Roger Penrose on Twistors and Quantum Non-Locality

it's not the easiest subject to talk to a general audience about but I'll do my best in fact I'll use a slide here which is as you can see from the kind of grub you look about it is a very old one it gives you the general idea of what a twister is in a certain sense here we have that is supposed to represent space time here we have time going up the picture which is usual in my pictures and space going off horizontally it's supposed to be three-dimensional this way so I you can see the three arrows there but of course it's hard to draw all the dimensions especially on the flat piece of transparency here now here here we have a light ray and basically that gives the initial picture of what a twister is like so space it first of all you have to get used to space-time already so we'll assume that you're not unhappy with the idea of space and time all being fitted together into one thing which is a four-dimensional space-time as we call it well what's this this is the history of a light particle it's a photon so it whizzes along at the speed of light and you scale things so that space and time are well if your measure of time is one second this way then it has to be one light second that way that's a long way actually but don't worry about that we like to write up draw our pictures so that light rays are at 45 degrees that's just handy because otherwise you know if you had a normal they'd be more less horizontal so okay you'll notice that there's a sort of corkscrew attitude to this thing where you might wonder what that's doing well that's because a twister really is not just a light ray its characterizes the spin of the particle as well so photons actually do have a spin and they can either spin one way right-handed or left-handed or combinations of the two and this is supposed to indicate that this thing is is spinning in the direction I think it's meant to be right-handed in this case now what's this over here that part of the picture is the space of twisters so one entire light ray is the single twister and so that over here is represented by a single point so that point is supposed to represent this thing here and the idea is to think of this space which is the twister space as being in some sense more fundamental than the space-time and you might say well how do we reconstruct this space how do we reconstruct space-time which is this space over here from the twister space well you say take a point here and look at all the photons going through that point and all the light rays through that point and they will represent a sphere that's really the sphere well I've got another picture down here so if you look out into the sky this is your light comb which is showing the light rays coming into your eye as you look back down and you will see a number of stars perhaps and those the world lines of those stars that's the histories of the stars in the space-time picture which is here and they will intersect your past cone that's the light rays coming to your eye each one will be the point you actually witness when you see that star and this is the field of vision that you might see I've got four stars here and it so happens that these four stars lie on a circle there's a good reason for that which I'll come to in a minute but this thing which I've called a Riemann sphere and I'll try and explain why it's called that represents all the light rays through that single point so that point where it's a point in space times you call it an event an event is a point in space-time and that is represented by that sphere so if you knew in this space where all these spheres were which represented points here then you would be able to reconstruct this space from that one so the idea is it's a sort of correspondence you can go backwards and forwards then you might say well why bother I mean you could do it here why what's the point of this well the point is that it reveals all sorts of structure that you don't easily see in space-time and I've listed a few of the things here complex now I don't mean complex in the sense of complex I mean complex in the sense of complex numbers which I'll come to in fact I might as well dump them down here while I'm talking about it here we have I'll come back to that but complex in the sense of complex numbers non-local well that's part of the title of the talks I ought to be saying something about that and it's the sense that that things there are phenomena in nature particularly revealed by quantum mechanics which are not local you can have things which were related to each other they look as though they're a long way away and somehow they're all part of one thing and you can't regard them as independent separate things it's a very mysterious feature of quantum mechanics which is the point of this talk trying to grapple and look at this in a different way which somehow makes sense of it in a way that you wouldn't really make sense of it if you just had this picture so that's part of the idea and I'll try and say a bit more about that sphere and the sphere appears again over here it's the same sphere really you see the sphere represents all the light rays through this event and now if you take this event here and you look out you will see the light coming into your eye and you see various stars and that's if you like your celestial sphere sets this the sky you see when you're out in the dark night and you look at the stars out there but especially if you're out in space and you could see all the way around the earth tends to get in the way if you're on the ground but apart from that and there's a rather strange feature that these spheres have you see here I've got a picture of two different observers looking out at the same sky they just manage to miss each other and adjust at the same point basically and they look out at the same sky but they're going at different speeds very hugely different speeds and mention though you know one of them might be going half the speed of light or something and looking out at the same sky okay they seem to say see the same stars the skies distorted in a way because in the direction you're moving things tend to crowd in it's a thing called aberration but it's distorted in a very particular way which is such that the circle here which I've got those stars which happen to lie on the circle they'll still lie on a circle for the other observe okay well I'll come back to these features later I think I just want to leave this here for the moment so you'll see the general idea of twisters and let me say something more about this other thing which I don't think if I commented on yet maybe because it looks a little forbidding mathematically sophisticated see that is one of the things that it depends if you're a mathematician that looks nice you say oh that's good but if you're not a mathematician it's a little bit daunting so I'll try and explain some of these ideas in a way which I hope is reasonably accessible okay first of all we need to know about these complex numbers so we have numbers which involve a square root of minus 1 well the square root of minus 1 is called an imaginary number