The reason for antiparticles - Richard P. Feynman

on behalf of the Faculty of mathematics I would like to welcome you to the first Dirac Memorial Lecture Paul Dirac Dirac was Lucasian Professor of mathematics in this university from 1932 to 1969 a period of 37 years which witnessed tremendous developments in quantum field theory and elementary particle physics some John's college of which Dirac was a fellow in collaboration with the Faculty of mathematics has generously established the Dirac lectures which will be given annually within the field of theoretical physics and particularly the field of quantum physics which Derek did so much to create we are extremely fortunate to have Professor Richard Fineman here to deliver the first direct lecture professor Feynman is professor of physics and has been at California Institute of Technology since 1951 his distinctions are far too many to enumerate but among them there is the Einstein a world of 1954 the Nobel Prize of 1965 I think he's of course well known to every physics undergraduate of this university and indeed the world over for the Fineman lectures of physics published in 1963 and standard reading in this University Feynman is an outstanding physicist of the present generation renowned for his work in quantum electrodynamics liquid helium and the theory of beta decay and many other areas of theoretical physics he lists among his interests in hers who Mayan hieroglyphics opening safes and playing bongo drums a man with these interests can be all bad it's a great honor for me to introduce professor Richard Feynman [Applause] when I was this thing work when I was a young man Dirac was my hero he made a new breakthrough a new method of doing physics which to guests an equation previous to that Maxwell got his equations but only in an enormous mass of gear wheels and idlers and so forth in order to understand what he was doing but Dirac had the courage to simply guess at the form of the equation and tried to interpret it afterwards I feel very honored to I had to accept after all since he was my hero all the time and I think it's kind of wonderful to have discovered myself giving a lecture in his honor he was the first to as he said read quantum mechanics and relativity together in his relativistic equation for the electron at first he thought that it was necessary and that spin was a consequence of relativity he also invented some ideas called sit above a gong and other things which turned out not to be very useful in the interpretation of the equation but after some time with the puzzle of a negative energy he finally solved by filling them all up and making hot bubbles which predicted the existence of anti particles and then it was realized that the idea that was necessary to Wed quantum mechanics and relativity together was the existence of anti particles when that was added you could do it with any spin as paulien vice cop proved and therefore I want to start the other way about and start with their anti particles and tried to explain why there must be anti particles if you try to put quantum cash with relativity it also permits us to solve another problem which is very mysterious pre relativistic time one of the grand mysteries of the world is the Pauli exclusion principle that when you exchange two particles you get you should put in a minus sign it's easy to demonstrate that in non relativistic mechanics if nature started that way it'll be that way all the time and so the problem would be pushed back to creation and God knows how that was done but with the existence of anti particles we make new pairs and therefore new electrons and the mystery now is why does the new electron that has just been made have to be anti symmetric with respect to the others and can't get into the same state as the others that are already there and therefore the existence of particles and antiparticles permits us to ask the question in a practical way suppose I make two loop heirs with two electrons and I compare the amplitudes but when they annihilate directly or when they exchange before they annihilate why is there a minus sign all these things have been solved a long time ago in a complicate in a beautiful way and back to the simplest way in the spirit of Dirac with lots of symbols and operators and so on and I'm going backwards to Maxwell's gearwheels and I'm going to try to tell you as best I can what I think is a way of looking at these things so that they appear not so mysterious I am adding nothing to what is known before it's only exposition so here we go as to how things work why there must be antiparticle in ordinarily in quantum mechanics if you have a certain disturbance on a particle which starts in a certain state Phi naught then it'll be changed the state will be changed and the amplitude that it sends up in a state car is as you all know the projection of Chi into a Phi there's a better bra and ket rotation for that I will suppose that's true when we go to relativistic quantum mechanics do now suppose there were two disturbances one at a time T one a you know a little later time B at time T two and we would like to know what the amplitude is to restore the original state Phi zero the amplitude that we go from Phi zero to Phi zero is one the direct amplitude and I'm doing this by perturbation theory the next thing is the slowest an order in which first AE goes into some intermediate state and which lasts for a little time with the usual exponential and then the intermediate state n is put into Phi zero by the operation B this is the form in which two successive Operations appear in quantum mechanics now if a and B are local that is if they only exist in small area of space and time they're not very widespread and we're going to make some simplifications I do that the way I'm going to do is first two very simple examples and then maybe talk a little bit more generally I hope you understand the simple examples because if you do you'll understand all the generalities right away this that's the way I understand things and any rate one finds a term the one plus the other term in this case suppose it the intermediate states of free particles of momentum P and if they have momentum P they'll have an energy given by their relativistic formula which is the positive square root of P Square plus M square and these that the braylee waves which are rushing from point one to point two with momentum P and energy E and we're going to suppose something all the energies are positive if the energies were negative we know that we can keep yet we would solve all our problems wouldn't we we could keep dumping things into this pit of negative energy and get the extra energy out and run in the world but we know we can't do that so we're going to suppose that all the energies in intermediate states are positive now here's a surprise if we put together these ways with some amplitudes as any function of P whatsoever this amplitude to get from one to two is not and in fact it cannot be zero when two is outside of the light cone of one and that's a shock to anybody that doesn't know that that if you started a series of waves out they can't be confined in the light cone if the energies are always positive and it's a very important thing and that therefore it must be the result of some sort of theorem and so I made up a mathematical theorem the proof of which I