The Unreasonable Effectiveness of Quantum Physics in Modern Mathematics -- Robbert Dijkgraaf

hello and welcome to perimeter Institute's public lecture series presented by Sun Life Financial Canada my name is John Matlock director of External Relations and public affairs with the Institute Gregg dick your regular host and director of educational it reaches away tonight a little bit under the weather so we hope bread gets well very soon and meantime it is my pleasure to thank you for attending on this chilly night and to welcome Ian Bancroft of Sun Life Financial Canada our public lectures sponsor the entire public lecture series all season long is made possible by Sun Life so please join me in thanking Ian very much for your wonderful support now a little bit about dr. Robert digraph someone we've been hoping to share with you for three years now dr. digraph is director and Leon levy professor of the Institute for Advanced Study in Princeton his research focuses on the interface between mathematics and particle physics in fact dr. digraph is a mathematical physicist who has made significant contributions in string theory dr. digraph is also a very strong advocate for science education and he is a distinguished policy adviser he is past president of the Royal Netherlands Academy of Arts and Sciences in fact that is the nation's leading advisor in science to the government and he is currently co-chair of the inter academy Council which is a global alliance of many science academies that advises the UN and many other international organizations for his contributions to scientific research his leadership in educational outreach his support of the sciences in the arts and for his advocacy in the public policy space dr. digraph was awarded the Spinoza prize which is the highest scientific award in the Netherlands in 2003 a very high honor and more recently in 2012 he was named a knight of the order of the Netherlands lion and he is a member of the American Academy of Arts and Sciences and the American Philosophical Society so tonight he will take us on a rather mind-bending an interesting journey into the unreasonable effectiveness submissive quantum physics in modern mathematics please welcome dr. Robert digraph good evening ladies and gentlemen what an absolute honor to be here and with such enthusiastic and supportive and engaged public so tonight I will actually take you through an adventure about the interaction of two of my two great loves physics and mathematics and how they are interact and that's of course goes back down down history not every application of mathematics was successful as you see in this picture and my title has actually taken from a beautiful essay written by Nobel Prize winner Eugene Wigner in 1960 where he talked about the unreasonable effectiveness of mathematics in the Natural Sciences and it's a great article and starts actually with a little anecdote and the anecdote as follows this is owner of a shoe shop who's complaining that he never knows know how many shoes in which size he has to have in his store and then a mathematician comes in says no well there's a simple formula for that that's the Gauss distribution and he shows the curve and the equation and then the shoe salesman's goes and says well all very nice but what is this funny symbol here well it's the Greek letter pi what's that well is the ratio of the circumference in diameter of a circle and then a shoe salesman says what do circles have to do with shoes and it's a very deep point how does that Greek letter pi end up in that formula describing the world around us it's unreasonable that mathematics is so extremely successful time and time and again and at the birth of modern science scientists actually we're very much aware of this Galileo wrote about the book of nature and described that it's written philosophy gross physics is written in this grand book I mean the universe and it's written in the language of mathematics it's characters are triangles circles and geometrical figures Euclidian geometry was of course almost synonymous with mathematics in that time and without which it's impossible understand a single word one is wandering around in a dark labyrinth and more recent I'm Richard Fineman famous particle physics who actually wasn't a great lover of mathematics but still he said without those who do not know mathematics is difficult to get across a real feeling as to the beauty the deepest beauty of nature so if you haven't seen many mathematical equations in these talks you actually have missing something you have to understand the language when another hand fireman also said if all mathematics disappeared today physics would be set back exactly one week so I also see this is like okay physicists win until very very distinguished mathematicians gave the perfect answer is that that was the week that God created a world so my score is two to one math to physics I think it's fine but let's go to down for this I will find a very emotional thing to show this is a clay tablet from Babylonian times it's more than 4,000 years old and in it are written numbers and one of these numbers is this number it's amazing large number 343 million seven hundred sixty-eight thousand six hundred eighty-one just remember four thousand years ago probably an engineer was walking around and in the stupidest number and why is a special number well it's actually square of 18,000 541 which is itself a sum of two other squares and of course you know what I'm talking about is Pythagoras this was something that people use to measure lands to actually do engineering and physics they knew these numbers were important and and actually the old Greeks too were fascinated by mathematics these are the Platonic solids five perfectly symmetrical figures and the Greeks actually thought these were very important in fact they thought that there is an association with the four elements you know the four elements which was ohms of their idea of atoms earth fire water and air and since there were five of these symbols and Phi is 4+1 they had an association so for instance earth was associated with little cubes because you can stack them together and fire you can burn yourself and it is indeed then given by this this tetrahedra this kind of pointy shaped pyramid and then turn out that there was one figure that couldn't be associated with any elements and it's the middle one the dodecahedron and then plaintiff said well perhaps this is the stuff out of which the heavens are made the rest of the universe it's the fifth element the quintessence and from that time that figure the beautiful figure you see it back in arts in old possible ways here from Leonardo here by an Italian monkey was kind of the inventor of accounting kind of very boring but there's a little dodecahedron there to say way I'm aware that was more than just life on earth and artists of other life