you might think it's not really there if it's called an imaginary number but you have to bear in mind that it's just as real as any other kind of number it's an I mean these things were first thought of many many centuries ago and it and ok then mathematicians began to realize that they were useful in many ways but they're still kind of you might think of the inventions just to help you but in fact they turn out to be deeply embedded in the way the world works at the level of the small things that are governed by quantum mechanics so we'll come to that ok so complex numbers mathematically are very usefully and able to solve algebraic equations I've written one sort of randomly here the thing is you can solve all of them that's what's amazing about them you just start with a simple equation like x squared plus 1 equals 0 what x squared minus 1 you see and that would be the square root of minus 1 so then the miracle is that all you have to do is have something which solves that peculiar equation which is the square root of minus 1 and then you can get all the solutions of these equations no matter how complicated this is so you said to the hundred that's a pretty big equation there ok now what is a complex number it's a thing like this said equals a plus I I is the square root of minus 1 times B and a and B are ordinary what we call real numbers well even then a question why are they real you see only real because you sort of uses them somewhat but they're infinite decimals and is that real and you worry about it on your compute your calculator or your computer you can only put down a certain number of rows decimals do all those decimal places mean something so you might ask the same question are these so-called real numbers real or not so that's the sort of question we mathematicians like to answer and say where they're all just as real as each other we like to think they're all real so that's that's fun okay now it's not just that as I said these numbers are fundamental to quantum mechanics absolutely fundamental up until that point the numbers have been just useful in many ways they're useful for example with electrical circuits if you want to understand about electrical circuits you use these complex numbers but it's just a sort of trick whereas here they seem to be revealed actually in the foundations of the way the world operates they're not just a trick they're really there that's a subtle point that it does seem they really are there in a sense see here is a this when I write that funny kind of see that stands for the complex numbers and they play important roles in many attempts at physics basic physics such as in string theory and we'll see in twister theory so I try to reveal some of that and we'll come back to this slide because it gives a sort of summary of the ideas of Twista theorem sense I go back to the complex numbers this is sometimes called the argon plane and the Gauss plane and all that but it really should be called the vessel plane because it was Kasbah vessel it was a Norwegian surveyor and mathematician who apparently first thought thought of this way of representing these complex numbers you represent them in the plane the real numbers going this way and the multiples of the square root of minus 1 that way and then you fill out all the other numbers which is a sum of a purely real number and a purely imaginary numbers it's called and then you can add them and you can multiply them and it's quite nice if you have a geometrical turn of mind that each of these is a nice piece of geometry you see to add W and 0 just form this parallelogram and there's the sum if you want to multiply W and Zed where you need to know where the origin is and where one is when you look at that triangle and then you make another triangle which is similar in shape it can be bigger or smaller but it's the same shape as this triangle and there is the product there W so you can add and multiply just using nice bits of geometry there's also this notion of a holomorphic function which is really very important in this subject ok it's a rather complicated sounding word what does it mean well the way I'd like to look at it's quite simple really it's built up from adding subtracting multiplying dividing and taking a limit but what you can't do is Express the thing called the complex conjugate so if you have X plus I times y and that could be this point zero up here then the number which is called the complex conjugate is replace I by minus I you see if you have a square root of minus ones we have here you might know about square roots if something is a square root of something then - that's something is also a square root of it so plus or minus so that - what - I would have done just as well as I so you could say well those we've taken minus i instead well this operation which flips you over from the number to its complex conjugate that's reflecting it in this real axis that's not holomorphic so we think of Z&Z bars in some sense independent things that's the way you're supposed to think about them and if it doesn't involve the barred one then that's holomorphic that's another way of thinking of ok that idea will come back - and that's a bit of this mathematical sophistication thing but you think of it this way it's not so complicated to think of in fact it's one of the things when I was learning about these complex numbers and things at university I thought they were just incredibly magic and this is some of the thing about them is that one of the reasons that I feel that it's really nice to see them as part of the way the world operates is there is a magic about them and I like to feel that the laws of physics are magical in a way and somehow this magic is governed by the mathematical magic of these numbers okay now let me say something else you see I've got adding and multiplying or subtracting it's not so different from adding what about dividing okay well once you know how to multiply you you basically know how to divide so let me put that in here it's just the same picture as the one I have here but I'm doing something else to it I'm going to think of not just multiplying Z's and W to get wz to get multiplied together but I'm thinking of dividing Q which is going to be W Z by Z so if I divide it by széll you just go back down the picture it's that one so that's not very difficult if you've got this triangle here and it's the same shape as that one they're similar then you can get you can see what the division of this number by W is by Z is but you know when you divide there's a snag because you're not supposed to divide by zero and that leaves you into all sorts of trouble okay so let's see what happens if I push then right down and make it get smaller and smaller and smaller till it gets to zero what happens to W over said well this point goes shooting way out like that and it goes off the picture when you get to zero it doesn't much matter you can go into zero and all sorts of different ways it just goes off to infinity in different ways now there's a very nice way you can handle this in complex numbers and that is we take this vessel plane as I have here and you sort of fold