
It turns out that on the basis of some other mathematics we know the total number of degrees that such a system has to have. It has to have 2,147,483,648 bosonic degrees of freedom as well as the same number of fermionic degrees of freedom to be consistent with special relativity and quantum mechanics. This is a mathematic result. What I showed you before only had three. Something is missing.
What is that something? Well, nobody knew how to find it out. This is a problem that is so hard, typically because the number of degrees of freedom, that, you know, not just – not most people would say I am going to tackle that problem. It has been unsolved for decades. Well, some of us are a little bit off our rockers. When we see hard problems, we try to figure out how to solve them.
One thing about hard problems that you should understand is when you have a hard problem in science and some of the best scientists in the world for decades are unable to solve the problem, then one way to achieve a breakthrough is to change the point of view about what is the problem. That is going to be key to our innovation.
I showed you Maxwell’s equations. That is how most physicists have tried – the generalization of those equations is how most physicists have tried to attack this problem. They write things like this. You can see an equal sign here. You know this is mathematics. Don’t let it frighten you. I like to tell the general public because I know math is not the most comfortable thing for lots of people. The way that most physicists have attacked this problem is by saying, well, there is some kind of an expansion. You can see the different terms of the expansion going on here. By the way, this expansion goes up to 32 orders if we want to get an exact answer. So, there is this expansion that no one knew how to figure out its detailed structure.
How did I and my two graduate students solve it? Well, I read a lot of stuff as you might gather from my giving this talk. I have been impressed with what is going on with networks and genetics, networks and information theory, networks and coding. Instead of thinking about the problem as a mathematical problem with a Taylor series expansion, I started wondering is there another way to think about this problem?
So, these are the kinds of equations that we start with. The question is are there crystals just like we found in the Standard Model. There were these mathematical crystals hidden behind the mathematics. Here is an expansion the old way of doing it. So in 2004, working with a physicist by the name of Michael Faux, we proposed a mathematical way to try to find the crystals that hide behind those sets of equations that no one could figure out the expansions for.
The idea simply is that you can – if you write the mathematics of a theory in a high dimension, like say 11, which is where M-Theory is, you can forget about all of the spatial directions and try to figure out what is left of that theory if there is only time. So, we imposed this condition on those initial equations. They take us through Einstein’s so-called light cone. I am just showing you a cartoon of it here. At the end of the day, we use special relativity. Again, just a quick matrix to show you. There is a light cone.
What we find is that we find a map. On one side of the map are a bunch of equations. On the other side of the map, there are mathematical crystals. This is a picture of the simplest one of those crystals.
Now, this crystal - interestingly enough, if the Standard Model is completed to super symmetry, this crystal will have something to do with electromagnetism. We call these crystals Adinkras. This is a word derived from an African language. When Michael and I started this work, he was in the Czech Republic. I was actually in Georgia, the country, not the state. We began an email conversation back and forth. About six weeks later, we had figured out how to find these crystals and we wrote a paper. Michael, who had stepped foot on the African continent a decade before I had ever stepped foot on the continent – I was in the West African country of Mali when we finished our calculations – suggested we should use this word.
These crystals sometimes can break apart. Now, for the simple crystal, we understand the rules for how this is done. But when we get to something more complicated that is still under research.
Here is a picture of one of those crystals. Let me show you how it breaks – hidden in these crystals are bits. Now, that is a very strange thing. Bits are part of equations in this hidden mathematics. This is how we break these crystals apart. Those bits have to line up with each other. This movie is going to show you how that happens. I am going to have to speed it up because I am a little bit slow on time. Let me just assure you that with enough motions this crystal turns into that. That is how we break these things apart.
The bits line up so that you find what are called error-correcting codes. Now, these are classical error correcting codes, not quantum error correcting codes. But that is what we found is that classical error-correcting codes control how you can cut these mathematical crystals into smaller pieces.
Why are there bits? Well, these mathematical crystals basically have squares buried inside of them. You can sort of see one square face here. This object, here, is one of these crystals that would exist if our world had four dimensions. This is what we find hidden in our mathematics. Here are some more just examples of these crystals. We don’t know how to do physics with the ones I am about to show you, but we understand the mathematics rules for constructing them. Some of them are quite artistic.
This made the cover of Physics World in 2010. Our discovery was the cover story. It turns out matrices describe these crystals. So, here are representations of sets of matrices. Here we are just calling the sets P1, P2, P3, P4, P5, and P6. The things that you see in the middle column are just some fancy names we have for the matrices.