form and shape are using this this disfigure effect in the Dutch old hole of the House of Representative there is a dodecahedron not everybody captures the essence of that symbol because in the Dutch newspaper said well is it appropriate to hang a soccer ball in the earth in fact a few years ago it was on the cover of nature that perhaps our whole universe would have the shape of this del de Quito that would actually would be kind of moving in that space will be the ultimate vindication of the old Greeks and fortunately as often with a good story it was destroyed by more research so now we think probably not oh no no of these structures but they were in the minds mathematics is a mind to physicists for instance he is the young Johannes Kepler who was really obsessed with these figures and when he was young he built his first compost cosmology he noticed that 5 plus 1 is 6 at that time there were 6 planets orbiting the Sun and so you want to understand why these particular sizes of the orbits and the ratio of the orbits can you'd explain them and he had a model where he fitted the five figures into each other and their outer and inner orbits and he got within 5 percent of the distribution of the orbits of the planets now this you know don't learn in physics books because it was actually alder many reasons for more than six planets etc and many other solar system but as it was Kepler himself who destroyed this because he noticed by looking carefully the measurements that the planets are not moving on circles but on ellipses now circle is perfect ellipse is kind of an ugly object now it can have be any shape these days we thought that the heavens were perfect life on Earth was a big mess and the heavens they were ruled by perfect order so he himself write to the writings diary I'm bringing in a cart full of manure into science but of course sometime after Newton showed that in fact the ellipse is beautiful all the orbits are beautiful because they're also called conic sections so there's great mathematics hidden in it now I want to show many equations but I want to show you one equation if you've seen one you've seen them all a equals B that's the typical format of a mathematical equation I want to point out a very important symbol that's all fur looked often that's the equal sign in middle it's the magic in equation the equal sign connects two different worlds world a and world B think of equals mc-squared energy related to mass it's a miracle philosophers call this principle the importance of the equal sign Clinton's principle because you know this this American president who once said it depends on what the meaning of is is it's a crucial step and a few examples I will tell you are about connecting two worlds which were basically not expected to be connect and to see where you have any kind of how you think of mathematics I want to do a little experiment so I want you to look at the screen intensely did everybody see something so now I ask you a question what's the triangle above the circle research shows that in half of the cases people say well I look and I remember the image and where was the triangle was it above no it was below the other half immediately when I say a triangle above a circle sees an image of a triangle above a circle and then says well minimum that's not the mental image is not the same as the image has you on the screen people the first category category are people who think basically with the left hand side of the brain they think in terms of of language of analyzing language and order these are people if you want to give them the directions you have to say left left right left the other one thing to you metrical to think in terms of images not text with images they much more help if you draw a little diagram how to go mathematics is exactly the same one part of mathematics deals with algebra that's kind of logical steps basically doing things in time the other half thinks about geometry something that you can walk around more space instead of time now if you think about it our universe also has two sides to its brain large structures in the universe are described a general relativity about curvature of space and time that's geometry the small world of elementary particles is described by algebra by quantum mechanics and these assumptions have to come together you know in some times you can say there are two ways to do this one way is to look at the macroscopic world around us and so where can we find mathematics so one school says we find mathematics at the smaller structures by reducing everything to little building blocks we find beautiful formulas to describe the big mess around us so for instance I'll give examples where we find some algebra that we pull out of geometry using Quantum's of mechanics another school says no no no if you want to find beautiful structures you shoot at a large structure if you think of a glass of water as billions and billions of little molecules you will never discover the beautiful laws of hydrodynamics and the properties of water so we'll see also examples where beauty is in the large structures and it emerges out of kind of chaos and small stuff so there are two kind of brands of mathematics two kinds of aesthetic feelings you can have applying mathematics to the real world now if you live in the world of elementary particle physics you have to explain pictures like this and that technical term is kind of garbage no this is this is just coming everywhere and is there mathematics in that can you calculate these objects and during times people have had mixed feelings about that in the 1960s people thought about perhaps there are so many particles around that these days hundreds and thousands perhaps data is a black box like how we most of us think about technology you know something comes in something goes out you won't repair your own iPhone or something a black box that couldn't be opened in fact at one of these days Freeman Dyson and he gave a lecture here is a very distinguished professor at my Institute he said I'm acutely aware of the fact that the marriage between mathematics and physics which was so enormous me fruitful and past centuries has recently ended in divorce five harsh words a remarkable thing as he was speaking early 1970s it turned out that the black box could be opened and in the black box was a tiny tiny formula here I used the full power of mathematical symbols to write an equation of the standard model of elementary particle physics in terms of a single equation I can lecture about this equation to a mathematician and explain her or him every symbol and I will say yes I understand these are elegant mathematical structures these are things we study anyhow now this is not the way you learn the formula in in in college affect you this is the way you get it's written in components let's just to kind of bring home the fact that this is