don't know but I'm sure it's right in this application I in this application that is if we put together a function only with positive frequencies defined this way the integral only over positive or measure with any weights whatsoever to produce a function of f of T then that function cannot be zero for a finite interval of time all right do you see ordinary you have a little bit surprised there because you know that you can take a piecewise function at zero over a range and Fourier and analyze it but then you'll get both positive and negative frequencies and I'm insisting there only be positive figures that got half as much data to work with I have two functions the real part and the imaginary part of F both be 0 now it may be that some clever mathematician can cook up some way that they could almost happen with some kind of a either the 1 over X or some Terra but I want it to be 0 over an interval of time you see in the inside the light cone and then if I change X which changes all the phases and everything else is still 0 I don't think you can make it work 0 outside the light cone so the point is it's definitely saw that this function cannot be 0 outside the light cone in other words there's an amplitude for particles apparently to propagate faster than the speed of light and no arrangement of superposition can get around that therefore if time t2 is late later than T 1 we have an amplitude to a connection like this in which a particle goes across faster than the speed of light but because of the principles of relativistic invariance there are axes if these are separated by a space like interval there's a speed at which time T 2 appears before time T 1 now we're not changing the formula we have the same process but we're going to look at it from this other coordinate system and what do we see we see a process by which what at two first thing that happens see time is this way now first thing that happens is a green thing and a black wouldn't come out and later on the green room annihilates with the black one too it turned a little bit it looked like this figure which is a new process this green line before used to go forwards and now it's going backwards so what we have to allow and we have to include it in order to get any type of relativistic invariance at all is a section of paths running backwards the possibility of a production of a pair and an annihilation by a disturbance in other words there must be anti particles in fact because of this difference of the coordinate systems and two different speeds so that which is ahead of which depends upon what which way you look at it we can say this that one man's virtual particle is another man's virtual antiparticle therefore first of all anti particles and pair production must exist and second anti particles behavior is completely determined by particle behavior is absolutely no freedom you know what the particle does you know what the antiparticle does how you just look at it from the other end because one follows one way and the other the other way to summarize this very crudely an output do this very much more accurately and fancy later if we reversed everything we would reverse the sign of X Y Z and T then we should be hit using antiparticle I call C conjugate charge conjugate or whatever it is the difference between a particle and antiparticle and this change of the sign of time and the three directions of space let P represent what's called the parity operator which changes the three directions of space so P T is C right but we're finished we understand why they're anti particles and we understand that wonderful theorem that CPT leaves everything unchanged okay and therefore since I've already done what I said I would explain why they're in deep articles I'm finished with the lecture but in order for the that's all there is to it it has to be that their antiparticles in order to get relativity in order to determine the sizes of the amplitudes this didn't talk very much about the sizes there are lots of relationships among the sizes of the amplitudes and this leads us in the new direction in which we're going to get a clue about the connection of spinning statistics because the anti particles are making new particles and we'll see how it works and the key to that is another proposition and is that the total probability I don't never can spell probability right because I haven't got the patience all the possible events of all possible events always has to add up to 100% the let us analyze this in the lowest order suppose there's a potential disturbance a we already talked about how a particle coming in at momentum 1 can be deflected to some momentum P with some amplitude a the absolute square of that is the probability of making a deflecting a particle to momentum P actually I include an a the wavefunction coming up if you look back at it now we can also calculate the probability of not doing anything that is leave the object the original particle where it was before and this is the amplitude is 1 plus alpha where alpha it was that diagram that we rent made before and the absolute square is this thing plus that thing it's complex conjugate I mean because 1 plus alpha absolute square is 1 plus alpha plus alpha star plus higher order and so there's a definite connection in size and and sign that has to be right so that this these two terms that added together it will cancel this in other words that these two add up to a negative amount I left out a lot of eyes and and things like that in here oh we will always put them back if you want them but the point is these two things have to be this that the probability of not doing anything is reduced from 1 by the same amount as the probability of doing something is increased so if the sum of doing something and doing nothing is still 1 so that connects this so aizen sign and everything to that well we can prove this way that B has to be a star but I don't need to bother you with that so I won't even bother you with that ok now that was the nonrelativistic case where we only had particles now we have a problem we have added a new diagram a new possibility we not only have the one for the product to go straight through but the property to do nothing is one plus the root twice the real part of this picture which I'll call hi and twice the real part of this pea production picture which I call sly all right now it's it's easy to check very simple that they're twice the real part of this one cancels the probability to make a transition from zero to P that works that's just the same as they did before what have we done however is we've added another piece and so we have to find out what it is that this absolute Square is correcting for all right well if you work out what C the size of this and that are completely determined as I we just pointed out several times a particle antiparticle absolutely related so the size of this is completely determined and this thing you can work out I'm talking about spin zero particles now it cancels the probability to make a pair with the electrons in all and the anti particle has a momentum P that is to say this is the negative of this probability now what does that mean you see though it sees you say these are all the probabilities are producing pairs but it isn't really because you could produce pairs with any moment for the electron and thats all cancelled by