Lecture 4 | New Revolutions in Particle Physics: Basic Concepts

[Music] Stanford University this is Sanjay hey could we get Sanjay on the aisle record recorded yes okay we just want to know that we're trying to collect up all the old lectures the video lectures that are on YouTube in one kind of a collection and the idea is for everybody who's taking classes online and offline we can actually have an easy way to navigate through all the lectures it should have to hunt for that right so we're passing out a little sheet that's got a web address on it right now this is a temporary housing for it and it should be good until whenever us but it when we do get a final home we'll have a link I'll directly go to it as well so this link should be good for hopefully all eternity until we get arrived in and all that good stuff all right okay so this is the address let's see HTTP colon slash slash new packet Tech one word new packet tech top come /resources slash and that's the address where it's sort of the website for the class so will be the website for the class and it's for anybody outside inside where they can easily access without all a confusion that's taken place up till now easily access all of the lectures that are online all right we we have been discussing a simple quantum field and we're not finished with it I have to take a two year course and believe me a real course in quantum field theory is genuinely two years of work it really cannot be done in one year sensibly I have to take that two years of quantum field theory and condense it down to a couple of lectures we've already taken a couple of lectures but I think we've had some forward motion I want to take the very very simple version of a quantum field that we've already discuss first of all to remind you what it is and discuss how it is used again the describe particle processes we've done a bit of this we've talked a bit about our the quantum field can code or codify scattering processes or creation and annihilation processes of particles but I want to go into it in just a little more depth so that you can see where some of the really interesting aspects of quantum field theory come from and how they influence questions like energy conservation momentum conservation how do those things how are they related to these quantum fields that we've discussed ok before I do so we need a little bit of mathematics a little bit of formal mathematics not much it's mathematics that we've done before in these series of classes but I want to get them up on the blackboard the first thing is the Dirac Delta function are the Dirac Delta function just to remind you what it is is a function which is the sort of a limit of a genuine real function it's a function of let's say its coordinate let's for the moment this call is coordinate anything it could be X it could be K let's just call it Y so that we don't prejudice whether it's something that I've called the that I've defined previously the Dirac Delta function is a function which is a sort of lump it's concentrated someplace that's concentrated someplace and not other places but it's the limit of an infinitely sharply are concentrated function so we imagine that we can go to a limit where this lump like function is infinitely narrow now I can't draw infinitely narrow so I'll draw with finite width but imagine in your mind narrowing it narrowing and narrowing it now of course if you narrow it without raising its height the area under it will decrease in decrease in decrease to the point where there's no area left under it so what I want to do is to keep the area under this function fixed as I decrease the width so as I decrease the width I'm going to raise up the height of it in such a way that the product of height times width stays constant how constant one just one an area of one located someplace infinitely narrow that's called a Dirac Delta function if this point is y equals let's call it a just R to locate the particular point then the Dirac Delta function is Delta of y minus a and Delta of y minus a is zero whenever Y does not equal a over here over here and equal something infinitely sharply peaked at y equals a that's the Dirac Delta function and it has the property by definition that the area under it the derivative the integral with respect to Y is 1 so it's so narrow and so high that the area is 1 and it's concentrated at the point where the argument of the function is equal to 0 in other words when y equals a that's the direct Delta function ok now I want to show you how the Dirac Delta function emerges from a certain integral an integral that we will come on many many times let's take the function e to the I KX K oops e to the I K X now again as we discussed last time with discussing this on an interval which is a periodic interval which has total length all around it equal to L so the distance around here is equal to L I don't know if this just a distance around that is equal to L that's periodic and so functions that live on this periodic space here should be periodic meaning to say they should come back to themselves after one full loop okay now what is K first of all K is one of the allowed values of wave number on this in other words one of the values of K where e to the ikx is periodic so let's assume an allowed value there and let's take this function and integrate it over the entire cyclic X dimension here we could take X going from 0 to L or for symmetry I could take it to go from minus L over 2 to L over 2 in other words instead of starting X at 0 and going to L I could start it at minus L over 2 and go to L over 2 put 0 at the center here nothing special about this it just Scimitar eise's things learn nicely so that the negative 1/2 and the positive herefor symmetric when you leave the space if you're marching along and you come to L over 2 you pop back up at minus L over 2 ok so we're taking a function a periodic function e to the ikx and integrating it from minus L over 2 to L over 2 what is the answer ok first of all what does the answer if K is equal to 0 L this is El what's right is equal to L if K is equal to zero and what if K is not equal to zero zero because if you take a periodic function that oscillates and you integrate it then you get zero because it's positive as much as its negative only for K equals zero in that case of course this doesn't oscillate it's just equal to one then you get L so now let's think of this as a function of K let's draw the K axis let's go to the K axis here's the K axis now K is not just any number it's one of the allowed numbers so it's discrete does anybody remember what the allowable values of K are two pi in over L right in particular as L gets bigger and bigger the distance between neighboring values of K gets smaller and smaller and eventually as L gets infinitely big these discrete intervals shrink to zero okay so now what's the what's the distance let's put K equals zero K can be positive or negative incidentally let's put K equals zero right over here here's K equals zero all right what's the interval between neighboring values of K two PI over L that's the distance from k equals 0 or from N equals 0 to N equals 1 for example 2 PI over L now let's take this function this function is a function of K of course we've integrand sorry I've integrated it over X it's only a function of K what is it equal to let's plot it on here it's equal to L when K is equal to 0 so right at K equals 0 here it's high and it's equal to L let's let's just give it a slight width just for the purpose of drawing it just for the purpose of drawing it right at K equals 0 here it has a height equal to L but it K not equal to 0 its 0 that's what we concluded at K it's equal to L if if K is equal to 0 and it's equal to 0 if K is not equal to 0 so how high is it its height L how wide is it well one interval you can't think of anything smaller than one interval on here it has a width overall what's the area if I imagine giving it that much of a width what does the area under it 2 pi so that means that this function is 2 pi times the Delta function this function here is 2 pi times Delta of K Y Delta of K at K equals 0 it's not equal to 0 at k equal anything else it is equal to 0 now what is the meaning of this this is the meaning of this is in the limit of very very large L this becomes a very very narrow very very high function whose area is equal to 2 pi ok we'll keep that in mind as we go along this is called the Dirac Delta function and we can now say let's now go to the limit of very large L if we go to the limit of very large L in other words we're making these intervals progressively smaller and smaller we're really approaching the situation that really does define the Dirac Delta function then this integral goes all the ways from minus infinity to plus infinity so just think of this as a formal prescription for an integral of e to the ikx and the rule is it gives Delta of K to PI times Delta of K this is equal to 2 pi times Delta of K that's something of importance that when you integrate over a function of e to the ikx like this DX you get a chromic are a Dirac Delta function of K let's see if you can guess supposing instead of doing the integration over X I didn't I did an integral over K exactly the same integral except it's not at all the same