a serious business but then on the other hand actually can put the whole standard model on t-shirt 16 particles are now 17 particles with the hexer that explain everything we see around us and in fact most of these particles don't even play a role in our present situation they were important in the very early universe beautiful equations that describe all this isn't it amazing why is that formula not on every street corner and the amazing thing is that it describes the world not by giving very precise prescriptions but basically by giving probabilities a chance that something can happen particle again go straight through they can exchange the particle you can exchange two particles or three or do something much more complicated and the mathematical formulas allow you to compute a probability that this will happen and these numbers are among the the most precise predictions and verifications in all of science some of them into 12th decimal that's the distance of a hare measuring the distance of the moon to the earth within a hare with absolutely amazing it's really the small particles in some sense where we have great confidence and understand things now to understand the role of mathematics we have to go to something painful which is extra dimensions and I will actually there will come several times in this talk and of course you go to Einstein who famously said time is the fourth dimension now thinking about higher dimensional spaces is difficult so I have a little exercise here for those of you are afraid of extra dimensions so this is two dimensions that's a square and you can think well how will the square look in three dimensions well it looks like this so you can admire my animation but I actually think in perhaps it should have brought you with three dimensional cube but you see even if I had a three dimensional cube and with rotating here you would still look at something two-dimensional because you are looking right now at the retina at the back side of your eye which is totally flat everything you see is two-dimensional the third dimension you kind of construct in your mind to interpret moving images like this animation so here is a four dimensional cube so perhaps we can all look at it and perhaps some of you your brain will say click and you will see the fourth dimension actually had a colleague who was a professor biophysics who for a full year with stereoscopic views looked at pictures like this many placing etc hoping to say that that something would say click well a lot of things happen to him but he didn't see the fourth dimension you know that perhaps you have to detritus for two years I'm always looking for a volunteer here huh effect another way to think about this is more in terms of time so here is a movie and Afghan the fundamental idea of Einstein is if you have a movie it's a collection of photos of images take these images and stack them on a big pile from top from bottom to top so we going up in the stack you're going and moving in time and a single particle you see you the Reds and the green particle which are rotating around in space-time over here they become these strands these strings that go up and down and that's the language in which modern particle physics things space-time diagrams now this can actually be used to answer some deep questions and one of the deepest questions that perhaps not all of us realize physics has answered it's the following question now you learn about electrons and you can read about the mass and the charge of electron but is every electron exactly the same is there a factory in the universe which makes perfectly identical copies of electrons how and why why is the never one which has a little scratch for something so this is a deep question and it's also you can time you know exactly when the beginning of the answer was formulated and it was a telephone call in the early 1940s between John Wheeler famous physicist inventor worked on the hydrogen bomb invented black holes worked in many many areas in quantum mechanics and his then graduate student Richard Feynman fireman actually in his Nobel lecture explains this telephone call Saturday night and so your advisor is calling you in the middle of the night and we know said I know why every electron is exactly the same because there's only one electron in the whole universe okay please explain this was the image of wheeler here's our space-time there's the electron now usually should go up in time we all going up in time it's no way in which we can do it but suppose that I could go back in time and time go walk-in come up to the stage and stand next to me it would be two identical copies of me right I could do it another time would be three would be four so weena said suppose the particles that only go up but also go down in time and they could go up and down and up and down and make this big big knot suppose you would take one of these slices with one of these pictures he would do it somewhere in the middle you see many many electrons going up and down positively and negatively charged particles they would have exactly the same properties because they are all the same electron so that was his theory while Hyman said father but that why are there more electrons than antielectrons the bla bla bla but these particles trajectories are now called Fineman diagrams in fact after the war firemen actually started to work with this and this is a beautiful picture of one of his notes but the first time he was thinking about sending these particles back in time you see he is doing these particles going up and it's going back and he noticed that by sending them back in time they exactly are given the properties that we wanted so some sense this he really ran with the idea and in fact he was drawing these beautiful diagrams Fineman diagrams and finalists describing he's making diagrams while everybody else is doing difficult computations he hope one day perhaps all the physics magazines and journals and books would be full of diagrams instead of formulas and he's right that's actually the case in fact some time ago a young physic physics graduate student was walking in LA and saw this van and it was lady driving the van she stopped at the red light and he noticed that there were all these final diagrams on the on the van so he stopped I said well mrs. Lisi I really have to tell you that you're the symbols that are on your van they actually mean something in physics they called Feynman diagrams he said yes I know I'm mrs. fireman a beautiful picture also the fireman family with the van you know in fact it's even more crazy particles are not going up in time they can for instance split make a little detour and go back you know this is the fundamental principle in quantum mechanics at all basically anything can happen as long as it happens fast enough before you can detect it I always feel this is the principle that Dutch