something else which is in a vacuum that we can come back to but here is a special case in which we produce a pair with the electron in the same state as all we have somehow in order to make this all work because this is a negative contribution to say in order to make it consistent it has to come out that the probability of producing a pair with an electron and all is enhanced when there's electron present in all and that enhancement is what's being canceled and that's the principle of bosie statistics that makes the lasers work that if there's a particle in the given state the probability of producing a new particle is increased it's doubled in fact and this extra 1/2 is 1 and 1 the extra 1 is the thing that's cancelled by this and that's how it balances and so we can get a clue as to why the bosie statistics are the statistics for spin zero all right now let me just outline it again we've added an extra diagram told us about a new process called pair production we discover when we try to check the probabilities that this diagram makes a negative contribution has to cancel something and that canceling is an extra probability for producing a positive antiparticle and an electron in the state oh when as a particle present originally in state oh and that phenomenon that when is a particle already present the probability of producing another one is enhanced is a characteristic of bosey statistics and that's where it begins that's to explain to you why it's possible and if all you have to do is follow more details more diagrams more cases and the whole thing will come pouring out I've just given an example to show you which direction we're going to go I'll just do one little example if we didn't have any particle but started in a vacuum then you can make a pair Q and P and the probability to make you see you can make pair production so that makes the particle Q and P and the probabilities to make those pairs to sum over all Q and P of this absolute Square and that must be compared to the amplitude to do nothing and the way that you balance that one is that you have another diagram in which you created a pair and then annihilated it and if you take twice - twice the real part of that or rather twice the real part of that is the negative of this and so that's the way that balances and the extra diagram we saw in a previous page is external on top of that so here's how all the pairs balance and their finish it all off what happens is that you find that you have to add all diagrams together and then you can see the exclusion principle they with the positive sign this way if suppose you have for disturbing potentials and you produced two electrons with their positron and they annihilate it again and now suppose you compare that to what would happen if you produce these two pair but instead of an i li in their own positron they want to cross them I lay the other guy's positron that would make a figure like this well this is a single loop you see and these are two loops and you're supposed to add all the diagrams so it'd been a head this and that and they'll all fit together and everybody knows that and it's very simple and it makes bosey statistics or Z statistics as a matter of fact is not so very mysterious as soon as we deal with anything like oscillators such as the vibrations of a crystal which makes sound waves we find that those sound waves obey bosey statistics then we're adding all the amplitudes together and so we don't find that quite as bad what we're going to do is we're going to discover later that for each loop when the spin is 1/2 each group gets a minus sign and therefore this has two minus signs for this two loops multiplied together which is plus and this is a single loop which is minus own we changed two loops - one loop we changed from trubiner signs multiplied to one and it's an opposite sign and you get family statistics so I'm just focusing now we have to look and understand why with spin 1/2 we're going to understand later why there's a minus sign in the formula for the loops ok now I'm going to return now because I have to be more careful to go a little bit more into detail and to explain the relationship between ENT particle behavior and the particle behavior of course is true and still true that the anti particle is completely determined by the particle behavior let me analyze this a little bit more in detail in the case of spin zero we noticed that for T 2 greater than T 1 that is above the plane here the amplitude to go between 1 and 2 was this integral I've chosen a special case of scalar couplings and so on because this form is relativistically invariant the 2ep under the d3p makes a relativistic invariant volume element and in momentum and energy you have to check that someday this e energy times time minus momentum times position is of course the typical relativistic combination of the product of the time components minus the product of their space components and this is a relativistically invariant form and i've chosen that one to study now because that plane could have been taken anywhere we know that all over the outside light corn I just took the plane for if you want this point here tilt the plane far enough down take the coordinate system this way and you'll get this function it's always the same function business relativistically invariant so this function is not only true above the plane T equals 0 it's true of the whole negative light outside the back light cone that is the space like region and the future like on this formulas right on the other hand for T less than T 1 and and also space like separated since energy must be greater than 0 always we have to have a form that the same amplitude is g21 I mean the amplitude in that region either T 2 less than T 1 in the bottom light cone or else inside with T 2 less than T 1 but in the space like region will have to have the opposite sign sequency with a whole lot of weight so i'll add it together in some complicated way the question is oh and in the region when t2 and t1 a space like we could use either one of these two formulas to obtain the amplitude it's like a patching and that makes the functions have to be equal there and that's going to determine what this function has to be this G because we know it's equal F in that region so the question is if a t2 and t1 space like we have an expression with a negative frequency can we express it also as a function of positive frequencies alone what an error you can't do it it's magic but for this particular function which is relativistically invariant it's possible let me show you why first for t2 equal T 1 F 2 1 is real because this isn't here and when you integrate over P you have both positive and negative P and the complex conjugate is just the negative P so your average both when you integrate and it's real f is real but F is relativistically invariant and if it was real for the 2 x equal in another system at 2 x their equals somewhere else and so forth and therefore since it's relativistically invariant it must be real for any t2 and t1 which is space like separated fine now since it's real for any t2 and t1 space like separated its equal to its complex conjugate if t2 and t1 is the tight space like separated and therefore one solution is that g is the complex conjugate of F that has the right sign on the frequencies and that's equal to F in the space like