The question is why do the matrices group in this manner? You can say matrices actually fall into different sectors. I am showing you the sectors here. This is another way to show the matrices, a mathematical object called the Permutahedra. This sets of objects here are shown in color. Everything in green is in one set. Everything in dark blue is in one set. Everything in purple is in one set. Everything in light blue is in one set. You can see they are distributed in an almost random manner, but not quite. It turns out that if you use this mathematical object called the Permutahedra, which mathematicians have known for decades, one of the sets looks like the flashing nodes you see here. Another one of the sets looks like the flashing nodes here. A final one looks like the flashing nodes here. So, the question is what mathematical property is determining this?
I am going to speed up a little bit. So, here is that object. We have 24 nodes. That is the name of those matrices. We have four different operations we can perform. We can start at a node and go nowhere. That is this operation. Or we can move along either the green, the red, or the blue links. That is what you see as these operations. The question is can you find operations that connect those quartets of objects?
We call these things hopping operators. You can see them listed here. How did we find them? Well, we looked at the object and we say, well, those two objects at the bottom are part of a set. They are connected by the black lines you see flashing. That corresponds to a red move and a blue move.
The black lines here also correspond to one of these objects. Those correspond to a green, a blue, a red, and a green move. The matrices are there. Finally, we look to the other member of the quartet. We found it was this. That corresponds to a green, blue, red, green, blue, red move.
Now, the question is all of those quartets actually have the same moves? Once we have learned about these moves defining the super symmetric representation that was the key to us solving this problem.
The first thing we did was say throw out functions and use pictures. They say pictures are worth a thousand words. So, we actually went from using functions to using these pictures. These are called Young Tableaus.
Remember the spins that I told you about? Well, it turns out the pictures I showed you actually have a mathematical equivalence to spin. So, now, we have captured spin. The other thing about these pictures is they have multiplication rules built into them. Again, this is mathematics that has existed for almost a hundred years.
We take these preexisting pieces of mathematics and we ask can we construct crystals that will solve our problem. We had a success with a simpler crystal. This one only has 36,000 – sorry, this one only had just over a thousand different moves. The problem that we wanted to solve has this thing as a crystal. This crystal is the thing that had the 4.2 billion nodes. That is counting the degrees of freedom inside of the nodes. We found the fields that were given to us back in the 70s, here and here. But we also found all of the other fields that are necessary to complete this.
How do we do this? Clearly, we are not going to sit down and draw these figures. We teach computers how to draw these figures. My two students wrote some programs. I guided them in the algorithms. We set the programs up. These are just our results. So, we found the entire crystal.
Now, this story about finding a way to solve a math problem by not doing mathematics is one example of innovation. I am using graph theory here. I am using algebraic geometry and topology. I am actually also using information theory. I told you there are error-correcting codes in this. We are using computer-aided conceptualization. Maybe at some point in the future, it might be that evolutionary theory is actually going to contribute to this.
As I said, the word quantum is everywhere. This – I just got this picture because it is a nice picture of one of the quantum computers at Q Station out of Santa Barbera. I was fortunate enough to actually go visit a laboratory about two years ago and see this device face to face, so to speak. I like to tell students learn to use coding. It is like putting on a math version of the Iron Man suit. It is with codes that we can solve these multibillion problems.
I would like to acknowledge The Teaching Company. A number of my animations are in a product called Superstring Theory: The DNA of Reality, which I did for The Teaching Company, now called The Great Courses. I have a whole list of students and collaborators.
I have been developing these graphical methods since 2005 with my work with Michael Faux. You can see Michael listed here. This young person, here, is my current graduate student that I continue to work with. These are the two students that were critical for us to solve this problem with four billion degrees of freedom.
So, thank you for listening. I hope this talk and the use of visualization, networks, and graphs will be useful to some of you in the audience. Thank you.
One final acknowledgement. If you want to learn more about my stuff, I am going to just leave this up. We can start the Q&A now. Thank you so much.
JOSHUA GORDON: Thank you, Jim, for really a fascinating and inspiring talk. I want to ask you a question that I asked in our one-on-one we had before this meeting and then I will turn – there are several questions in the Q&A.
One of the remarkable things about what you just described is that most of this work happened for you in the last five or ten years. We often hear about physics as being a young person’s game. You described to me what you think contributed to this major discovery in terms of what you have done to engage early career folks, early scientists, early students. How did that contribute to this leap that you described, realizing that you had to turn from math into geometry, into shapes, into pictures?