society is based on you know we're kind of genetically a favour to understand quantum mechanics and in fact you can even have this funny thing but out of nothing a particle antiparticle are born and then they annihilate again or as wheeler would have liked it a particle going up in time back in time and that's kind of closing loop making these endless loops just in vacuum in fact vacuum is full of these particles that are created and emulated again that's what we call the vacuum energy and certainly you must have heard in talks here about dark energy the energy that's inside empty space time of our universe and it must have something be a reflection of all these crazy processes so what has this to do with mathematics I want to bring you a mathematical problem that methods have been struggling with for centuries and that's understanding not now these are knots which are embeddings of circles in space or not a knot that you tie your shoelaces because you can another's but you you kind of you tie you somebody's shoe laces and then glued it to ends together you know that's actually kind of silly because then you never can untie your shoe but that's the whole idea so and how many different ways can you do this if you're allowed to move the knot around but not to break it is there something like a big book of knot and there is in fact there are is an infinite number of knots and we know if there would be something like an alphabet for not to even to order them like you order names in a telephone book the alphabet would have an infinite number of letters that's one of the really most difficult problems in mathematics and to understand how difficult it is that till the nineteen these of this infinite amount of letters that you should use to describe them only one must found around 1920 nuttin not a second was found so difficult is it but then if I solved it resolved using physics and you will now be appreciate what the physics solution was because people said let's look at a not like wheeler or firemen would look at or not they would think of this as a particle moving in a spacetime a funny space-time because it would have two space dimension and one time dimensions or two spatial dimensions and time pointing up and would be particle going up and down in space-time and it would be something that could possibly happen and the rules of quantum physics will give a number to this process which is the likelihood that it can happen so it could be like a point seven percent chance of this process so you would associate this number to the knot and in fact you can go around and if many things can happen you can send the quark around and then you can have a process where this will be your final diagram of your one particles interchange the glue one or two or three or four so you can use all the Fineman diagrams to describe all the knots in fact this is so successful this program that within twenty years ten years it led to the conclusion of this program and the mathematician Kentavious who basically made them got the Fields Medal you know there's no Nobel Prize in mathematics there's something much more difficult to get which is called the Fields Medal it's only awarded every four years and you have to be under 40 so it's like with your Olympics you know you can peak wrong like Andrew Wiles who proved Fermat's Last Theorem and turned out to be 41 whenever so bad and too bad you know you next life my second problem is actually that I want to deal with you is even more in some sense more elementary it goes down to the purest form of mathematics which is counting Kronecker the German mathematics famously said you know God created the integers 1 2 3 4 5 all the rest was human efforts it's it's very important to realize that mathematics is what mathematicians do or you you can walk around everywhere but you will never find a mathematical equation carved in a wall or something it's just impossible I must say I had one experience oh I have to tell you the little anecdote I was working in beautiful Woodlands and certainly I saw a sign and the sign had the equations of particle physics there in fact it had the equations of string theory in colors and symbols it's for standing there right in the middle of the root so I said what is this you know is there some divine intervention and then I noticed that there's a castle close to the roots and it was bought by the Maharishi University and no that's they had a sort of program and they embrace the laws of physics so they put these signs in the boots so people would actually know look at it and perhaps be inspired that's the only time I'll actually grow a fountain formula in the in nature usually you have to write them down yourself counter manometer geometry and that's actually a field which is very respectable in mathematics but it's sometimes it's also frowned upon as being kind of little bit recreational mathematics now it's just counting things because just to show of what you can do and but actually now it has a very different status and a more modest because of the element I want to tell you so for instance here I've drawn it's very difficult this actually is six dimensional manifold or it's three complex dimensions it's given by the solution of what's called the quintic equation so it's this equation over here it's a little bit like the formula Pythagoras a squared plus B squared C squared but then with fifth power and not three but five of these elements so this is a beautiful equation and need you the wonderful thing is it slices outer space the space a six dimensional space and the six dimensional space called the calabi-yau manifolds is very special because it's one of the rare spaces that allows a solution of Einstein's equation but yet it's compact it's like a sphere something so it's the anal analog of a flat sphere but then in six dimensions it's a beautiful space and mathematicians like it because they can count things on it for instance they can count curves as they call it they call they count curves of the greedy now I'm not showing you many patients but for those of you are interested it is basically the following you have your five variables X 1 2 X 5 each of them is a polynomial of a certain degree D in a single variable Z so it has all kind of coefficients a 1 D a 1 these coefficients are all unknown your 5 of these polynomials 5 set of equations with all these coefficients you plug them in in equation try to solve for the coefficients a all the A's over here and if you figured out you see there are as many equations as variable so there should be a finite amount of solutions clearly if D is larger you get more solutions to give an example one number every algebraic geometry knows is the number 2875 because obviously that's the number of lines on a quintic so the lines is degree one curves it's a classical result obtained in the 1880s the next one is conics already much