region and that's a unique solution because if I had two of them and subtracted them the difference would be 0 over a finite interval and I said there's no solution to something with a one sign of energy and a different zero over a finite interval so this is extrapolate about into the times t2 less than t1 in the past light cone and this is our answer for the amplitude in the past light cone so here's the answer for the whole amplitude for every where if t2 is above t1 in the forward light cone it has to be this expression in the intermediate where t2 is outside the light cone at t1 it's either this one or that one they're equal and forty two earlier than t1 in the back light count this is the right formula we started by supposing something was relativistically invariant and they're able to deduce what happens when you go all the way around back back on the other side and that's not so mysterious suppose we had four dimensional Euclidean geometry X Y Z and W all equivalents are the sum of the squares of all those four things was a constant and we started discuss rotations in that space that's the analog of relativistic transformations in for space ok I can vary X Y Z and W by a rotation because for instance I can rotate in the XY plane and change X to minus x and y 2 minus y another allowable operation is to rotate in the Z W plane and change Z 2 minus Z and W at the minus WS the fourth dimension so you can change all of them around and therefore once you've supposed the thing is invariant you know where it is everywhere even upside-down the only difference in the in Kowski space is that there's a kind of a no-man's land where t2 and t1 are outside the light cone and all the transformations can't really move you through there but we have obtained the correct continuation because the continuity is maintained by supposing that the energies are always positive that limits the solution in other words this operation PT which changes the sign everything is really a relativistic transformation I should say more correctly or Lorentz or rotation transformation kind of extended through the across the mysterious space like region by the demand that the energy is greater than zero so it's not mysterious that we can work out the whole function and it's not mysterious at relativistic invariance produces the whole work okay so that was alright for spin zero and now I'd like to do spin 1/2 and see what happens I'm there's a way of representing once you start knowing that the thing is spinning here and they have Lorentz transformations you can work out by a nice arguments about group theory exactly how all the states change and everything out without inventing the gamma matrices of course you end up finding them but it's all a group theoretic business which I won't to go into it's a lovely exercise but let's start out and I'll give you some results of it rather easily anyway first we just deal with ordinary rotations and we know that if a thing has angular momentum em along the z-axis and you rotate around the z axis the phase changed by imy and when it's spin 1/2 m is plus or minus 1/2 let's say plus then it would be e to the I over to Phi and therefore a 360 degree rotation ends up by multiplying the function by minus 1 at this point all attempts to do anything by instinct by a intuition fail of course because it's hard to it's hard to to understand that you know I mean one of the things that's the hardest part of this is to keep track of when you should have the minus one and when you shouldn't have the minus one Dirac had a very nice demonstration of this fact that rotation one time around can be distinguished from rotation when rotation twice around is about the same as doing nothing and I'll show you something you can find in dancing girls do here I'm gonna rotate remember which way around all the way around until you can see the blue mark again and I've rotated a 360 degrees but I'm in trouble I can't under it but if I continue to rotate it still further which is a nervy thing to do under that I do not break my arm I straighten everything out and so two rotations is equivalent to doing nothing [Applause] and but one rotation can be different so you have to keep track of whether you've made a rotation or not and the rest of this talk is a nerve-wracking attempt to try to keep track of whether you've made a rotation or not I mentioned something else I'm going to need for shortcut to give you an idea of the nature of the formulas if you want to put in this thing to have half angle formulas for example the projection if you like to know with what amp up to if it's up in a direction Z with what amplitude is that up in a direction Z prime which is at an angle theta to Z the answer is cosine theta over 2 which is the square root of 1 plus cosine theta with an effective now we're going to do the same thing adding relativity and going through this group theoretic analysis I'm not going to do I'm going to use a scale of coupling I have a constant square minus P Square or M square is a constant and that's like cos squared minus sin squared instead of cosine squared plus sine squared in this relativity of cautious and cinch is where you would have cosines in sine and the other theory and the energy would be M cash in the momentum and cinch of W now it turns out that these know this should have been a W and I can say it and you don't have to actually do it that the amplitude for instance if you started with spin up here and you ask with what amplitude you go into the correct a spin state in a direction p2 from starting from rest at p1 p1 is M the time component and no momentum p2 is e the tanker ball let's say it's moving in a Z direction the amplitude that you get is typical of the amplitudes for a half-angle spin except that I use have to the governor or a rapidity and the answer isn't there this problem is cosh w over 2 which is within factors which I'm not going to keep by twos and M's and things is the cost of V plus 1 square root the amplitude to do that is this and the cost is really e so this is the square root of e plus n if you'd like it in a more relativistic form consider P 1 dot P 2 plus M square by the dot product of two vectors for vectors I mean a product of the times minus the space components if you multiply the times you get any and all the spaces gives zero so P 1 P 2 P 2 is Emme and here's a nice relativistically invariant looking expression I mean it's a very expression for this amplitude in terms of P 1 and P 2 now here's the thing that I wanted to say that we don't we hadn't said before but it's another important thing I talked about turning everything over but it's also possible to imagine that one of these vectors is turned over to be the opposite direction and then we'd have a positron going into an electron we had pair production and we calculate the amplitude for pair production by a continuous transformation of the formula that we have in terms of P 1 and P 2 until one of the pieces backwards for example if there one of the pieces the positron it would be coming in the other way and if the positron at any momentum P and energy minus C II then the P 2 vector that we represent it bikers were drawing it upside down or backwards is minus C minus P where P 1 is the same old thing in so P 1 dot P 2 is minus M e and it turns out that the amplitude for pair productions sure enough is