integral e to the ikx but instead of integrating DX i integrate BKE what must that be it's a function of X but it's exactly the same structure here except I just interchanged X and K right X and K appears symmetrically here so if I interchange X and K this doesn't change but I've changed the integral over X to an integral over K so this is now some function of X but what is the what is the function 2pi times Delta of X so these are two little observations that we will see happening we'll see their utility occurring in a number of places well that's the Dirac Delta function that's our first little bit of mathematics tonight what was that next a little bit of mathematics I taught you what a cat is a cat is a symbolic notation for a quantum state I also think I told you that I didn't what do I use I did I did I did I did and I told you it was half of a bracket yeah okay it's the other half of the bracket now that we want to talk about the back otherwise known as bra these are just notational devices in there for our purposes these are simply notational devices and the question is when do you use the bra and when do you use the cat for our purposes we will think of cats as initial States or a process of a process starting with some initial state is described by some quantum state will describe the initial state as a ket vector initial in final states we will describe by bra vectors and when you put a bra back the next to a ket vector you make a bracket this is simply a bunch of symbolic manipulation but at the end what comes out of the symbolic manipulations are numbers okay numbers numbers experimental numbers so I'm going to teach you now some symbolic manipulations for every let's go back to the harmonic oscillator for the harmonic oscillator we characterized the quantum states of the harmonic oscillator by occupation numbers n the number of excitations of the harmonic oscillator the number of times the number of units of energy that's been put into the harmonic oscillator we can also describe it that's a ket vector that's the ket description of a particular quantum state we could also write it in terms of a bra vector so far I haven't told you anything I just told you there's two ways to write the same thing you say what's the difference between them not much okay but if we are clever in our use of this notation it will help us do some bookkeeping that's interesting all right now first of all there's the notion of the inner product between a bra vector and a ket vector now all of this is in our past quantum mechanics classes and I refer you back to it we're going to go through it lightning-like tonight if I have a ket vector N and a bra back to M I can put them next to each other and putting a ket vector next to a bra vector in this way always gives a number the kid vectors or abstract things the bra vectors are abstract things but the product of two of them back to back or front to front I'm not sure which and that form is a number this number now n stands for some quantum state the nth quantum state of the oscillator and in stands for the end state of the car of the oscillator this number this is definition now is equal to 0 if M is not equal to n and it's equal to one if n equals in so the ball vector and the ket vector for the same value of n have a product which is one it's called the inner product the inner product between these two it's one if n equals M it's 0 if n is not equal to M or to write it in a unified form we can write it as Delta n n whose definition is it's 0 unless N equals m and 1 if n is not equal to M it's kind of like the Dirac Delta function it's a discrete form of the Dirac Delta function all right this is notational this is just notational tricks for things now let's come to creation and annihilation operators what I want to get at before we go further is how creation and annihilation operators act on bra vectors I've told you how they act on ket vectors and I have not told you how they act on bra vectors now of course you can say it eivin told you in fact I haven't even told you the rules which will allow you to deduce how they act but I'm going to show you how they act and then show you why this rule was particularly nice alright so let's take creation operators first of all what does a creation operator do when it acts on the nth quantum state of an oscillator it multiplies it by square root of n plus 1 times the n plus first state what about the annihilation annihilation gives you square root of n times n minus 1 so one of them raises the quantum state one of them lowers the quantum state and in each case they multiply by the appropriate square root of n now we can ask death then we're asking for a definition this is not a theorem which we want to prove this is a definition of how creation and annihilation operators operate when they act on raw vectors so how shall we take a plus to act on the enth bra vector and I'm going to tell you now what the rule is the rule is that however it acts on the in vector it gives another vector another bra vector of course let's circle that bra vector and now take its inner product with the mcat vector however this acts whatever it gives it gives another bra vector that's the rule when an operator acts on a ket vector it gives a ket vector and acts on a bra vector it gives a bra vector and the notation is such that when an operator acts on a bra vector you put the operator to the right and hit it to the left when we acts on a ket vector so it's a neat notation the rule is that you get exactly the same answer as if you did Y times you play a quest and allow a plus to act on em and then take the inner product within C there's two distinct operations however a plus should act on it and we'd like to find out how it should act on in we're going to define it so that you get the same answer for this bracket as if you allowed a plus to act on em well I've given you the rule for how a plus acts on em and then take the inner product within so let's see if we can figure out from this from this set of abstract principles or abstract definitions how a plus acts them in let's see let's first take the bottom line here the bottom line let's calculate it it's N and now a plus acts on em what does that give square root of n plus one that's a number we take it outside square root of n plus one and then what n plus 1 what does this give this this factor gives zero unless n is equal to n plus 1 if n is the same as n plus 1 then you get the number square root of n n plus 1 which also happens to be the square root of n of course so that's one way of calculating what that's this over here but notice it only gives an answer a nonzero answer if this is one unit higher than this sorry yeah sorry it only gives an answer if M is one unit lower than n if M is one unit lower than M then a plus comes along increases M by one unit and then we get a nonzero answer so the only way to get a nonzero answer is if n is one unit lower than n well let's look at this over here supposing that a plus acted on in to raise n then we would only get an answer if M was one unit bigger than n but according to this rule we only get a nonzero answer if n is one I think all right let me say let me say let me say it accurately I get my OK from this form we see that n has to be one unit bigger than m to get a nonzero answer n must be one unit bigger than M if on the other hand a plus increased the value of n over here then we would only get a nonzero answer in the opposite situation where in was one unit what less than M so this can't be the right rule that when a plus ax to the left that it increases the index here what it must do is decrease the index here and in fact that's the definition that's the correct definition when when a + acts on a bra vector n it doesn't increase in it decreases in n minus 1 and what about the numerical factor there the correct and Americal factor is just square root of n if you want it if you want you can work this out all right what if it what about n a - what does that do well the answer is I've given you enough rules that you can determine yourself what it does what it does is it increases n and plus 1 times square root of n plus 1 in other words to make the story short when a + acts on ket vectors it increases in and multiplies by square root of n plus 1 a - decreases in and multiplies by square root of n when a plus and a minus act on the bra vectors they just interchange a plus decreases in and multiplies by square root of n a - increases in and multiplies by square root of n plus 1 that's the rule just that's the rule which leads to a very lovely calculus that's useful in quantum mechanics or calculus meaning now tricks for computing simple things let me give you an example let's calculate in two distinct ways the following quantity let's take a plus a - remember what that was what does that stand for yes it stands for it's a quantum mechanical operator that stands for the occupation number the number of quanta in the state and let's calculate this quantity here a - I'm sorry a - let's calculate it in two different ways now for those have studied some quantum mechanics you'll know that this expression stands for the average value of whatever this quantity is in the quantum state in but for the moment let's just calculate it as an abstract