more complicated these are conic like circles parabolas hyperbolas and they're already a large number of them 600 9250 and that was found in the 1980s and using the full power of modern mathematics the next step would be the number of cubics curves given by a cubic equation and there are 370 million two hundred six thousand three hundred seventy five of them now it's a funny story about that number because mathematicians try to compute is wrote a massive computer program got the number out in fact physicists asked them can you compute so they started writing this big program code of the code got a number come back to the number to the physicist if this is sir hmm are you absolutely sure there's not a mistake in your code okay they went back and checked and indeed it was a mistake and so they came back so sorry or you're you're right thank you for pointing out this is the number 370 million blah blah blah yeah yeah physicist that's right so okay well wait a month is a massive effort all the great minds you know endless computer time what did the physicist have for the physicist had a little list and this was the list dead all the numbers so just imagine you know that you know you take a century go from one to two it takes the massive advent of computers to go from two to three going from three to four is like the Large Hadron Collider of of mathematicians you know you wouldn't do that and then certainly all of them are there it's very difficult to convey the emotion this brings to a mathematician you know this this is like you're not just beaten to one results you know it's like you your whole life doesn't make sense anymore you know uh apparently you're doing something completely trivial where does all these numbers come from and these beautiful beautiful numbers well here's the story it has to do with string theory string theory is a generalization of particle physics where instead of single particles we think of little vibrating strings well a lot of things to talk about string theory in the mathematics really as beautiful elegant structures in fact you can look at the movements of strings and if a string would move just as little particles would give this kind of spaghetti strands these kind of knots in space-time a string moving in space-time gives a two-dimensional surface or you can think of it as a complex curve in turn uses the language of complex numbers which is really what mathematicians like so they think of this as a complex curve and so or a Riemann surface and if you someone you can think of these these are instead of the Fineman diagrams we have kind of Feynman diagrams on steroids so they're not like one-dimensional they are two-dimensional or like a fat thick Fineman diagrams the Riemann surfaces and what you can do if you want to another so that's the first time at the settlement second element of string theory it has many extra dimensions not yet space and time which makes four dimensions but it has so-called internal dimensions had small dimensions that are curved into each other in fact six of them and if curved in a way that they satisfy Einstein equation and so calabi-yau manifolds or this quintic in particular could be the internal space of string theory so basically of every point in space and time there is this internal manifold curved into itself of tiny tiny dimensions so string theory is quite interested how a string moves in that space and in fact it is that the shape of that space that in string theory would relate to what we found on our t-shirt the standard model of elementary particles in fact adding loops and handles to the to the to the space the club' are manifold means adding particles and they're changing their properties in terms of the four dimensional physics so not only string Theory's studied these spaces but he also studied the movement of the strings in these spaces now I said the string is moving around so in doing something very naturally it's doing what we call the sum over histories the fundamental law of quantum mechanics in quantum mechanics you don't have something happening everything happens at the same time only with varying probabilities if you leave the audience the lecture hold tonight you will you will go through the door I hope and just enter to the streets this is small probability goes through the wall and I think non events will actually no experience this because we not elementary particles this is just almost zero probability but it's a very small chance and so these numbers can be computed and in fact they belong to the most precisely computable numbers in physics in particular in these kind of mathematical settings these numbers can be computed exactly and in fact if you compute the numbers you get all of them the coefficients is this list of numbers I showed that kind of totally take took away the mathematicians they're just the coefficients in this formula here they are all together they make up one nice thing which is the quantum mechanical amplitude what is the probability of a string moving in that space now this gives it a context but didn't yet given solution there's another thing the physicists that they made they looked at what kind of spaces do your have or you have to quintic but are there other manifolds because they want to study physics that we want to say what kind of possible worlds are possible within string theory and what they do to make a plot they made the plot of all the possible spaces that could appear and they notice that there's a beautiful symmetry in the plot now it's kind of left-right symmetric so for any space they could find there was like a mirror manifold there was a manifold that shared some of the properties but was fundamentally different had different geometry different properties different particles but they were like paired in this deep way and it turned out that in particular for this quintic that all the mathematicians were interest there was a candidate for the mirror manifold and if you knew the mirror manifold the calculation is very difficult calculation but namely finding all the numbers n could be translated them as was an easy complication an easy calculation by the way on the on the mirror manifold so this kind of sounds like magic to mathematicians like okay well first of all you sum everything up to big formula and then hocus-pocus plus there's a mirror manifold you know and that actually captures all your formulas and this is the answer but clearly the numbers were right and in fact what happens is that mathematician just as with the nothing variants they took this idea and they said let's try to prove this not by kind of constructing all of string theory in physics oh no no no they take the hint they take the formula and they prove it in their own means using everything that now that they know where they were heading they were actually able to make the proof so this is now an essential part