it up into a sphere so you add a point right out is infinity which will help so that you can actually say what you mean by dividing a number by zero it's infinity you might think well infinity is not really a number well it is here it makes that's the nice thing about mathematics you can you can cheat you see you can you could say that you're not allowed to do this and then you think well what happens if I do that and if you may find okay may land you in trouble immediately but it may not and if it lands you in trouble you may think well there's a way of getting around that and this is one of those things so you can see that you can get around the problem by constructing this thing called the Riemann sphere this is the Riemann sphere right here it's called the Rhema sphere because Riemann thought of it I think that's good enough reason and how do we do this now okay we take a sphere perfectly nice sphere in ordinary three-dimensional Euclidean space and I take the equator to that sphere and I put a plane through the equator and that plane is going to be the vessel plane or the complex plane so I've put the plane here there it is and there's a thing called the unit circle that goes through one and I and minus 1 and minus I so it's this circle here very important it's the unit circle and that units for circle is supposed to coincide with the equator of the sphere and then I take the South Pole and I project it's called stereographic projection it's a very beautiful kind of piece of geometry I project from the South Pole and I'd say take appoint you that's some complex number you see that's a point on the vessel plane there I join it to the South Pole and see where it goes it makes this point up on the sphere and I'm going to call that u on the sphere zero projects right up to the top that's the North Pole anything on the unit circle goes to itself but then I have infinity you see infinity if I tried to project under the plane it would go right out to infinity on the plane that's where I got to when I was trying to do my construction over here so infinity when it's the sphere it's right there it's an honest point on the sphere so the Riemann sphere includes infinity now we're going to come back to that because it's really important in not only quantum mechanics but in relativity and to me this was a link you see we have something which is fundamental in quantum mechanics and that's this Riemann sphere now of trial explain why and something else which is fundamental in relativity and that's the sky which we think of as also a Riemann sphere and the link between them has to do with these fundamental complex numbers and that is really the basic driving force behind twister theory ok so now I'm going to come to this but let me say a little bit more about relativity because that was that's the easier one if you like it's relativity maybe confusing but it's easier than quantum mechanics I'll tell you what we need to know is the light comes I can hear you see this is the sort of basic structure in space-time space-time is four-dimensional you get used to that by throwing one of the dimensions away that's the usual way of doing it so you think of the picture I've just thrown a space dimension away I have them all in this picture here but this is the light cone and you think to see what does it represent a flash of light coming in to that point and then you think of an explosion there and that's this flash of light going out and the history of that you think of sections through it that's now that's the next moment that's the moment after that so that's now that's the next moment think of that as the sphere now not as you see it looks like a lips or a circle there but it's really meant to be a sphere and this is meant to be another sphere because of the extra dimension and it's going out with the speed of light okay so that's the light cone and here is a picture of special relativity and lotsa light cones all over the place and they're uniformly arranged and light rays have to go so that they're always tangent to the cone and massive particles have to have their histories within the cones but if we now think about the picture that I had before and I've now got a slightly different version of it this is the celestial sphere and you're looking at these four stars and there's the past light cone intersects the points and I'm really just saying that thing all over again differently moving observers perceive this sphere in such a way transform from one observer to the other so that circular patterns go to circular patterns well if you know more about these complex analytic functions and the whole of holomorphic functions and so on you find that if you want to have a transformation which takes this sphere and now we're thinking of that as the Riemann sphere that's point we think of that sphere the sky as this Riemann sphere what's the point of thinking of it that way well the point of thinking about it is that if you look at the transformation which transform the sphere into itself but which preserve its structure as a holomorphic thing that is the equations that you use are holomorphic there they don't involve the complex conjugate they only involve these adding multiplying taking limits this kind of algebra that kind of analysis which shown trying to promote here and if you ask for the holomorphic transformations of that sphere see if you're thinking complex it's really not a sphere it's a curve that doesn't look much like a curve does it but you see that's because we think of curves in terms of real parameters a real number will tell you how far you've gone along the curve but if you think of a complex one okay it's two dimensions for a real person it's a the vessel plane you see it's a plane but now when we fold it up into a sphere it's a sphere but it's really only one dimension because it's one complex dimension so that's the idea you've got to get your mind around that a complex dimension counts as two real dimensions so this may look like a two-dimensional thing but you think of it as a one-dimensional thing if you think complex that's the idea and you think complex you then got to think holomorphic too and holomorphic sending this into itself the only transformations you can do are these ones so the only transformations of the Riemann sphere to itself which are genuinely holomorphic are the ones which preserve circles and they are the ones that you actually can get from one to the other by transforming from one observer to another so that this complex geometry the homomorphic geometry is really revealing something about relativity and it's not an obvious thing in fact the fact that circles go to circles when you think about it a bit it's a bit surprising but it's you think about it the right way it becomes obvious and usual thing is to think about it the wrong way of course and then it then you have to wonder why it's true I'm not going to explain that because that would be going too far afield okay now I think I'm going to say something about quantum mechanics what is the most basic thing about fundament I talked about the most basic thing about relativity it's this group the Lorentz