the exercise but let's calculate it two ways in the first way let's allow this operator to act to the right first a minus ax to the right what does a - give what it acts on this it gives square root of n times n minus 1 we still have a plus which we haven't used up yet n now numbers come on the outside numbers like square root of n they come on the outside square root of n now what is a plus du with an axe on n minus 1 it pushes it back up and multiplies by what one integer higher than what appears here so that means again square root of n it gives us two square roots of n which means n times n what n a plus n minus 1 brings us back to N and n n what is that that's just 1 so it just gives us in if it acts a plus a minus 1 sandwiched within the quantum state in just gives us the numerical number in what would happen if we did it in the opposite in the opposite order but by acting to the left on the bra vector let's do it what happens when a plus acts on end to the left it gives us n minus 1 times the square root of n but we still have to act with a - what does a minus do when it acts on n minus 1 it raises you back up raises you back up to N and gives you another square root of n so you see with this definition that the creation and annihilation or the raising and lowering get interchanged when you go to their action on bra vectors and ket vectors then it doesn't matter which way you imagine these operators operate we'll get the same answer that's a useful useful notation and so when you see a thing like this you don't have to ask should you operate with this to the right and then take the inner product with the left hand side or should you operate to the left and then take the inner product to the right hand side you get the same answer that's the beauty of that particular definition here all right that's good now what are we going to do with all of this well of course we're going to study quantum fields which are objects built out of these creation and annihilation operators yeah yeah let's let's go right there let's go to the quantum fields go back to a simple quantum field that we've discussed already and then come to the description of quantum processes scattering processes creation of processes and violation processes and so forth how they described by quantum fields or how they describe in terms of mathematical expressions involving the quantum fields why are we interested in processes like collisions and creation and annihilation processes involving particles because in the microscopic world that's about all we can do if we want to do experiments the only real handles we have on experiments is colliding particles together and see what see what comes out and describing those processes how the initial state of the particle of two particles for example morphs into some other state involving 5 7 9 particles 4 particles whatever it is the tool for that is quantum field theory and we're setting up some simple examples all right so let's go back to the definition of a quantum of the simplest quantum field the simplest quantum field we took it to be a function of only one coordinate namely X there's no reason why we can't think of x y&z here and make position into a three-dimensional thing if we do so momentum also has to be Oh incidentally will work in units in which H bar is equal to 1 I don't think tonight the speed of light will come into anything but what's the connection between momentum and K if H bar is equal to one they're the same normally you would have an H bar over here it's bar K but if H bar is set equal to 1 K and momentum are the same thing okay right okay so we concocted a thing which we call the quantum field associated with the point of space X as I said X could be a three dimensional point of space if it is then K has to be three dimensional if we're living in three dimensions then the momentum is three dimensional meaning to say it has three components and then K will also have three components but other than that the formulas I'm going to write down are pretty much the same in three dimensions whatever number of dimensions were doing so I of X was a sum over all the allowed values of momentum if we're talking about the universe on a circle of a periodic universe then these are the caves which are two PI over L times an integer if we're talking about an infinitely big universe then K can be any anything any number sum over K summation of the K of the creation operator for a particle of momentum K times e to the minus I K X that now has become a quantum field there is a conjugate quantum field and now we pretty much if we're quantum mechanics people we talked about talked about the hermitian conjugate if we're classically oriented people we simply talk about the complex conjugate you can call it size star or side dagger of X the complex conjugate which is a similar thing involving annihilation operator Ruess times e to the plus ikx yeah is this a definition or this is definition this is definition but you know definitions the question it's always appropriate when somebody gives you a definition say why is that the definition and then the usual answer is wait wait till you see how we use it and you'll see that it's a useful definition okay so I'm afraid that's the situation here why is this the definition because this is a useful definition I could have put something else here and it would have been a useless definition so it's it's premature to ask why this is the definition but it is a it is a nice simple expression not very complicated sum over the allow about allowable values of momentum creation operator times e to the minus ikx or the complex conjugate which has anihilation operator times e to the plus ikx now I'm going to do something even a little bit fancier I'm going to give these I of X's some time dependents so far they don't depend on time the definition does involve does not involve time but I'm going to introduce some time into the game and how do I introduce time why do I introduce time well remember that there's a connection between the momentum the the wave vector K and the frequency of an oscillation these are fields these are fields fields oscillate as is written now it doesn't oscillate it doesn't have any time dependence if this has anything to do with a real field waves and so forth we better introduce some time into it well that's not so hard because we remember each value of K there is a frequency frequency tells us how things change with time supposing I have an object a mathematical object which has a certain frequency Omega how do I write the function that goes that oscillates with frequency Omega e to the I Omega T e to the I Omega T which also happens of course to be cosine Omega T plus I sine Omega T this is a function with a definite frequency the frequency is omega alright now in past lectures I pointed out to you that for various kinds of waves each K K is a wave vector it's inversely related to the wavelength through H K there is a frequency so we can say that there is for each K there is an Omega of K and each oscillation each time we see an e to the minus ikx we can put in front of it an oscillation like in front of it or behind it an e to the I Omega of K times T now this thing has not only space dependence but it has time dependence and moreover it's time to pen that we do the same thing here of course e to the minus I Omega of K T let's uh let's erase side dagger for the moment we'll come back to it I want more room on the blackboard here's an object which has both space dependence and time dependence the time dependence has been arranged in such a way that for each value of the wavelength okay it oscillates with a time dependence which is just the right time dependence for that wavelength this is now a function of space and time it truly is a quantum field now it varies in space and time ok I want to take an example and show you that in the example there is an equation of a wave equation for psy of X and T and see if we can figure out what the wave equation is see if we can find the wave equation for psy of X and T knowing the connection between Omega and K I haven't told you what the connection between Omega and K is but let's suppose that we're talking about a non relativistic particle in other words I'm not talking about a photon I'm talking about a species of particle which moves with much less than the speed of light that we're working in the approximation in which everything is nonrelativistic okay when h-bar is equal to 1 or over here H bar equals 1 Omega has another name what's there the name for Omega energy remember the energy of a photon or energy of a quantum of a single quantum we're talking about a single quantum now the energy is H bar Omega if H bar is equal to 1 Omega and energy are the same momentum is equal to H bar times K so when H bar is equal to 1 energy is frequency and momentum is K is wave number okay what's the connection for a slowly moving particle slowly means only slowly moving compared to the speed of light what's the connection between energy and momentum [Music] right energy of a slowly moving particle is the square of the momentum divided by twice the mass this is of course the same thing as writing energy is equal to one-half MV squared and P is equal to mV all right if we if we solve for V