of geometry and it's it's it's done it's proven in a mathematical way the final subjects will have to do with something which is absolutely crucial to modern physics and also very dear to the heart of any mathematician which is symmetry actually going back to the old Greeks one of the amazing thing is that the Greeks were so fascinated by this symmetrical figures yet they didn't invent group theory which is the modern language we use to describe symmetry and the application of symmetry in the real world now again symmetry is crucial in our understanding of elementary particles you might have heard that particles come with all kind of properties like for instance quarks they have colors there are three colors there's a red blue and green of course it doesn't mind you know it's just a name or they don't they're not really have visible visual color and it's a good word because if you think about how you perceive color say on your television screen you think of 100 pixel disco it's actually made a color by actually getting green red and blue light to you in certain proportions if all three are absent it's black if all three are maximum it's white and you can make all the other colors into it the moment you are picking a color you're moving in a 3-dimensional space now cork is also kind of moving in a 3-dimensional space you can think of a quark as having a little arrow that's pointing in some some space color space and if that arrow goes around it kind of changes his property and the theory is that quarks are continuously doing this quarks as they are tied together with three of them in a proton for instance are continuously exchanging their colors these kind of arrows are wiggling in fact there is a kind of a field of quarks and I hope again you admire my PowerPoint skills because this is one way in which you could think of this is this is just like the color BL space it's everywhere in space-time actually as we speak is there's this quark field permeating every nook and cranny of this lecture hall in fact of the whole universe and the so does the symmetry where you could rotate all of these but this is not what happens in nature this I would say is really a kind of a communist way to move oh nature is much more liberal I say so every space point is free to move in its own individual way and so there's a huge symmetry because you can take any space point and in any point of space and time and move around the direction of the color of the of your quarks in fact it's this kind of waves in these fields that we in the end see as as gauge field set for instance the well the photon would be the exam of the four lights but in this case four quarks we call them gluons which are the the force carrying particles of the strong nuclear force that holds the the particles in the nucleus together in fact in using the laws of Feynman diagrams something like this happens you have a red quark and a green card the red park in the Queen Kirk they come here and then they exchange color so the red becomes green the green becomes red and they do it by exchanging this glue on which has two colors red green so it's a particle that can turn a red quark green and a green quark red and so it's our three colors you can imagine or nine three times three of these gluons well this is a small detail whether or not nine or eight but I won't go into this detail but that's actually what happened our eight of these blue moon flavors and in fact they will what they do is actually they are the particles that make these color changes possible so how is this again related to modern quantum physics well again string theory comes so to say to the rescue or I mean gives a totally different way of thinking about it now you see another picture is that you know we could be I talk to you about extra dimensions but the extra dimensions can be of various kinds for instance the certain physical model where our three plus ones or four-dimensional space-time is like a flying carpet floating in a much bigger dimensional space so there is there is an extra dimension but you can't reach it because you cannot tie down to this flying carpet now one of the way in which that carpet could have extra structures is not one layer bits many layers if I would give an old-fashioned talk with plastic transparencies you know you could have like two or three together you wouldn't even notice so you can have various of these she'd so you could pull them out as open strings would actually be attached to these strings and you could pull them apart you would see that the string is attached for long sheets and goes to another sheet so I actually should have given me them colors you could have red blue and green sheet and you the red green blue one would be something that would connect right through green sheep in fact you get a natural way a matrix of subjects a open strings which is a very elegant way to describe gauge fields in using this language and in fact if this thing would break apart it would break apart in two of these open strings which are always be connected to the same kind of intermediate point so you would have the rule of matrix multiplication which is the essential way to understand the symmetry groups of nature in in a very very simple and elegant way now why is this again important well it's important because all this is quite essential in a modern way of understanding black holes so black holes in string theory it's again a long story but the what we'll do actually the closed strings will actually curve space and time but you will have a black hole the black hole will have a horizon and the things kind of leaving on the black hole horizon are these open strings so in in a model language where you find gauge fields we find basically the analogs of quarks and gluons on the boundary of this black hole horizon and that has very very important implications of our understanding of space and time and how it all kind of fits together I would say this third example is something where it's more difficult to point out to the kind of formulas that food sweet mathematicians away but I'm sure that will be coming and there are already a few of them and and that's actually one thing that is a being a continuous pattern over the last kind of decades what we have seen is that like physics and mathematics are in some sense speaking different languages and what we are building and rebuilding a dictionary where the elements in mathematics have a physical integration I can draw nice diagrams and figures and the same figures and diagram and up also as mathematical formulas but you some sense you weren't aware in describing the same the same stuff I think one of the great thing that physics is doing this context is putting mathematical problems in the natural context I think that's something that you know mathematics is of course much more ambitious than physics in other two famous sayings