obtained by just substituting this value for P 1 dot P 2 in here and by doing a little arithmetic with in a sign it's I times the square root of e minus N and therefore the probability of getting a pair is e minus M times some units return ologists to the scattering amplitudes and this is a very interesting thing because it shows you how pair production rates can be calculated in terms of scattering rates by just shifting the vectors about and of course this is greater than zero that's absolute squared now when we go and look at our pear case where this through screen last time but I haven't got a green pencil so it goes here it goes this way that's the one term and then the other term goes backwards and we tried to connect them last time by a kind of continuity this time because of this if we went to the same state again we get the square root of e plus M twice and we get E Plus n here how will it continue over here do you remember if I didn't have the e and either place that these two functions are the same in the space like region if they're the same that derivatives are the same so what fits together is the derivatives you take the derivative let's say with respect to t1 for a t2 for instance that would produce an e and this were in the same place reproduce a minus C that is the signs pay well that's wrong okay this one is backwards this is t2 minus t1 right and the derivatives have opposite e so this continues into that now what happens now I can't move yeah everything is forced on me like no trees now we go to look at the real part of this one and it does the same thing as it did in the case of the boson and it produces the right rate which is the square of this which is E Plus then but then we look at the real part of this one it doesn't produce the right rate it produces the right minus E Plus M instead of e - and it's the opposite it's the reverse sign of the Bo Z K therefore the real part of this number two diagram this time is no longer the negative of the probability of producing a pair it's the positive of it it's a reverse sign I mean relative to the bosey case now when you remember what we had to have for the Bo zk that because this thing was coming up very strong negative we had to increase the probability when there was a particle present now it's coming up with the wrong sign it means we have to decrease the probability one is a particle present and this indicates why it is that the thing works it won't fit together only if you say that when is a particle in state 1 the probability of producing another particle in state 1 by a pair production is decreased instead of going up to 1 plus 1 equal to is in the bows each case it goes up to 1 minus 1 equals 0 in the Fermi case and the rule is that they have a particle in a state you can't make another particle in that state by a pair production and the fact that that particle is preventing something that you expect it to happen from happening ships the probability the other way and it fits with the fact the sign is reversed and so you see at this level we have a completely clear clue that we are going to be able to demonstrate in fact we have four specific examples the connection between spin and statistics and that is different for spin 1/2 than it is for spin 0 in fact what happens is it turns out I mean it is rather interesting it's very curious that you can make any functions that match over in that region there was the lock that we were able to do it if you take more complicated functions they won't match that is the positive frequencies can't be matched exactly with the negative fingers because the positive frequency piece has two components the real and imaginary and the negative frequency one will have to fit both which is too much there was just luck that it worked the only thing that will continue to work is if the functions are matched their derivatives match and higher derivatives match so the only thing you can put in here polynomials in the numerator and that's one of the reasons that for years people have been stuck with quantum field theory in which all the couplings and all the propagations involve gradients more gradients more gradient but never non-local couplings because they just drive them nutty because they can't make this thing fit so that we learned something else in there while we're at it okay and the first thing that happens you see the the stale a case is just a constant then if it's linear in the moment that means the square root of momentum on each side spin 1/2 is sort of splitting the linear thing if it's quadratic that's a spin one object going across and so on and it alternates like that and depending on whether it's an odd number of powers or an even number of powers this sign gives me trouble or it doesn't it alternates and it's all nicely fit together and again I should be finished and it would be wise to stop but perhaps you'd like to see a little more general detail about how that sign comes about in a more clear fashion and so what I would like now to is right is the general rule that connects the particle and antiparticle we said it quite clearly that you look at it backwards to be more precise the following is true an antiparticle going from an initial state of momentum P I and energy e I in a certain particular spin state you I even if it's not spin 1/2 whatever spin it is pin 1 whatever 0 that's starting in that state and do various things for example could emit a photon of polarization a I just made a little example that the momentum Q energy Omega to end up in a state of momentum PF that's this one with a sanity EF and with another spin you earth this happens with the same amplitude that a particle does something and you have to do the particle by looking backwards right the particle starts in a state of momentum PF EF and the spin which is a reverse the PT reversal of us actually the pea isn't gonna make a new is abit just mainly the time reversal of the spin it absorbs a photon of polarization comes out minus a star I'll explain why momentum QA it has to be a star bakra see when you emit over here you have the coupling a star but you're going to talk about absorbing it why pursue Momentum's go any other way you see it's picked it up instead of dropped it out and so it's an absorption instead of an emission which is the time reversal okay it's other words this time reversed and it's it to end up in a state of momentum P ie I and here's the key and the spin state the inverse of P T times u I why is it the inverse because in this first thing we're turning one way and then the second one the other way so if we made a PT turn this way and then we make the unpitied turn Oh everyone's a time reverse on the others the inverse of a time reverse sounds the same done it alright we'll come back to that or me as a matter of just let me make some notes here first first if I had a object of momentum P the time reverse state would have known momentum minus P yeah right but then if I use the priority that I changed all the axes that minus P would be flipped back to P so that's why it's still the same momentum in other words what we're doing here is the C that is changing from particle antiparticle is equivalent to priority change and time change everything is done in the reverse order in time and if you work out the time reverse polarization you discover prints if you had circularly polarized