and plug it down into here we get this relationship here okay that also tells us now the relation between Omega and K K squared over 2m right so now we really do know the precise form of psy of X and T it's a sum over the allowed values of momentum and Omega of K is not just some arbitrary Omega of K but it is K squared over 2m with that proviso psy of X and T cells are satisfies a differential equation a wave equation let's see if we can see what the wave equation is first of all in wave equations derivatives with respect to space and time enter and are equal to each other in some form that's what a wave equation is equations for how space variation is related or how time variation is related to space variation so let's consider first the time derivative of SCI what's the time derivative of SCI I'm going to call it side dot standard notation for time derivative so I thought we get we obtained by just differentiating the time dependence with respect to T that's equal to some on K a plus of K e to the minus I KX times what times I Omega inside the summation I Omega of K e to the I Omega of K T all I've done is say every time this is still under the summation here every time I differentiate with respect to T it pulls down a factor of I Omega right so that's I dot now let's look at the space derivatives what is the space derivative decide by DX well we do exactly the same thing every time we differentiate with respect to X it pulls down a factor of minus I K so this is equal to the same kind of thing summation of minus I K a plus of K times e to the minus ikx e to the I Omega of K T so differentiating with respect to X always just pulls down a factor of I K differentiating with respect to time pulls down a factor of I Omega now there's no simple relationship between this and this why not because Omega is related to K squared if Omega was simply related to K for example if Omega had to be equal to K then we would just say side dot is equal to decide the X for example imagine a world or a kind of particle where the frequency is in fact equal to K that's a very simple situation in that situation I Omega K is the same as mine is apart from a minus sign is the same as I K and we would just say that such a field satisfies a equation that side dot is equal I think the minus decide by DX is that clear all right but that's not the case here we have to have Omega equals K squared so how can we get K squared take another derivative each time we differentiate it brings down a factor of I K so let's differentiate again that brings us minus I K squared what's minus I K squared I think that's minus K squared right minus K squared minus or the minus sign but now K squared is 2m times Omega K squared is 2m times Omega so let's write that 2m times Omega and let's divide the left-hand side by 2 n 2m is just a number yeah altitude is the minus sign relative speed like I thought you get white eye let's see let's check when I differentiate with respect to time it gave an eye Omega so that's all I Omega now when I differentiate it with respect to K it gave them a minus I K and I did it twice so that gives minus I K squared what is minus I K squared minus K squared this is minus K squared so yes as far as I can tell the minus sign is there the there was a there was a minus K squared and then I substituted for minus K squared minus 2m Omega and that's where this came from okay I'm more familiar with this in the form in which we divide both sides by 2 m this doesn't matter of course let's divide it by 2 m and we get 1 over 2m times the second derivative of the psy with respect to space squared is omega x all this stuff but that's clearly proportional to side dot what's the what's the relationship here we have it yeah we better but let's let's yeah there's going to be an i in the formula we can divide by I which is the same as multiplying by minus I okay to make a long story short the right equation should be minus I sigh dot that's the kind derivative of psy divided or is equal to 1 over 2m times the second derivative of psy with respect to x squared now do I have the sign right not but of course I don't I never get it right okay how was your right I'm not sure what are the two em then not one over 2m squared 1 over 2n 1 over 2i me okay oh let's see is the minus sign there or not let's put a minus here and then there's a plus here so it looks like this it looks like it's the plus sign far as I can tell it's the plus sign yeah well of course that will do that that will do the trick right okay does anybody know the name of that equation it's the Schrodinger equation right but it's not the Schrodinger equation for the same thing as an elementary quantum mechanics an elementary quantum mechanics sigh is just a function of position it's not an operator it doesn't do things it's just a thing whose square is a probability here it is an operator when it acts on States it creates particles it annihilates particles this is an instance of the relationship between particles and quantum fields quantum fields are operators they happen to have the same equations as the Schrodinger equation of elementary quantum mechanics they are very closely related but they're quantum mechanical operators they're observables you can observe them under what circumstances do they behave like classical fields in other words that you can measure them and the same way you measure the electromagnetic field the answer is that they behave like classical fields when the number of quanta is very large the number of quanta is large then the magnitudes of the fields a law arge and the quantum fluctuations are small or quantum uncertainties are small exactly like a harmonic oscillator a harmonic oscillator if it's got a big motion behaves classically if it's only got one quantum unit of excitation it behaves very quantum mechanically so this is an example as I said of a quantum field and it has creation and annihilation operators in it nevertheless it's a thing which satisfies an equation then the equation is the Schrodinger equation this is obviously a more advanced notion of the Schrodinger wave function then just saying it's a thing whose square is the probability for a given particle it's something a little bit different okay and as I said it is a quantum field it's the simplest version of a quantum field [Music] how do we use it how do we use it to describe processes so I'm going to give you any and we've already talked about this a little bit what I didn't talk about in which I want to come to now is energy and momentum conservation and how energy and momentum conservation are codified or codify above the right word is in the various dependences of sigh and the way that we describe various processes I want to I want to imagine now describing the very very simplest process in which a particle scatters off a target the target in this case is a thing which is so heavy that it doesn't recoil the particle comes in either reflects off it or scatters off it in three dimensions it could have its trajectory changed in other words it could be coming in from the left and go off straight ahead but what can you say about a particle scattering off an ordinary target are just stuck there it doesn't move stays there forever and ever which of the conservation laws would you expect to be true for that particle how about momentum the refraction reflection is what's a refresh reflect reflection refraction and absorption well let's forget absorption for the moment all right the particle comes in and goes out so it's not absorbed but yes it could be reflected bing-bang it could be refracted in which case its direction of motion has changed in any case the momentum of that particle generally is not the same coming in as going out of course secretly what happens is momentum is really conserved but what happens is of course the target absorbs some of the momentum but let's just fix the target let's imagine nailing down the target and the target is not part of our mathematical description or the motion of the target is not part of our mathematical description then in the mathematical description of a particle scattering of a target the momentum of the particle is not conserved all right what about the energy yeah the energy of the particle is conserved if you take a tennis ball well a tennis ball is a bad example because it you know it heats up when you throw it against the wall but an idealized tennis ball where you can ignore heat you throw it against the wall and it bounces off what can you say about the way that it bounces off you can't say that momentum is conserved the momentum has gone from that direction to that direction but the energy is conserved so how can we see can we see that in some simple model of the scattering process that momentum is not conserved and energy is conserved using these wave fields so we talked about this last time a little bit and I showed you how you could describe a process of scattering creation and annihilation and so forth using these quantum fields so here is a model here is a space as horizontal time is vertical and the axis over here represents a target it's it's fixed in space it neither and it goes on and on forever into the past it just sits there and a particle comes in it hits the target at some point and either bounces off or goes