one is that reality is the first approximation to mathematics I like that one and second well there's a famous story where a mathematician was describing to a large audience what the difference was between physics and mathematics he says well physicists describe the laws that God has decided that nature should obey mathematicians study the laws that even God has to obey and a typical mathematician will feel that they're not bound to reality so they can go in any direction but time and time again we have seen that putting things in a physical context is extremely useful for the development of mathematics in fact you can say that in some sense every context enriches a mathematical concept a famous mathematician who unfortunately died mr. young bill Thurston has written a beautiful article about the use of proof and mathematics and he comes with the simple example he says suppose you have something that everybody learns at any calculus course which is the derivative now if you go to a university math course you get a definition with Epsilon's and deltas it's pretty complicated and he asked the question is this the right way to understand the derivative and then he goes only gives auto definitely to say one velocity we all know the physical experience of speed and we know we can move we know what it means to go 50 kilometers per hour only for five seconds that makes sense it takes some time to get used but do we understand this intuitively and we understand that our velocity could also have a direction we can move in many ways many ways and becomes definition to think of a slope now you can think of a slope is a very beautiful way to think of a derivative you can see why you can have a slope in a mountain so it has two directions or you can think of it as an algorithm those of you were famous are familiar with calculus now the X cubes if you differentiate becomes 3 times x squared you can do that with other things than numbers you can think of this an approximation now think of as a linear approximation or think of it looking through the microscope and Thurston goes on and on and certainly as definition number 37 which is very complicated about the sum principal bundle and a flat section and I won't bother you with the details we said once I needed definition 37 and then he makes this point it says what we think of our our our key ideas and mathematics they come alive because they have so many applications and contacts and it's actually that what actually makes mathematics so powerful it's it's these kind of tools that have this unreasonable effectiveness now mathematicians are very neat so they they kind of cleaned up their toolbox you say well this is geometry this is algebra this is number theory but any commonly the simple-minded physicist comes they just wants to solve a problem now when he goes to look in the toolbox and takes whatever you want and just you know hit it and try to correct the problem and then certainly add note that certain mathematical problems actually come together so can that physical intuition be taught can you should you take any mathematician say well first no be trained as a quantum physicist and then become a quantum mathematician apparently doesn't work it's almost very strange that it seems to be that mathematical rigor and logic and physical intuition are in some sense complimentary very difficult to have at the same time none of the problems are described today have been solved by making the physical intuition more rigorous it was basically a path that was too long and winding and overgrown and nobody wanted to go there it's good for intuition but actually the proof was done in a very different way using the skills and intuitions of mathematics so perhaps again the left and the right hand side of the brain kind of in in conflict and finally there's an another point but of course physicists are very much aware that all this mathematics is extremely beautiful and tempted and seducing now and sometimes like or dis says that I have to tie themselves to the mast of the ship not to be seduced too much by the beautiful mathematical formulas because it's not obvious that everything that I described today is actually describing nature but it could in a very kind of indirect way of course be and it's it certainly is kind of all hangs together and in fact going back to the very first part of my talk where I told you about Plateau who had this idea that the dodecahedron the fifth element could be described the world outside the earth which we now know there's no need to we actually know with great certainty that the physics at very distant stars and planets it's the same physics that we were just finding here on planet earth actually Plateau had a something happened to him which typically for a theorist if you look in his text he writes it's only a suggestion it's in don't take this too serious later generations to be extremely serious and then it became a canola fide theory it was just a suggestion as a theorist you you try different ideas and of course plants all had a good image for that you know he had this famous plateaus cave where there are two worlds you know we are kind of we are the the prisoners in the cave who can only watch the shadows on the wall of the cave but somewhere up there in the Platonic realm are the pure and perfect mathematical figures and so it's the poor shadows of these objects that we are struggling with like you I can show you a picture of a dodecahedron I can show you an artwork of a dodecahedron but none of these has the absolute symmetry that it would have in your mind it's only the Platonic concept which really has the perfect symmetry but the crazy thing is that in these days we are living in kind of the upside down plat Plato's cave no it's it's not it's more physics in some sense that's kind of up in the air and has very grand ideas and complicated extra dimensions strings brains it's all there it's perhaps a dream world but a remarkable thing of that dream is that it's projecting shadows very precise and clearly defined shadows in the form of mathematical equations and conjectures that mathematicians then kind of take and work on and prove and enrich mathematics by it so I would say actually this dialogue this Devore divorce that was pronounced by Freeman Dyson I think it's I would say the two have really fell in love again you know it's a it's a very intense period of the interaction between mathematics and physics and seriously I think looking back in this time I think it's a golden age for the two fields and I think it's one of the intellectually most stimulating and exciting phenomena and modern science thank you very much thank you very much for extremely accessible and interesting and highly visual and even humorous lecture that was outstanding thank you very much we're going to open up the floor to questions now so if anybody has a question and please