light with the electric field going around one way you have a X equal one and you y equal i if you take the complex conjugate it goes around the other way the electric vector and that's the time reversal and so on so everything just fits together to make this exactly the statement that this is what's happening instead of Claddagh kalantay particle that's the C but everything backwards in time and everything reversed three in space and I'm not going to go into details to prove it any further now the little difficulty here we have a state PT UI and here we have P T to the minus 1 UI I mean UF and is PT the minus 1 UI later on we'll want to go from the state to the same state so it's much better to have the same system so if I had a state UI to go in UI I want that ptui to ptui so I'm going to change this beat T to the minus 1 UI to ptui because I'm going to show that for spin 1/2 see if the states are the same the in and out states are the same now we can check the signs of the real plots very easily because they are the same previously however for spin 1/2 I'm going to show you that PT reverse negative inverse is the same as minus PT in other words it's the by the way you can choose the P phase the way you want but let's choose it so that P Square is 1 so there's no problem with the inverse PV equals P so what we're going to show T to the minus 1 equal T multiplied nasty multiplied by t v--'s we're going to show that T squared is minus 1 for spin 1/2 and plus 1 for spin 0 and when I change this to PT I have am extra minus sign for spin 1/2 and that's where it all comes from why should it be that the sign that changing around T twice changes the sign of spin 1/2 because a 360 degree rotation changes the sign and if you can think of space-time huh weight point T twice right turn it around twice it should be 360 degrees oh yeah in some crazy x ZT space but smells right on it I mean if I flip the x axis twice I would be rotating 360 degrees maybe it's true of the T axis too and that's what I want to prove I want to prove now that it's true that if you make a time reversal twice on a spin 1/2 particle you'll get the same state with a minus sign in other words two T's is the same as a 360 degree rotation and changes a sign that's where it all comes from this is in fact so elementary and there should be an elementary physics textbook which is so much fun having to do with time reversal I have here a list of states and here that time reverse states and here they are twice reversed let us call a state a the state might be particle doing this oh that's too complicated particle doing that then the state T a is a particle doing this ok well the product with momentum this way would be a then the party a is a particle of momentum that way all right so ta B's this is direct notation okay and this is the name of the state and what I mean by this is simply the name of the state I'm describing the name of this day the time reverse state for instance the particle look and if I do it twice I get that a particle located at X time reversed is still located at X no big deal so this is this a particle of momentum P we just said was a particle of momentum minus P when you reverse it so but a particle of momentum P can be made by the superposition of particles at different X with the phases as given and a minus B would be with the opposite phase and so we discovered that time reversal is what's called an anti unitary operation you must take the complex conjugate of the coefficients whenever you see them for instance if there's a state alpha A plus beta be a superposition with coefficients that need not be real then the time reverse state is alpha star times the time reverse stated a plus beta star times a time reverse state of B and that's the general principle of time reversal now if I could take the time reversal again with X it goes Dax with - speed change the sign again it's PE so that's okay nothing happened there and with this one it's alpha times T ta + beta t TB why because I take the complex conjugate twice and if you're very good at algebra you know doing that is a waste of time now this state of course being twice time reverse must be the same state again sound like a 360 degree with you you do it twice it's the same physical state and the damn quantum mechanics always permit you to have a different phase so you can't say that this thing is the same state as that but you can't say that this thing is a phase times that and that means that TTA is e to the I Phi that is I wrote badly a phase times AE to the I Phi times a and all the states have the same phase so that this interference that I have here will be the same interference there so is a universal phase shift when you take double time chip now let's look at spin 1/2 let's take the state which is spinning up in the Z direction well first let's talk about spin it's spinning what's the time reverse spinning the other way all the spins have to be reversed every state goes into the opposite physical state let's start with a state up in the Z direction it'll go into the physical state down in the Z direction with some phase it's easy to prove that this you can start with any phase and it all comes out the same in the end this space I'll choose to be one so it just comes out that the time reversal of the up spin is the down spin yeah yeah now the time reversal of the negative Z spin is a phase times the positive z spin of course then we're going to show that the phase must be negative in just one moment because let us consider the linear combination which is plus and minus this should have been plus Z and minus Z if I wrote that out so carefully if you're super put a pose equally this one in this one you'll get the state representing positive X if you do them with opposite sign you get the state negative X all that should be known to the elementary student of spin now let me suppose that I try to say that the time inversion of minus Z is the state plus Z without that minus sign then do you see that the time reversal of the state plus X this plus will go to minus and this minus ago to plus will produce the same state again which is not right it's a to produce the opposite state and therefore I have to change the sign here otherwise when I did the time reversal I would produce the same state well if you put the minus sign in there and watch what happens the plus goes to minus and the minus goes to minus plus if you understand such horrible thing and if you look at it a bit you'll find that that's the negative of the spin in the minus X direction and that's what the negative is just a phase and the spin in the minus X direction what we want to reverse hence it's necessary with spin 1/2 that if the time reversal of the plus state up state is the down state that the time reversal of the down state is minus the UP State it's so much fun now we do it again ah well we just learned that the time reversal of the down state was minus the Upstate and here we learn that the primary versal of the up state is the down state now to keep this minus sign going through so you see that we V due to time reverses we change the sign so it's just necessary and as nothing it wasn't read space but we did it two reversals have spent two time reversals and you change the sign it's equivalent to it's just what you would do if you rotated 360 degrees for a particle of even spin