forward or scatters into some other direction so we described the initial state by saying this one particle with a momentum ke let's call it K in for initial initial or incoming and then a particle goes off having scattered off the target and let's use a language now it's absorbed by the target and suddenly remit it by the target okay think of it instead of just thinking of it as bouncing off the target let's in the back of our mind have a picture in which it is absorbed or annihilated by the target and then instantaneously recreated by the target but possibly with a different momentum let's call that final momentum K final let's just call it K I and K F initial and final and what are we interested in we're interested in the probability amplitude the quantum mechanical probability amplitude that the particle scatters from momentum K initial to K final now one more ingredient I'm going to assume that the scattering can happen with equal strength equal at any time always at the same place we're going to place the target at x equals zero but we're going to assume that the scattering process can occur at any time whatever time that particle gets there it will scatter or not scatter or do whatever it does but with no special dependence on the time at which the process takes place that's going to be our graph our basic assumption how do we describe the event of the particle being absorbed at position x equals zero well we describe that by means this should be side dagger shouldn't it I think this should be sorry dagger it's the daggers or the go with the plus sides here I think in my invitation that I use I think a plus always what with a minus here I think so and then this one would be plus if I remember okay we described it in the following by a kind of bookkeeping it's all bookkeeping but it's useful bookkeeping you absorb the particle this is a piece this is a field which creates particles let's write down the other conjugate field which annihilates particles that's the complex conjugate or sign of X and T which is made up out of annihilation operators e to the plus ikx e to the minus I Omega of K T it's just the every place everything is conjugated e to the minus ikx becomes e to the plus ikx and so forth and these are to be thought of as complex conjugates of each other or hermitian conjugates of each other alright let's imagine first of all absorbing the particle at the origin how do we do that we absorb it by an operator where we start with the initial state the initial state has no particles with any particular momentum X set one particle with momentum ki so if we describe it in terms of occupation numbers only the occupation number associated with momentum ki would be nonzero we can also just label that by saying this one particle with momentum ki and be done with it we don't need to specify all the particles that aren't there but nevertheless it's useful to keep in mind that when we specify the initial state with specifying all of the occupation numbers of all the various quanta and only one of them according to assumption here is present so there's our one here and all the others are zero and we just label it by saying what was one particle momentum ki then that particle is either is annihilated at the origin if it's if it's at the origin and let's describe that by psy of X and T but X is equal to 0 0 and T why is x equals 0 and assuming that if the process happens it happens at the target and the target is at x equals 0 but it can happen at any time so that's the process of absorbing the particle at time T that's what this says but then the particle is immediately recreated that's why you say why does it have to be it doesn't have to be that's the model that I'm making a particle absorbed and we immediately remit it that's a mathematical model it's not necessarily a law of nature about any particular kind of particle in fact in the real world a photon can be absorbed by an atom and emitted later or earlier so what we're talking about here is a simple mathematical model in which the process happens all at an instant particle absorbed in Rhian aisle and readmitted now what's the omission described by its described by the creation of a particle also at point zero so that's side dagger of 0 and the same time T but what time what time should I put there the same time but which time it will matter so why why 1 or 0 how about let's say it can happen at any time any time that the particle gets there okay whenever the particle gets there that's the same as saying integrate this over all possible times in other words there is a process in which the particle absorbs absorbed at this time at this time at this time and so forth so let's not prejudice what time it happens by just averaging or integrating over all possible times this is a mathematical expression which we can work out I will show you as we go along with the implications of this integral over time R but the basic physics of it is that the process could happen at any time no special time no particular time is singled out is special and we get rid of the specialness of the value of time by just integrating over all values of time what does this give this gives the final state here's an initial State these are operators which operate on the initial state and give some final state how do we calculate the probability that the final state consists of one and only one particle moving with some other momentum all right if you remember your rules of quantum mechanics we take the inner product of this state with the final state so let's say one particle with momentum K final this is a number remember a ket vector operator bra vector give us some kind of number is this number the probability for the scattering not quite we have to square it we have to take the absolute value of it and square it now I've left out one important ingredient in this expression we're going to evaluate this we're going to go through it and we're going to see what it says e because I'm I'm purposefully not purposefully I would rather be able to explain all of this in complete logical order but you know given an hour to do all of quantum field theory we have to draw on things that we've done in the past so in the past I explained that probabilities are squares of amplitudes and amplitudes are inner products between initial States and final States ok so here's the thing we want to calculate but I've left one thing out I've left out the strength of the scatterer the strength of the scatterer is a measure of the strength of interaction between the scatterer and the scan scatter e so let me give you an example you could have a charged particle at the origin which is scattering a photon a photon comes in is absorbed by the scatterer and re-emitted what is in that case what corresponds to the strength of the scatterer the answer is the electric charge of the charge the bigger the electric charge the more probable it is that the electron that the photon will get scattered there is a measure of the strength of the interaction between scatterer and scattered particle the strength of the coupling between them which which has to be codified somehow and it is simply by a numerical number put here it's called the coupling constant G G for what I don't know what G is for it's a measure of the strength of the coupling or the strength of the interaction between the scatterer and between the scatterer or the target and the particle itself I'll give an example some examples the scattering of a meson from a proton is a strong process meaning to say that if the Mazon is absorbed if the Mazon arrives at the location of the proton it has a high probability of scattering that's indicated by a large coupling constant a photon in the vicinity of a charged also could be a proton a photon is also absorbed and re-emitted by a proton the probability for that scattering the probability that the photon gets redirected is much smaller than the probability for the Mazon and that's indicated by the coupling constant being much smaller for the interaction of a charged particle with a year you think of something even smaller yeah on the tree now interacting with a proton has an even smaller constant for scattering off this that's what made for the target being an electron of the well right now we just right now we're just making a model in fact of an electron or something like an electron scattering off target so this is don't think of this as being valid for any real thing this is a simplified model for a cell for a number of different situations what about the scattering of a graviton by a by a nucleus graviton is the analog for gravity of a photon well it can happen but it's an extreme small constant I won't even try to tell you how 10,000 minus some large number by comparison alright so the strength of the interaction and the probability that this actually happens the particle gets redirected is indicated or described by the coefficient G which appears here it's just a number now it's called the coupling constant what's called a coupling constant there are many coupling constants all right so let's see if we can calculate this what are we telling to calculate we're trying to calculate the probability or going from K initial to K final that's all yeah is there any logic for G be in a constant as