don't be shy about that please raise your hands our microphone runners will come down to the front of the hall here please raise your hands high will find you and ask you to stand up and share your question here's one right down here on my left so we'll start right here thank you good evening so question I have is that this what you talk about this effectiveness this has happened numerous times in history like graphic Algebra group theory development of it this has you know if physics applications in for mathematic mathematics in the 19th century so I mean are you like a theory are you advancing I mean is that like something the reason or because it seems to be like happening continuously throughout history like you know physics and you're quite right so there were I think there were curious of thank you very much for that question there have been periods of ups and downs between that dialogue I clearly now the 17th century was absolutely spectacular but what I think is perhaps the most unreasonable of this if you think of but almost all our mathematical concepts are geometry calculus now they all have come from the our everyday experience no it's it's our brain our brain has developed over hundreds and thousands of years dealing with stones and branches and just the laws of nature's as they apply to ourselves we have never lived in the world of quantum mechanics so the fact that we with our brain developed at room temperature can go to these extreme circumstances of the very small or high-energy of the very large and that our mathematics is still working more than that I think that it's even kind of enhanced that the kind of mathematics that is relevant to these realms is even more powerful than the one that describes classical mechanics it's it's not obvious for instance I think in the 1960s when poor people were see that the world of elementary particles is this big mess just if you look at the for instance the data of ordinary some of you appear from seeing the particle handbook which describes all the elementary particles and not the 17 of the standard models but all the other ones it's like a thousand-page thick telephone book you know results it's not at all clear that this is being described by a handful of equations so I think that is not to be expected but you can at some point perhaps draw a lesson and one of the lessons I draw is that nature likes to work in this way see is kind of fond of equations so famously said that I guess Darwin that these fond of beetles you know there are many but also equations and then that and no we never know you could run out of luck and perhaps at some point we do but it's extremely powerful that in the shortest distance that we have done measurements and the largest distance we have very simple equations describing this and I don't know what's unreasonable afterwards it's perhaps a tautology you say of course it's described like this but let me say there were points of existential fear that our luck would ran out thank you another question over here and a reminder show of hands anywhere in the theater and we'll bring a microphone to you do you think that because of the other exchange particles the patterns they have with the string theory do you think you could prove what ever on the graviton exists ah uh well it's a kind of a difficult question uh because it's some sense what do I think actually there's kind of a different kind of proof which you mean it's it's really kind of finding the experimental evidence for profit owns and I think actually we will see a lot of evidence in direct evidence in cosmology but actually detecting a graviton you know with a classical graviton will actually need a gravity detector gravitational wave detector that that are being built I think there is a there is a very powerful theoretical argument that they should exist now we see indirectly there are waves of gravity just as we see electromagnetic waves we know that these waves come in energy which can be smaller and smaller small at some point it will be kind of a quantum theory will say there will be minimal packets of energy that you can store into it so it's I still find it very difficult to avoid the concept of a quantum gravity wave so a graviton I think actually in terms of the mathematics that's involved it's not it's actually some pretty straightforward it's you know it's it's just by analogy of the photon so I don't think that we need to solve very difficult mathematical problems there thanks very much for that question I'd like to follow up on that with a question yes emailed in here and it's actually a beautiful segue to our next public lecture next month and the question was what sparked your love for science so maybe you could take a few minutes on your journey from elementary to high school to post-secondary and how did how did you come to be the mathematical physicist a you are well I'm sick part late and I think quite many people are quite late to mathematics and physics because of course you got it at high school but I never realized that this was just the tip of the iceberg in fact the first time I really realized this was I think I was like 11 or something and I somebody gave me a book about a very different topic about DNA and genetics so I would still very vividly remember reading the book and it's a sort of crystal clear and I felt I was there was a big conspiracy out there that prevented me from knowing this oh I ran into the kitchen my mother was cooking this as well this is just a good disgrace you know this is this is how we work nobody told me and uh this is basically the first thing you should learn in elementary school and and the same way I'm still upset that people think if they want to teach physics to young children that to be of little things that around you or something I think you should know that the world is made out of atoms so let's see that's bet that there are elementary building blocks of the world it's the most fundamental thing so I got in that I think it's only when I was like 15 16 that I started to discover not only that I was this kind of you know went kind of naturally but also that there was a whole world around and I think some of my high school teachers were extremely helpful in I know I got an old mathematical encyclopedia and in sometimes they'll think very fondly of these days where you weren't taking any official classes you were just figuring out yourself I think that's that's the greatest pleasure of being in being a physicist or a mathematician I think you you can just you need just a piece of paper and you can just work yourself thank you thank you for that

Channel: Trev M

Views: 483,252

Keywords: Mathematics (Field Of Study), Physics (Field Of Study), Perimeter Institute For Theoretical Physics (Organization), Robbert Dijkgraaf (Academic)

Id: 6oWLIVNI6VA

Length: 59min 55sec (3595 seconds)

Published: Sun Mar 29 2015