group we can be understood as a holomorphic thing okay what's the basic thing about quantum mechanics it's quantum linear superposition which is both beautiful and very puzzling if you have two states of a system and I'm calling them a and B then you can form combinations of W times a plus Z times B where these are complex numbers they've got to be complex numbers or it doesn't work and that's physics they're out there in the world now we note another property of these state things these things I've written with green capital letters and if X is a state that if you multiply it by a complex number it doesn't change the physics so these things are really what are called state vectors they're a little bit more than the physics there there are the things that you use in the formalism of quantum mechanics and if you multiply one of those things the state vector by a complex number which is not zero you don't change the physics so that means if you took this thing here W a plus Z B and I multiply it by Q I get qw+ qzb and it's the same physical thing because multiplying by Q all the way through doesn't make any difference but what I haven't changed is the ratio the ratio of W to Z is the same as the ratio of Q W to QZ the Q's just disappear so the ratio of these things is what's important so in any two state quantum system it's the ratio of these two numbers they're called amplitudes complex amplitudes I may use that word from time to time but there are these multipliers that you have here that have to be complex numbers now here is where Riemann sphere comes in and I'm going to reveal what's underneath here this is one of the simplest quantum systems and I've always found it an amazing one every time I think about it I think it's amazing if you have a particle like an electron most of the basic particles which constitutes matter electrons protons neutrons and the quarks which are the constituents of protons and neutrons they are all things which I have what's called spin half they spin at a certain rate which is half in basic units now a spin 1/2 particle is basically built up out of two independent states and I'm calling them up and down that means spinning right-handed about up so you thought think of spinning about an axis which is pointing in the direction so the spin is right-handed about that direction or it could be spinning down right-handed about that and those are the two basic states if you like and all the combinations that you can make this is this super superposition here's an instance of this basic principle of quantum mechanics and I was regardless is amazing all these combinations are another state pointing somewhere else so you can produce other states as combinations of these two and there's nothing special about up or down after all what's wrong with right and left what's wrong with that way in that way they're all just as good as each other and the thing is that you can see for that sort of reason at least with this kind of system these two basic states are no more no more particular than any other combination they're just as good as each other okay now there's a nice piece of geometry here and you might have guessed what this nice piece of geometry is long as I can find my transparency oh it's just here is it no wonder it's there already it's the Riemann sphere here we are you see how do we know which way this is pointing when you put W and Zed in well what you do is you form the quotient I said it only depends on the ratio so you form this ratio which divide one by the other of course it might be infinity if W happen to be naught but we know who deal with that now because it's just a Riemann sphere and you take your point W Z over W it's a point on the vessel plane there it is you do the projection I was just talking about before it's a point on the sphere the direction out from the origin to that point is the way it's pointing so you have this beautiful connection between these funny parameters these numbers in quantum mechanics which in mysterious numbers that the mathematicians like to play with for many centuries and suddenly they're part there they're sitting there in physics and they're there in the spins of these basic particles you have particles which have more complicated spins but you can still see them there they also have nice geometry I don't want to go into that now but this is where it's most manifest and you can see that Riemann sphere really playing a role and you see also this also used to strike me it depends on space being three dimensions I don't know that I have to be a little nervous if there's string theorists in the audience who tell us that space ought to have lots more dimensions that's one of things and one of the reasons I never liked that idea particularly fine strings are fine what I don't like isn't having to put them in this huge number of dimensions because of things like this which seem to tie these basic ideas to three dimensions of space now I'm going to tell you something that's nice but it's very strange and this is part of the point of this talk quantum nonlocality see there are these very strange things well it's basically Einstein who started worrying about these things first he was I think trying to poke a hole in quantum mechanics because he even though he's one of the people started the subject he didn't like the way it went because that's not the only case where that's happened in science but in this case he thought there must be something wrong and so he showed pointed out things that were very strange implications of quantum mechanics thinking that they were going to show quantum mechanics was wrong unfortunately in this case it showed that he was wrong that quantum mechanics was right but they were very insightful things to point out and they these contributions from Einstein his colleagues but all Skien Rosen and David Bohm which is more like this specific case I'm showing you here he pointing out the quantum mechanics does have these very very strange implications now here we have something starts off in a state of spin zero that means it's there's no direction it's got to it at all is this symmetrical thing and it splits into two particles each of spin 1/2 so they're like the ones here and you spin over here and here have this property that their opposite now I should say this is one of the very strange things about quantum mechanics it may be the thing is spinning in some direction now how would you ever know well you could say well let's measure its spin but the thing is when you measure the spin there's only one bit of information you can ever get out of the spin of a spin 1/2 particle so you choose your direction say well let's try this one well if you tried that one and you got it right you see if you tried that one you tried that direction you say it says yes I am indeed spinning that way if you tried something else it would only give you a probability if I tried the opposite one say no I'm spinning the opposite way if I try this direction it would be 50% chance that one way or the other way that's all you can get it'll just say yes or no this way or that way it won't say no what which direction it's