opposed to being a function of Omega okay hmm that's a good question the answer is there is no logic to it no there is logic to it but there are situations where it's a function of Omega all right I will I'll show you some situations as I said this is a model of a particular kind of simple scattering there are more complicated kinds of scattering a more complicated kind would be and this would really be for the case of a photon in an atom for example photon comes in atom gets excited and then D excites by emitting the photon okay here is a little gap a time gap between the time the photon is absorbed the time it's emitted this happens to be equivalent to saying that the coupling constant is Omega dependent okay so as I said we're talking about a simple model and just trying to find what the consequences what the consequences are there's going to be one important consequence and it is very general the rest is not so general but the one important consequence that I want to get to so let's just plug away we're going to just go straight ahead plugging in for sy its value written in terms of creation and annihilation operators and then use those creation and annihilation operators to annihilate the initial particle and recreate the final particle and calculate the expression that's up there the square of it will be the probability so let's do it okay so yeah if G is equal to one yeah all right um that's a good question well you could ask what happens if G is much bigger than one does that mean that the probability is much bigger than one no it doesn't it doesn't it just means that you have to go ahead into a more complicated calculation this calculation that I'm doing is only correct for a very small G okay it's correct for very small G what you have to do in G is larger is extremely interesting and will be very important to us and what follows the answer is you have to do higher order perturbation theory okay but we're not doing that now well we're just doing the simplest thing all right so let's do the simplest thing and just plug in for psy its value all right sigh the first sign up a side dagger but the first side is a summation over K naught K initial which is K a summation index a minus to absorb the initial particle of ke e to the ikx e to the minus I Omega sub K T and that acts on the initial state which happens to have one particle of momentum ki yeah hmm thank you X is zero at the scatterer so let's leave it out e to the ikj Rho is just 1 which terms in this sum will contribute only when K is equal to ki will we get anything right why because the annihilation operators gives zero when they act on states where there are no particles the only particle around has momentum ki and so only when K is equal to ki 2 we get anything but nevertheless let's leave it in this form just for the moment now what about the other operator side dagger that's also a sum over momentum I better use a different index because I don't want to use the same summation index twice get confused so I'll call it let's call it L and now we put a plus of L that creates the new particle with momentum L times e to the well X is still equal to zero X is still equal to zero so we don't get anything from here but then we have e to the I Omega sub L times T what about the final state the final state is the state with a particle of momentum K final okay which term are first of all the term in the sum of a k which contributes is only when K equals K I right what about the sum here over L what does l have to equal K final remember that when a plus ax to the left it annihilates it has to find a particle to annihilate a plus of L only gives something when L is equal to K final so really both of these sums collapse only K equals K initial and L equals K final the taller is what did I leave out G but I've also left out something else where is it integral DT it's the integral DT which is the thing that I'm really interested in all right so the only contributor here is K equals K I what does a minus of K I do and it acts on a state with momentum K I it just creates a state with no particles right it annihilates the particle with momentum kie what about this sum this sum is only nonzero when L is equal to K final so we can put K final here and when this acts on a state with K final oh sorry yeah okay right we're okay this is good what happens when a acts on the state with K initial it just gives a state with no particles right so that's just no particles what happens when this acts on the state with particle with momentum K final it just gives no particles again what is this number one so that much is one so all this operator nonsense of creating and annihilating particles all goes away and all we have is a number G that's the probability for the scattering or that's the amplitude for the scattering G nothing else except that we have this integral not in this expression I think you're thinking about a relativistic problem no no there was only one particle here because there was only one particle yeah yeah it was only one particle right okay but the important thing the real only thing that really I'm illustrating here is the integral over time and what is it it's an integral DT e to the I Omega let's just call it final minus Omega initial this is K initial times T that's the whole upshot the time integration gives us an integral over e to the I Omega final minus Omega initial T all of the operator nonsense just gives us a 1 nothing very interesting the coupling constant gives us a G and now we have to evaluate this integral what is this integral Delta function this is an example of the Delta function an integral DT of e to the I something times T let's see was it 2pi times the Delta function if I remember I think it was 2 pi G times Delta of Omega final minus Omega initial is this ring a bell Delta of Omega final minus Omega initial it's only nonzero when Omega final is equal to Omega initial Omega final is the final energy Omega initial is the initial energy so somehow magically now notice that if we had not integrated over time we would not have gotten this Delta function of energy so somehow there's a connection between the fact that that there's a conservation of energy and the fact that it doesn't that there's no preference for any specified time we could have made a model in which the scattering only happens if the time is within some boundaries then this integral would have only gone over some limited amount of time it would not have made a delta function so the ingredient here which is closely connected with with energy conservation is time translation symmetry that every time is like every other time yeah Oh Oh the square of the Delta function well yeah yeah yeah the probability the for the probability we have to square this this is the probability amplitude the probability itself will have the square of this but who cares about the square of a delta function the Delta square of a Delta function is also zero if Omega final is not equal to we have to worry about squares of Delta functions but in any case this will be zero unless the initial energy is the same as the final energy let's assume the initial energy is the same as the final energy then this is the operative important part of the scattering amplitude let's call this by its name it's a scattering amplitude and the probability for the scattering is proportional to 4pi squared g squared right the strength of the scatterer or the probability for it to scatter contains G squared they always wind up containing PI's but the important thing is the G squared and which which fun which final momenta can happen well I didn't specify anything special about the final momentum here this final momentum could have been anything as long as the energy was the same so this scatter has the property that it takes a particle and with a probability proportional to G squared changes its momentum with equal probability to any momentum with the same energy that's it so this is a scatterer which can scatter into any direction with equal probability okay any direction with equal probability that of course is not true of all scatterers this is this very simple model that scatters with all with equal probability in all directions and the coefficient G squared 4 PI squared G squared in this case is the probability so this is I really I've shown you many things so far tonight in particular the definition of a coupling constant the fact that the integration over time is the thing which ensures energy conservation which is just another way of saying the problem has time translation symmetry there was nothing specif special picked out about one time or another time and and I've Illustrated the idea of a scattering amplitude or the amplitude the thing which becomes squared in order to calculate a probability this is a fairly generic set of ideas but the sped the details can differ you can always find stands by by defining 0 x is 0 say mama can you always simplify this by redefining the position so X is zero yeah good question yes yes in this particular case yes the question of what would happen had you not put x equals zero here right that's that's kind of the question okay what you would have found is it would have been another factor the other factor in here would have been in the amplitude e to the