which can have a state with a component along the z axis of 0 the time reversal of that is still zero along the z axis see any positive spin will turn negative but the 0 will be 0 so therefore the time reversal of this state some J that's even I mean integral with M equals 0 will be some phase times the state with the M equals 0 but if you do the time reversal again we I got the time reversal of this is e to the minus I Phi times the time reversal of that because of this alpha star but the time reversal of that is e to the I Phi and these Phi's cancel out and they get the J M equals 0 so therefore two T's on a integral spin is +1 in other words to tease has the same effect as a 360 degree rotation okay it's - for spends ahead and half integral spin and it's plus per integral spin how did I get here so fast okay then I got them in no long water nope now when we look at the the sign of a loop I'm going to talk about the loop that had the contribution that was supposed to stop the vacuum you remember if you have a potential present it can produce pairs therefore the probability that you don't produce pairs must be less than one and you have to get the sign of the loop right so that one plus those loops is less than one in other words the real part of the loops must be negative and what you have to have to do that is when you have bows you have two particles coming out and going back into the same state yet that exactly the same thing with the positron and the electron going out and coming back in the same state as indicated here and I've written it out more carefully the amplitude is 1 plus X where X is the contribution from this mu it's real part times two must be negative and cancel the probability if payers are produced so the sum of the probabilities overall turner's remains unity and that's going to determine the size of the thing now they get the sign of the positive positron propagation to be right just like a particle going from a state to the same state we saw that we have to multiply by an extra TT otherwise we'll have the wrong contribution of positron out that extra minus sign in other words we have to multiply by minus 1 that is the X is the loop trace x minus 1 one way of seeing it is that the way you want to go coming out and coming back is to change the time once and change the time again as you go around the loop and that changes the sign I pointed out earlier if the loops had -1 that we would have family statistics because you have two loops or one loop and if there's a minus on each loop when you check when you made the exchange you change the sign it's interesting to realize that when people first figure out this loop there was an ambiguity that no one seems too worried about it scores something like this you're supposed to start in some state and then propagate around with all the couplings and everything else and then they ask are you in the same physical state and everybody puts the U bar there you conjugate or whatever and sums over all states that becomes place but the same physical state might be - you may be we've turned by two T's and indeed we have because in order to get this the way we want it we have to have the two T's in succession and so it should have been - you that we multiply in other words we have - the trace of the propagator the kind of a circle that I mean here is that two particles are coming out in some state and coming back in the same state I can represent that my pants don't fall down by a belt of this kind where this one went around one way and this went around here but you see that that is two time reversals it's a hunt three under 60° audacious and so when there's an odd number of pairs of T's I count that as one TT are not number 360s the contributions should be positive that's the case I want and when there's an even number of T TS as extra TT in it the contributions should be negative because the extra TT makes it made the signs plus and minus are relative to what happens with the bosie case in which they're always positive so what we have discovered is that we have to put a minus sign in front of the traces of the propagators for the spin 1/2 particle and therefore that produces as indicated before a minus sign between two diagrams I think I have another picture of those where you have two loops or one loop and the minus sign comes twice with one case and once with the other when you exchange something connected with two loops it becomes one loop mr. Finkelstein David finger Stein always makes the following demonstration which bothered me for a long time because it sounded like was very important but I couldn't get all the signs right and straighten it out but suppose you have two loops I can't make it out of one belt but you imagine the other one here and instead of connecting them that way I can make them crosswise you see here this goes over to the other loop and this loop comes over to this side right and you put them together how many turns do we have okay well see there's an extra one in there the loop I wanted went around straight that's the plus one if it has an extra turn it's - and so this one is - - what you think of it and that's the reason for the sign change when you exchange the particle if you want to understand that it really is two turns you try to go if I can ever get it back to where it was yeah but here's the way it was on my arm I think and you see that if you go around you go around once twice to see tea tea tea tea which is four and there's two extra teas and it's a minus sign so everything's okay and I think I understand mr. Finkelstein demonstration finally I would like to show you an example in which we have a spin 1/2 object in which we know where the angular momentum comes from that's fun suppose that we had a thing that Dirac invented then it's therefore appropriate dyslexia magnetic pole and suppose we have a charge Q and we'll suppose that these particles the magnetic pole and the charge are both spin zero so we don't have to worry about any intrinsic angular momentum of either of those two things but there in each other's presence and here's the by netic pong here's the charge in their springs from one magnet field in all directions enough from the other electric field in all directions and they cross each other and make a a pointing vector which is ain't which is momentum which goes around this way and sure enough the thing has angular momentum and the angular momentum is independent of how far apart they are because if you started this way and move them in and out there are no forces so there's no angular momentum and you can figure out what the angular momentum is very many ways now leave is an exercise this suppose that it's along the axis I'm going to show that it's emu I'm to MU choose the charge and muse that they buy this suppose you try here's the magnetic pole here's the electric charge suppose you want to move that axis if you move that axis you've got to move the charge through a magnetic field at a certain velocity and that produces a sideways force which produces a torque and then you multiply by the time interval in activities the change in angular momentum and that turns out to be just the change in the direction times emu so the angular momentum is emu and I'll let you prove it I wrote eat you that's very wrong it like change my charge gee whiz that's horrifying it should be q okay the