spinning so that's one of these funny things the state tells you which way it's spinning but you don't know what the state is and you can't tell by doing a measurement to work out what that state is so it's quite subtle to know that these things are spinning oppositely you might choose to measure it spin this way and then you know if somebody else measured this one it would come out this way but they might measure it some other direction you see well john bell introduced these things called bell inequalities I don't think he called them that that's what other people call them these are to do with ways of distinguishing this very strange behavior of quantum mechanics from the way in which a classical system could behave you could imagine that these particles coming out here are some piece of machinery so each one and they start off together with some rules about how the machinery on this one is related to the Machine around that one and what Bell shows is that the way quantum mechanics behave simply can't be explained that way if these things are not in communication after they've separated there is no way that they can imitate what quantum mechanics actually says they do and these this is this is quantum nonlocality this is the it was really John Bell who showed that these were real things quantum mechanics behaves in a way and it's it is a very subtle thing but you might say either these things are in communication or they're not if they're in communication this one could send a signal to the element says look I've just been measure had my spin measured this way be careful to measure yours you know the other way something I lack see you could have messages like that that's communication but it's not like that because if it was like that you could violate relativity you could send a signal faster than light and that would violate relativity because these things could be photons and not spin half in fact they usually are photons in which case you can't get a signal back to the other one just he couldn't say look and be measured this way you better make sure you do it the other way it's too late you see because the signal cut get over there so but nevertheless they are what's called entangled that well that word entangled was introduced by Schrodinger to explain the kind of system that's the situation that comes here now there are examples using photons I'm not quite sure what the record is at the moment but the distances you see you think of quantum mechanics as being about small things okay the things are small all right these could be photons that's small in a sense but they can be awfully far apart in fact as far as we know there's no limit to the distance apart that they can be and there are experiments which have been performed in Switzerland I'm not quite sure what the record is here but at least over 100 kilometers so photons separated by 100 kilometers and they have these entanglements between each other at which they they seem to be part of one system they can't communicate this information to each other but you can't explain them as being separated from each other they're still entangled well the nicest example the simplest example of this I'll show you that one is won by Lucien Hardy of Perimeter Institute I'm glad to say so let me try and describe Lucien's very beautiful example it's not quite what we have here it doesn't start for must been zero system it has to start from a thing which has spin one it's the next case art from spin half and it's this particular state you can put it in this state it I've put some arrows here to try and indicate which state it is if you have to know about these things but don't worry about it there's some state here and it goes off a long way way this could be you know miles apart and you might choose to measure the spin of these things now the rules of this game are going to be that the person over here the right-hand person is going to measure either up-down and it's going to say if you say up-down it's going to say yes I'm up or yes I'm down no to up if you like or right left there's a choice between those two up-down or right left and the one on the left is going to do the same thing up down or right left now let's imagine that these clever particles here have sorted out between themselves what they're going to do and then now they're out of communication so that's the kind of thing that you would think would happen if these were classical objects certainly now there are certain rules that you find and Lucien has worked these art so the rules will always hold there are four things that these people could do they could measure up down up down left right up down left right here and up down there or or left right left right whichever it was and the rules are that sometimes this is the only thing which isn't just yes or no sometimes it must be the case and it's about probability of a twelfth maybe out of twelve occasions where these things get emitted roughly speaking there will be one of these which does the following that if it happens to be the case that the two observers measure up down but they sometimes both get done and that has to happen about one out of twelve of the times that they choose it down so these little particle people down here have to decide that these things must occasionally one out of twelve about say that if they're measured up and down they will sometimes give down down okay but they don't know which way they're going to be measured so it might be that the one on the Left didn't do up-and-down but did write left instead now it's the rules come out quantum mechanics tells you that you never get left for this one and down for this one you never get down for this one and left for this one and you never get right for this one and right for this one so those are the rules okay now what's wrong with all that well okay sometimes I say it will do this so let's suppose that these things have come apart and the one on the left has decided instead of measuring up and down left right well we know that the rules tell us that if this one's going to come down this one has not allowed to be left so it's got to be right okay but they didn't know it was going to do that and they might not have done that they managed not the other way this one did do up down but the one on the right hand side decided to do left right instead now these rules tell you that if first one's going to be downs it has to be from this this one has not allowed to be left and so it's got to be right but then suppose neither of these had been up and down and this fella the people in here didn't know that it could have been both left right and since this we found out that this one's got to measure left and this one's going to measure left in this case but that's not allowed so they cut you they can't get it right there's you always can catch them out one way or another if they are what's called local realistic models if it's a local realistic model that is to say that these these pieces of machinery which are trying to imitate quantum mechanics okay they can confer with themselves right at the beginning and decide what they're going to do but from there on