I K initial minus K final times the position of the scatterer what should I call a position of the scatterer okay just let's just call it X of the scatterer X of the target X on the target is a number X of the target is just the position of the target when the position of the target was zero this didn't occur if the target was moved over to position X target you would get an extra factor but what will happen to the probability no change because the when you square meaning to say when you multiply by the complex conjugate this goes away multiplying this by its own complex conjugate which is what I mean by squaring multiplying by its own complex conjugate this goes away so this does not appear in the probability they're the same probability no matter where the target is but in an intermediate stage of calculation in calculating the the amplitude the position of the target would occur yeah so it's very simple models you can cram two years into our young but the mathematics unless I see it wrong seems to say if we said that it didn't matter where it happened but it happened exactly at time zero we'd get a different conservation yes for you guess the conservation momentum momentum there's no energy conservation right right a design a physical interpretation or is this more property well of course in the real world that we're studying for example the interaction of two particles and we take everything into account both particles and so forth the scattering can occur at any time at any place right ah but you could imagine you can certainly imagine an approximate situation where you might okay you're asking me is there ever a situation where here the approximation was we just ignored the recoil of the target okay and therefore the momentum was not conserved is there ever a situation where energy is not conserved and that sort of situation is what happens when you have time dependence in the coupling car or time dependence in the strength of interaction now yes there certainly are situations where that can be a good approximation to say something something makes a sudden change in the system from the outside and when a sudden change in the system happens that's when the scattering takes place yes we can think of situations where that happens I can where we can do we can try to concoct one I can't right now I'm a little fuzzy but yes we can certainly concoct a situation where something sudden happens in such a way that it makes the scattering possible at an instant of time and then energy not conservative right most of the coupling constants come from experimental data yeah well so you just said introducing and time dependence in the coupling constants well that usually means when there's a time dependence in the coupling constant it usually means is something that you've ignored in the system which is either moving or time dependent something whose dynamics and degrees of freedom you're sort of ignoring in the same way that you ignored the possible recoil or repositioning of the target here so I will try to think of a situation where where there's an interesting violation of energy conservation or where the bookkeeping was such that you threw some of the energy away but let's come back to it yeah hitting a moving target hitting a moving target right hitting a moving target there's an example where the target where they were again the motion of the target is thought of as completely I on certain skills you have to worry about the size of the target versus the yes the frequency of the the size of the wait yes so electron versus x-ray versus gamma versus yeah here in this model Atari the size of the target was zero yeah in this right and you're absolutely right something interesting happens when the target has a certain finite size and we can we can discuss that but let me just come back to the question about the moving the moving target if you think about it for a minute of course if you have a moving target energy is not conserved for example supposing that wall was moving toward me and I had a tennis ball and all of a sudden the wall hits the ball and the ball moves off the energy of the of the tennis ball we're not accounting for the energy of the moving wall we're not trying to worry about that so if the wall is moving that corresponds to a time dependence in this case a time dependence not of a coupling constant but at a time dependence of the location the scatterer that would be enough to make energy not conserved and we could see it in the in the mathematical formalism okay so to reiterate these creation and annihilation operators are the tools which allow you to discuss transitions of particles from one state to another state um we have not talked about well actually we could talk about creation and annihilation yeah let's let's discuss another situation Oh incidentally supposing psy was the field operator for an electron each kind of particle has its own separate field okay so electrons have a different field than photons when you put psy here you better specify what particle you're talking about um let's think of the electron sorry describes electrons now here was a process in which an electron came in and an electron came went out what happened to the total charge did the total charge change no one particle came in went a particle come out now let's imagine a different situation let's imagine a situation where one electron comes in and two electrons go out okay crazy I mean it can't happen but let's for a moment nevertheless try to imagine it how might we describe it by the same kind of mathematics one electron comes in two electrons go out how do i modify this and Theia my simple model is two electrons go out from exactly the same point yeah we might put another we might square this in other words sigh of zero Teesside dagger of 0 Teesside dagger of 0 t sigh of zero tea oops this should be a tea here right yeah okay what about now this is this of course can't happen in nature because electric charge is conserved this corresponds to the annihilation of one electron and the creation of - bad idea but nevertheless let's write it down what about two electrons in and two electrons out is that okay well at least it doesn't violate electric charge conservation how would we describe that another sigh sigh sigh sigh dagger side dagger how about two electrons in three electrons out sigh sigh side dagger side dagger side dagger okay which one's of these are allowed and what's the rule I'm not the writing a rule now we're just going to observe same number of patient in a laser right same number of size as side daggers this is not allowed this is allowed this is not allowed rule is same number of size aside anger automatic vertical yeah it's not a question of the order of the operators it's the number of size versus the number of side daggers when the number of size hi let's when the number of size is different than the number of side daggers versus when the number of size is the same as the number of side daggers how can you diagnose these combinations of course it's it's very easy just to count them of sizes inside daggers of course but let me give you a mathematical diagnostic that really does have a deep meaning let's imagine a transformation now side dagger does mean the complex conjugate of side it is the hermitian conjugate in the language of quantum mechanics but the hermitian conjugate is the analog of quiet of complex conjugate let me imagine that I take an expression like this just a mathematical expression like this and I transform sigh by multiplying it by a phase a phase means an e to the I times some number let's call it alpha times I I just I'm just doing now blindly playing a game wherever I see psy I multiply it by e to the I alpha times side what happens to side dagger if I multiply a complex con or quantity by e to the I alpha what happens to its complex conjugate e to the minus I alpha side dagger what happens to objects that have an equal number of science i dagger they stay the same what happens to objects which don't have an equal number of science are dagger they change so we could characterize the allowed processes by the ones which are described by operators which are invariant unchanged by the operation of changing the phase of the operator now that indeed but for the moment let's give it a simpler name invariance under changing under redefining the field so that you change its phase and overall everybody know you change the phase by a constant phase factor if the option if the interaction expression here is invariant then charge is conserved if it is non invariant then charge is not conserved what happens supposing I had side plus side dagger what kind of thing does that correspond to incidentally or I'll leave that to you to think about but is this is this unchanged by your base this is the real part of Sai and if you multiply by a phase you change the real part of sorry so that's not a quantity which is invariant inclusion should be done good then you have an electron in Picchu in eternity young live that many signal data electronically no wait if you have what kind of target target full of electrons well sure you can have yes yes yes yeah right oh you say what happens if you have an atom and you hit the atom with an electron and seventy-five electrons go off well