charge is Q and the pole is mu and the angular momentum is mu q I got it right here except to here I also have another way of checking that the angular momentum if we choose a mu equal to half H we can have a quantum mechanical system Dirac showed no other only multiples of that are allowed we'll take the smallest one they can't be any other value because I know momentum has to be quantized in quantum mechanics and so there has to be relation of the pole and the charge but it's kind of fun to test out where they're rotating or how 360 degrees changes the sign imagine this is the magnetic pole and there's a kind of a needle because I 4 miles that this is the way the pole is and here's the charge and I'm going to take this little guy on the charge and the pole and turn it all the way around and see what happened well when you carry a charge through a magnetic field the phase changes by e to the I times the charge times the line integral of the vector potential the famous theorem now if we carry that thing that's supposed to intimidate you that's Borman Aaron Hoff paid a lot of that now if we tell you this charge all the way around the circle we have the line integral of a around a circle and the line integral of a around the circle is the flux there's some kind of theorem you know hit the grade a is the curl of Oz over the surface blah blah blah it's the flux now what is the flux if I take the positive cap here it's half the flux that Springs out of mill because the other half goes down springing out of mirror the flux is 4 pi mu know mu over R squared times 4 PI R squared is the flux so it's 4 pi mu so it's half of 4 pi mu so e to the I Q times the fluxes is e to the I Q 4 pi mu over 2 and if I did it right it's e to the 2 pi mu q But mu Q is 1/2 the t to the I pi which is minus 1 in other words sure enough if you rotated 360 degrees it's minus 1 it's a spin 1/2 object no problem it's absolutely right by the way incidentally the reason why it has to be quantized is if I had taken the flux below I should get the same answer right no reason how do I know where to take the flux one is positive another negative so I could have gotten even the minus I PI which would be different oh no it's not it's the same e to the minus I pi is minus 1 all soaked so it's the same that's why you can quantize it that way now imagine two of these things one charge and pole which I'll call a charge and pour which I'll call B and exchange them nothing should happen because the charges don't do anything in the poles don't do anything cuz they're spin 0 yet the things have angular momentum 1/2 what happens what happens well watch as you carry this be around an a think of it relatively the charge on B is moving relative to the charge on a so if a were kept still to look for the relation you'd see B would move around to the other side I mean relative they'd go like that why you starts over here and ends over there but I don't want to move this so I just do that to picture it so we find a going around the charge going around this way meantime back on the farm the other one it's charged moving around B so if B is held fixed you find a charge a it's going around be in this direction and so there's a phase change which is the integral of the charge e to the I Q a you know that the line integral of the vector potential from here to here and an additional phase change from the other one which is the line integral from there to there no matter what the shape of this line is it's a little thought this line is a kind of mirror of it of course right and therefore you're going around something which splits the space 5050 then the line integral all the way around is again the flux and found out the flux was minus 1/2 flux half flux and so you get from these phase shifts from the mechanical efforts of the charges moving through the poles the next to factor minus 1 and so when you compare what you get straight and what you get when you move them you have to multiply by minus 1 in other words this time we discover that this time the sign comes from the fact that when you exchange two things one relative to the other plus vice versa has a 360 degree rotation in it it goes around it around altogether charges are moving 360 degrees around each other and then it works and so I would like to summarize the whole thing by saying when two charges are exchanged there's an implicit rotation of 360 degrees we have noticed it inside with those productions and so on in terms of the T's and stuff and we're across it but when you exchange two charges there's a effectively somewhere hidden in there if you analyze it carefully at 360 degree rotation and it's this if the spin is 1/2 then the amplitude must be changed by a factor minus n minus 1 because of that 360 degree rotation this experiment was to show the extra twist which I already did for you and that's the key to the factor of -1 that comes in the exchange of fermions relative to what happens in bosons but to summarize although we went a long distance in into detail it's interesting to remember what we did because it's not much if you suppose there's only particles and things can only go forward you'll find yourself impossible to avoid going into the outside the light con this cannot be for only positive time because that depends on coordinate system so therefore there must be something outside the light cone for the negative times 2 and therefore the path that went forward a different point of view has to go backwards and we find pair production then when we took a very simple example of a charge all we have to do to discover that there is hidden in this works then when you go to put it together the keys to spin statistics isn't considering in this well in this diagram that when you're calculating a sum of the probabilities that this diagram has to be effect in order for this to be here you have to have an effect on the probability of producing a pair that depends upon whether or not the charge is present an answer which comes out positive for bosons and negative for fermions and it comes out negative for fermions because Fermi on it's got a kind of a linear term in the propagator and the linear term changes sign relative to this as you flip over and so there's a minus sign that contains everything and the rest of it was just elaboration thank you very much [Applause] [Applause] [Applause] they I call on professor John Taylor to propose a vote of things I can imagine a history of science examination question ended at the end of the century saying compare and contrast Dirac and Fineman and he's given us some scope for beginning to answer that question today the work of Dirac and Fineman has been interrelated in the most extraordinary way and finally has given us today at all which beautifully shows a lot of these key points which they these two men have worked on he's come over nine time zones I think to give us this lecture and we'll presumably return over nine time zones and I hope there isn't a minus sign we're very grateful to him for coming thank you very much [Applause] [Music] you [Music]

Channel: mrtp

Views: 111,111

Keywords: physics, paul, dirac, paul dirac, richard feynman, feynman, lecture, elementary, particle, laws, memorial, 1986, antiparticles, particle physics, theory, cambridge, university

Id: MDZaM-Bi-kI

Length: 75min 2sec (4502 seconds)

Published: Sun Jun 17 2018