they're out of communication there's no way of imitating this behavior so that's a nice example nice enough that I can actually sort of explain it here can explain why does it but that's what the the rules of quantum mechanics come out with this answer and indeed this is the answer that experiments give you okay that's space-time seems to have this very strange kind of entanglement built into it and it's something people genuinely worry about for good reason now I want to try in order to get to the point where I can explain where twister theory has anything to say about it which probably only lift myself ridiculously small amount of time to explain it to you never mind I'll do my best I wanted to tell you all that because it's really rather fascinating I have to tell you some more about quantum mechanics I'm afraid okay I told you about the quantum superposition of two states which is what we see an example of here but it really goes on to lots and lots of states and you say you might have a particle in one place or another place or another place or another place and each one of these could is have its quantum complex number attached to it and so every position in space would need a complex number and I've tried to lump them all together here each one is labeled by a complex number this is what we call the wave function the wave function of a particle is really just telling you what these complex numbers are for each position it could have X is the position and size X is telling you what that complex number is at that position that's the wave function very fundamental to quantum mechanics it's the basis of the Schrodinger approach to quantum mechanics now what about several particles see if you have lots of particles then the particles can be here here and here and we've got a an amplitude for that here here and here well they're pushed before of them here that's right oh there could be here here and here and here and all of those different arrangements all together have to have one of these complex multipliers so the sigh the wave function is a function of all these numbers together and that's where this entanglement comes from because it's not just that each one separately has a quantum quantum amplitude for where it is it's the whole collection altogether has one of these quantum amplitudes associated with it so that the wave function is a function of all these things you might be lucky and find that it splits into this product but if it doesn't that's what's called an entangled State so in fact most of the states are going to be a great entangled mess in fact it's a great puzzle to me why we don't see that okay you can detect these things in quantum mechanics but but quantum mechanics doesn't really tell you why we're not all a great entangled mess we don't seem to be least I hope not now okay that's I've told you about wave function so it's not bad now I'm going to tell you about another part kind of wave functions these are these are positioned state wave functions the ordinary roading or idea but you can also talk about momentum state wave functions a particle here could be moving that way or that way or that way the momentum is the mass times its velocity velocity can have different directions so again it could be going this way or this way or this way this way and each of these possibilities separately has this complex number attached to it and so again you have this thing which is again a wave function but this time dependent on the momenta so that's a momentum state wave function and there's clever ways from getting from this kind of this kind which use what's called a Fourier transform a nice piece of mathematics but I'm not going to worry about it I am going to tell you something very mysterious though but the momentum P and the position X are what are called canonical conjugate variables and people doing quantum mechanics love these things because well tell you why sense and it's very mysterious but it's all part of the way quantum mechanics works but if you're working in momentum space and you want to know how to talk about position it's not allowed to be in there but the position turns out to be differentiation with respect to momentum now if you don't know what that means I'm not going to try and explain to you it's telling you how the the rate at which things change so that as the momentum changes the rate of changes is this position well that's a funny thing isn't it there's actually one of these has got a minus sign but I couldn't remember which was which when I was making the slide so I imagine one of them's got a minus sign in it but never mind too much about that okay back to twisters now I'm going to if you remember my original picture here like this why don't I just do that I'm going to say a bit more about this space here here is minkovski space that's the space of space-time a special relativity and here is a light ray in that space-time of special relativity and here is the point which represents that light ray it's just this picture yeah that's that point representing that library now I'm going to add a bit to it you see twister space is really a bit more than this space why is it a bit more what you see if we look over here the nice idea would be to say well that Riemann sphere you what these Riemann sphere is all over the place and remember what a Riemann sphere is it's a curve but it's a curve if you're a complex person and we're trying to do things as complex as we can complex in the sense of conflicts numbers so that means that this is really a curve and it has to be a holomorphic thing to be a proper point in space-time that was what I was saying here but it would be awfully nice if this whole space was a complex space if the whole space was a conflict space then these rima spheres would just automatically be sitting there and you know where you are and it would be a very beautiful piece of geometry it doesn't work for a very good reason because if you count the number of dimensions of these light rays it comes out as five it's just a number of parameters you need to specify with arrays so that means that this space here is five dimensional what's wrong with that what's wrong with that is that five is an odd number and every space complex space in real number terms has to be cut has to be even each dimension counts as to each complex dimension counts as two real dimensions five dimensions is no good well fortunately there's a you wouldn't have been talking about this there hadn't been a way out there's a bigger space which is six real dimensions and this six dimensional space is indeed a complex space it's a very well known one if you have to know these things it's defined in quite a simple where you take four complex numbers you look at their ratios and these are three independent complex ratios it's what we call a projective space complex projective space and that's this thing over here okay and I put these Ed's and then you have a if you want to know where this place in the middle is which I've called that's the space of the light rays you have to write down an equation like this which you see has these bars on it which means it's not holomorphic