charge is not violated of course what has happened is the atom has changed all right all right this is a situation in which you cannot get away with ignoring the dynamics of the target in the particular the the important dynamics of the target is the change of charge of the target you can't in this case treat the target is always a completely passive thing you've got to remember right what's that yes that means the target itself has to be described by a quantum field all right and we can discuss that I think we've gone far enough for tonight all right so what do we found we found that time translation invariance corresponds to energy conservation phase invariance we can call this phase invariance corresponds to charge conservation and we haven't worked it out yet we will work it out next time that spatial translation invariance corresponds to momentum conservation I'll show you how that works next time but we're making our way slowly into how particle physics processes are described by fields that's our goal for the for the next one more lecture on it I don't know maybe half a lecture on it and then we'll move into some of the symmetries and some of the more interesting laws of particle physics sighs yes yes but it's not hard to Rihanna that's great I've chosen always to put all the daggers to the left and the sides to the right remind me next time and I'll show you how to rearrange other expressions so that they become like this I know what you're asking you're asking what would happen if instead of this we put Sai Sai dagger sai Sai dagger and I will tell you next time but they still not this is allowed no not quite not quite not quite not quite remind me next time I'm too tired now to start going into it but remind me absolutely next time what is the difference between this and this and I'll show you next time all right I left I actually left time for some questions if you want to ask some questions I just have run out of steam in well it's not a trivial if you have as many sides of side daggers that means the interaction eats the same number of electrons as it spits out all right each side annihilates an electron each side Dagger creates in a way all I did was point out that all of the objects with equal numbers of size and side daggers will be invariant under the phase operation here if they have different numbers of size and side daggers let's take a case yes I side dagger side dagger you get it right I'm not sure what the rest of the question is if they have equal numbers of size and side daggers it's phase invariant and it conserves charge if has different numbers of size and side daggers it's not phase invariant and it doesn't conserved charge so we're just observing that when it's phase invariant it conserves charge when it's not phasing very they're Twitter energy-momentum in charge for the moment we haven't we haven't gone through a momentum conservation although was mentioned by somebody here asked me if the process can happen at any place instead of any time then yes then it conserves momentum but we'll come to it will do that case yeah in classical mechanics it's north is theorem in quantum mechanics it's even simpler but [Music] but here I'm sort of short-circuiting all of that discussion by just showing you how it works mechanically in in terms of these quantum fields well there's the availability of that first system that is uh PI Square D square that means when you empirically assign a value to the coffee classes that must be one over two days it must be less than one over something or you have or your model is very incomplete if you ask me in what way it's incomplete I will tell you in what way it's incomplete is that the real scattering process is not described by just a particle coming in and getting scattered out it's described by something more complicated which is a sum of problem of sum of amplitudes in which the particle comes in scatters off the target comes in goes back out comes back in scatters off the target twice comes back in scatters off the target three times it's a whole infinite series each one each term has a factor of G in front of it it's only when G is small that only the first term is important but we'll come to that this is this is a basic theme that when the coupling constant is small you can get away with the simplest minimal process when the coupling constant gets large one has to well let's just call it what it is one has to sum up an infinite number of finding graphs basically these pictures that I've drawn are Fineman graphs and the vertices of the Fineman graphs are described by these operators okay the vertices of the Fineman diagrams tell you the basic elements that can happen particle comes in bounces back out that's side times side dagger two particles come in and bounce back out that's side agus ID agus ëyesí or whatever and we'll we'll give things their proper name next time now are there any situations when coupling constant has imaginary component well yes fatality you want you who you want me the existence of a theta proper ammeter in QCD he the existence of a theta parameter in QCD I'm sure that means nothing to you yeah it is it does happen that there are complex values of a coupling constants it usually means the time reversal invariance has been broken complex coupling constants usually indication of a violation of of time reversal invariance that's really what it comes down to now you can't see that from what I've told you up till now at least if you can you're smarter than I am I can see it but only because I know how to see it but it's not obvious but that is the case complex coupling constants mean that time reversal invariance is is is broken in other words things don't look if you run the process backward as a movie going backward it's not a possible process this is that at one point also that instead of having me the scattering occurred the outgoing particle the same time as the inter particle can be before or after you also mention before I do not have not now right now yeah no no I did say that and I sort of regretted it the minute I said it but we we will we will now this is it's a fair question but it's not the time for it now yeah deputy frame that used to say that last before your ticket once then Jean Franklin would say it laughs before you tickle it okay problem to say you'll get this at some point how do you turn this into something that's relativistic ah again you're jumping way ahead I'm trying to go slowly and everybody's pushing me to get ahead okay are you really want to know you add this in this and put a square root of Omega K in the denominator nah nah leave it we're jumping ahead and you talk about charms conjugation conservation conservation with a phase factor yard but the best maxium that doesn't change also in a non relativistic process but in a relativistic process it can and there's nothing in basic quantum mechanics which says that rest mass can't change it's a combination of quantum mechanics or not even quantum mechanics even just classical mechanics together with an invariance principle and the invariance principle is Galilean invariance which I had which is an analogue of non relativistic analogue of Lorentz invariance but yeah it is true in nonrelativistic situation total mass can't change but it can in relativistic scattering so what do you want to ask me exactly well is this that logically we think of rest mass conservation of the same terms as charge no no no no no no not at all not at all one is absolutely true in nature the other the other is only true for very very slow velocities right there is nothing sacred in physics about the conservation of rest mass and then what's more is not true an electron and a positron annihilate into two photons there's no conservation of mass there is a conservation of electric charge so one of them is a sort of accidental nonrelativistic fact the other is a deep underlying symmetry of nature which happens in simple case just to correspond to this phase invariance next time we'll talk about fermions we'll talk about what happens if you have more than one species of particles supposing you have electrons and muons and quarks and so forth how do you describe all of this what's allowed what's not allowed the kind of processes all in the language of the scatterer again big Amos said it was an electron but I thought we only knew about photos ah good point good point good point good point yes yes yes yes good point yeah this so we have we're talking about a boson occasion of an electron yeah sorry I completely missed that you're right right but there are particles in nature which carry electric charge yeah right exactly a charged pine is on a slowly moving charged pine Mazon would be an example of a boson which you would describe in this way yeah yeah so good absolutely all right thank you for more please visit us at stanford.edu

Channel: Stanford

Views: 98,904

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Keywords: science, physics, particle physics, simple quantum field, vectors, field theory, particle, wave, momentum, occupation number, harmonic oscillation, position, reaction, annihilation, wave function, probability, atom, decay, photon, creation, energy, bra-ke

Id: eA8X8IU_fWY

Length: 111min 42sec (6702 seconds)

Published: Thu Feb 04 2010