SYLVESTER JAMES GATES: As I mentioned in my opening remarks, this is my 51st consecutive year of teaching. Part of my teaching is done in a summer program, which I originated about 20 years ago, where I work with undergraduates, graduate students, sometimes post-graduate students, and sometimes even high school students. People often say how in the world can you work with high school students on a problem in String theory? The answer is it is all in the pictures that one paints mathematically. If you have the ability to paint pictures that are accessible to people, if they are bright, no matter what age they are, they can make a contribution.
The other thing about working with young people that I find just amazingly regenerative is that sometimes a young person will ask you a question about something you have answered a thousand times before, but they will ask a question or explain an answer in a way you have never thought about. This keeps one on one’s toes not to dismiss crazy ideas because they might not be so crazy. I think that is an extraordinarily important – that was extraordinarily important for me in my work. I would commend that to any scientist who wishes to keep innovation and ingenuity alive in their work.
JOSHUA GORDON: Thanks for that. It does speak to the importance of supporting our young people in science. Speaking of young people, we have a question that comes through an attendee from that attendee’s seven-year-old daughter. I thought I would ask it. I am just going to read it. I am not sure if the seven-year-old daughter used this language or not, but I am going to go for it.
The electromagnetic waves at 90 degrees that you showed look like DNA. Have you ever thought of the double-helix as two waves at 90 degrees to each other?
SYLVESTER JAMES GATES: I have not, but the reason is because if you actually look at the structure of DNA, the winding of the DNA does not suggest a perpendicular structure. There are bands between – in fact, let me go back and point out the structure using my PowerPoint. We saw it very early in my presentation. Let me get that up on screen.
To this young protoscientist – well, first of all, let me commend you. You are seven years old thinking about scientific questions. I started thing about science at age four. We are the people who go to have these extraordinarily long careers. I hope you will stay in science.
The point is the following: the strands of DNA that you see illustrated here are two separate strands. They are connected by bonds, which consist of guanine, cytosine, thiamine, and adenine. Those bonds are actually what keeps the two sides of the DNA strands together.
In electromagnetic waves, which I am now going to go to that picture since you brought that up – let me do this very quickly. It should be here. In the electromagnetic waves, there are no bonds connecting these green and purple lines. You can see one wave goes left and right. One goes up and down. But nothing connects them. It cannot be that these waves and the backbone of DNA are in some sense the same. There are no bonds here.
JOSHUA GORDON: Well, something must connect them. They are coming from the same thing. No?
SYLVESTER JAMES GATES: Since you want to go there, Josh, I’ve got something for you as soon as I can get my slides to advance. Here. There is something that connects them. That something is called the photon. The photon is also a set of functions, what we call the scalar electric potential and the magnetic vector potential.
It turns out that the EMB fields, which I have actually written for you here, on the bottom, are actually derivatives of these same things. It is just like if you think of motion. You can be at a certain position. When you learn how to do calculus – seven is a little bit too young probably to learn calculus, but when you learn how to use calculus, you will learn how to take an operation called the derivative. The derivative of a position is a velocity. The relationship between the electrical field, magnetic field, and the photon is the same as the relationship between velocity and position. You have to learn calculus to understand that relationship. It is not a solid bond. It is the derivatives that connect them, Josh.
JOSHUA GORDON: Thanks. One last question we have time for. This I am asking because we think a lot about these issues about whether there are two discrete phenomenon or whether there is a spectrum. So, why are the weak and strong nuclear forces separate properties and not one property that is really part of a spectrum.
SYLVESTER JAMES GATES: So this is actually hidden in something I didn’t tell you about, but we saw the basics of it. Remember that whole discussion about these mathematical crystals. I mentioned the mathematical term SU3. It turns out that there are lots of these kinds of mathematical crystals behind the physics of the Standard Model. The reason that the electromagnetic interactions are different from the strong interactions is because it is a different mathematical crystal that controls the laws of electromagnetism different from the laws of what we call quantum chromodynamics. It is the fact that there are more than one kind of crystal at work.
JOSHUA GORDON: I want to thank you again, Jim, for a great talk. I think one thing to think about is the complexity of the crystal that you talked about at the end with the many, many, many numbers you flashed across the screen and thinking about that relationship of that and the complexity of the brain and the kinds of math that we will need to understand it. It is a little mind-boggling to say the least.
Thank you so much for your presentation today. Thank you to the audience for listening in. Thanks to those who asked questions. Again, this talk will be posted for anyone that you would like to clue into it. It will be posted on our website in several weeks.
SYLVESTER JAMES GATES: Josh, can I add one final thing? If someone wants to email me questions – I know I shouldn’t do this. I actually respond to emails all the time. Just Google my name and you can find me. I will respond as my career allows.
JOSHUA GORDON: Thank you. We will see everyone at the next one of these things.