okay clearly I'm not going to have time to spend all these things in detail but I want just to give you a general picture of this thing how do we understand this space well first of all I'm going to put oh it's getting complicated yeah this is the Riemann sphere which I like to draw as a little sausage because it's not this it's also a straight line you see it's a sphere in a straight line at the same time this thing I can do is making it look like a sausage but sometimes I'll draw it as a straight line and there's this important thing which is called incidence which is when this point lies on the sausage and that's the same as when this cone lies on the line so you can see when they're going to come together and that incidence relations is written on the other transparency it's the it's basically it's really the basis of Twister view but some this it's written at the bottom in a form which you probably have to know about matrices to know but there it is there it is it's a nice simple equation really but I want to try and explain something else which is this what does a twister actually look like why is it called a twister okay it's not such a complicated piece of geometry it does in the sense of depend on this but not too too hard here we have a point in this twister space now if that twist if that point happens to lie on this space in the middle you see that's this five dimensional space which I started with we know what it is it's a light ray but if it doesn't lie on that space suppose it lies in the bigger space out here which is the six dimensional space how do you picture it but one way of doing it is there it is sitting in the bigger space here is the line which represents the point in space-time and that point I can draw the plane through that line and through that point now that plane you see when you talk about twisters you find that there's a sort of duality between planes and points so if you've got that plane you take what's called its complex conjugate you put a bar over the top of it that's this complex conjugate and then it becomes a point again so W is the plane W bar is a point and that means it's a light ray so that means that given your line in this part of space-time which represents the light rays okay that's a point in space-time now so the green thing is the point in space-time though it is and the W is represented by that light ray now wherever I happen to choose that point it could be anywhere I have a light ray through it what is that configuration of light rays four point up here well it's that so if you take any point and it could be a point sitting here you see where the arrow is you has a circle going through it and that's telling you that the light ray is going off in that direction okay that's a little bit difficult to see and I admit that but this was really what started off twister theory and it's why they're called twisters because this twisting thing is zipping along with a speed of light and it represents the twister it also represents something about the angular momentum of a massive particle without any mess so it I could describe it in those terms but I think I won't do that now okay now a slap down this is really more for the expert you see this is just the equations don't bother with that if you're if you're not an expert but these are the equations that we need the basic equations for twister Theory notice that one thing I would do want to point out though is that the Z thing that was the coordinates I had originally for the twister and that was said to I tend to use Zed for the twister it splits into two bits and I want to explain what those two bits are those two bits are what are called spinners and you can sort of understand the spinners this is all part of why twister there is a little bit complicated because you got to learn all these things properly I'm just going to throw them at you a - spinner well you should something like a bit like a vector but it's very specially associated with the light cone it points along the light cone so it's the particle like a photon might be zipping along with the speed of light and so it's direction of motion and in fact it's momentum which tells you what the frequency of that light is it tells you how energetic it is the momentum of that proton is also gauged by the length of this vector here so you've got a little bit more information than which way it's moving you know how energetic it is what its frequency is if you like what its momentum is there's one other thing here there's this thing called a flag plane which you must think of as being tangent to this cone that's very hard to picture because you need though all the dimensions here really but to see get the dimensions I take a slice through the cone and that's a sphere remember that that was we did that before this point here representing the actual direction of the photon is a point on that slice through it and this flag plane is a little arrow at that point that describes the geometrical structure of a spinner there's an algebra of these things they have two components I've got SCI naught is the W and x i1 is the Zed and these are the same w stands Ed's that we saw before so that the spinner is somehow picking up these two complex numbers which tell you how to add spins together to make other directions of spins and there's a nice calculus of these spinners which I'm using over here you see so those of you who like all these things can look very carefully these transparencies and at the bottom I've got a formula which tells you how the momentum and the angular momentum of this spinning object which is going to be described by twister how it's constructed out of these two spinners which I shall now put down here which I call Omega and PI I know that the people who work on high energy physics with twisters of use of use lambda mu or something which I regard as a mistake because they got it from my first paper on twister Theory where I used all the wrong conventions and they picked up on this and they use all the wrong conventions too but the later it came to my senses and realize what the right conventions were but nobody list that I'm exaggerating okay what I'm claiming are the right conventions are the ones I'm using here where the two parts of the twister which I'm calling Omega and PI are telling you some nice clear things pi is tell you the momentum of the of the particle the massless particle could be a photon if you like and Omega is telling you basically the angular momentum of it well if you have this picture down at the bottom that gives you a sort of picture of it here is this twister thing zipping along with speed of light with its twists it's got this angular momentum its spin as well and here is pi pointing out that way and we're done we go well you take the light cone of the origin and see where this meets it and then Omega points out that way so for the this is actually for what we call a null twister which happens to be the simpler ones but never mind about that it's a pretty good picture of a twister that's pi is telling you where it's going and Omega is roughly speaking telling you the moment of it about this origin telling you where it is so those two things together tell you all you need to know about a twister now it took a long time to realize what's going on here so let me try and explain it or I can show you this picture now that is a nice demonstration of Qom ology it really came about because a long time ago there was some people trying to make a movie television and film they did actually make it but it was a trying to say something about twisters I don't know whether it was success successful or not and I said in a weak moment I said well it's something is to do with comb ology and they say what's that that I couldn't really explain that so what we might want to put that on the television program and I thought this impossible haven't you explained this television program but then I went away I went home and I thought hmm maybe this explains it so let me try and explain what comb ology is by means of this picture it's a precise non-local measure of the impossibility of this picture now you see it's not all that impossible because if I cut one of the corners off you could make that thing out of wood or I could cut this corner off you could make that out of I could cut that one huh I can even take a little bit out there you could make that too what you can't make is that so locally it's perfectly sensible the problem is in the impossibility of the whole structure okay another way of thinking about that is let's try and build one okay I'm going to build one I start with that corner I hope I've got it the right way I can't see these things very well yes okay okay that's fine and then I I build another corner and I glue that on there sit there instructions instruction manual tells me you've got to glue that piece on there okay so I do that that's fine and then I fight oh there's the last piece and I've made this this final bit yeah it is and I guess it's this way up and it tells me how to glue that on to that and it tells how to glue that on to that and then I try and glue it under there of course well then I find it doesn't work well you what you have to do is you get these instructions and that's what these functions are really doing you see they're instructions about how to fit your function together and does it fit or is there a glitch and that glitch is a non-local thing as I try to say it's it's not the problems not here I could glitch it glue it there the problems not here I can glue that the problems not here it's not there together the problem is when I try and put them all together and you do that fancy little bit of mathematics that I've got here and that tells you whether you can do it or not the co homology element is the element which tells you how impossible it is is it possible or isn't it now that's exactly what these twister wave functions are like now what is that to do with nonlocality in quantum mechanics well I'm going to give you an even simpler example I told you the Lucian Hardy example this is even simpler but it's not doing the same thing it's something simpler because it's an only one particle nonlocality and this is well here we have a source of a photon and this is its wavefunction spreading out over here it's called a wavefunction if you like because it's a bit like waves so you've got these waves coming up you could imagine them as being waterways perhaps here we have a screen now the screen the wavefunction tells you how likely it is to receive that particle here how likely it is to receive it here and you could say it's the intensity of the wave which tells you the probability of seeing it here or here but suppose you see it here suppose there's a somebody sitting here and says whoops I've seen it too late you can't see it I've seen the photon and so that signal to send it too late I've seen it would have to travel faster than light in order to get to every other point on the screen to say you're not allowed to see that Photon it's been seen that's non-local it's a non locality that's already there and a single wave function or possible wave function and I'm trying to say that that's a bit like this in fact it's very much like this because you see this is like your twister wave function it's spread out okay this is the space-time view of it but in Thrissur space or in the space of these light rays it's spread out and it's got this kind of glitch to it you make a measurement here and you somehow broken it the information is fed out from where you've made your measurement into something else which does the measurement and from there on the impossibility is removed so that there is this tension in this wave function which is this impossibility of the structure which once its measured somewhere it's released and the thing can't be seen anywhere else and it's a kind of nonlocality that is featured in this comb ology which you already see in the twister wave function which is reflecting this kind of nonlocality here okay well in the Lucy and Hardy example in the general and some Podolski rosen examples you're talking about two particles so you didn't have to say in twister theory how do you describe a two particle wave function well it's comb ology again but its second comb ology and I can't describe it it's simply like this in fact I set this I don't know how many years ago probably about thirty years ago or something to students as it look here's a nice example of first calm algae find me an example which does as well with second kamalji well nobody's come up with a nice example yet maybe there is one but it's the kind of thing when you have two particles and you want to see how this entanglement between those two particles is expressed in the kind of tension of this nature well you can mathematically you can understand that a second come on yes it's there but to see it in a more kind of concrete way is something for the future maybe somebody will come up with a nice way of expressing that and then maybe these entanglements will not seem so mysterious as they do now I'd say like the point I'm trying to make is that if you look at it from the twister perspective in a certain sense these non localities become rather more natural and not so I mean this kind of comb ology is the kind of thing that mathematicians when studying complex spaces simply come up with automatically I mean they've been doing this thing for years to try and understand complex spaces of higher dimension but to see that playing a role in physics is is really exciting and very nice if you like geometry especially if you like this kind of complex geometry and it's perhaps revealing something deep about the world operates thank you very much you

Channel: TVO Docs

Views: 143,059

Rating: 4.8472118 out of 5

Keywords: TVO, TVOntario, TVOKids, polka, dot, door, polkaroo, education, public, television, Elwy, Yost, Steve, Paikin, big, back, yard, ideas, Canada, Big Ideas, Perimeter Institute, Science, Physics, Mathematics, Quantum Theory, Twistor Theory

Id: hAWyex1GKRU

Length: 60min 24sec (3624 seconds)

Published: Fri Dec 30 2011

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[u/wintervenom123]

https://universe-review.ca/R15-19-twistor.htm

Some more info for anybody interested.