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New Revolutions in Particle Physics: Basic Concepts  Lecture 4
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Erkannte Entitäten
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Stecher University this His
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sounded at baguette Sunday on the recorder recorded yes we did not have the like all the old lectures the only in 1 election and the idea is for everybody online and offline actually have a meeting with navigate while lecture I can't write the work that has on right the way out for until now later the final hole will have only laid directly or arrived and not like right OK as this is the address http colon slash slash new loan package Tech 1 word packet said stop come flesh resources been that's the address where but it's so the website for the class will be the website of the class and that's for anybody outside inside where they can easily access where about Dole confusion that's taken place up till now easily accessed all of the lectures that are online we know we have been discussing a simple quantum field and not finished work I have to take a twoyear course and believe me I'm real cost quantum field theory is genuinely 2 years of work really cannot be done in 1 year sensibly I have to take that to years a of quantum field theory that condenser down to a couple of lectures relate taken a couple of lectures but I think we've had some forward motion I won't take them very very simple version of a quantum field that we already discussed the remind you with this and discuss how it is used again the describe particle process to to use we've done a bit of this we talked a bit about Our quantum field colder around codified scattering process use are creation and annihilation process of particles but I want to go into it just a little more depth so that you can see where some of these really interesting aspects of quantum field theory come from and how they influence questions like energy conservation momentum conservation how those things how they are related to these quantum field so we discussed OK before I do so we need a little bit of mathematics Aruba formal mathematics not much mathematics that we've done before a new series of classes but I wanna get em up on the blackboard the 1st thing is Barak Delta function either Dirac delta function is to remind you what is is a function which it is sort limit of a genuine real function it's a function are let's say it's coordinator for moment the score was coordinated their anything could be exit would be OK but Cisco wiII so we don't prejudiced whether it's something that I've called defined previously the Dirac delta function as a function which is that's concentrated someplace that's concentrated someplace and not a report but the limit of In the early show bleak are concentrated function so we imagine that we can go to a limit where want like function is infinitely narrow back can draw is infinitely narrow also drew 0 with finite with the legend your mind narrowing it now going in that that cost a funeral it without
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raising its height area under it will decrease in decreased and decreased to the point where there's no area left under it so what what I want to do is to keep area under this function fixed as I decrease the way so as I decrease with a raise up the height of it in such a way that the product of High Times with stays constant how constant won just 1 an area of 1 located some place incidentally narrow that's called a direct note the function of this point is why equals call any locate the particular point than the Dirac note the functions His Delta of white minors and why minors a is Zero or whatever why does not equal a here over here an equal something infinitely sharply peaked at White equals that's the built a fortune and have the property by definition the area under the derivative the integral with respect Hawaii is 1 so it's so narrow and so high but area's 1 and it's concentrated at the point where the argument of the function is equal to 0 0 other words when white equals a that's a Dirac delta function OK Now I wanna show you have a direct Delta function emerges from a certain integral and interval but we will command many many times let's take the functions each RIA K X k eatery eye candy X now again as we discussed last time we're discussing this on an interval which is at periodic intervals which has Tulane all around it equaled the L. this since around here His equal to L most the best and surrounded his equal that l periodic functions that live on this periodic space should be periodic meetings say sitcom back themselves if the wonderful look OK now I was 1st of all case is 1 of the or allow would values of wave number on this summer was 1 of the values of with the IKX Soto assume allowed values and let's take this function in a bid made all but the entire cyclic X mentioned he would get tank X going from 0 Mobile AL all the symmetry I could take it to go from miners over to tell over another words instead of starting XI at 0 going to hell started at minus Alomar to go it alone to prove 0 % here nothing special about this just symmetrise things line nicely so that the negative hair from positive hair forced when you leave this place a few more marching along and you can't tell over to you pop back are buying over OK so we're taking a function of periodic function IKX and integrating it from minus over to Elverta was the answer OK 1st of all the answer Katie equal to 0 L tried equal tell if K is equal to 0 and worst case is not equal to 0 0 0 used to take appear at periodic function are it's a new integrated then you get 0 because the positive as much as negative only a 0 and that case of course this doesn't oscillators just people the 1 thing you get out after now let's think of this as a function I OK West the kayaks axis West or the care access to the Texas that case is not just any number it's 1 of the or allow numbers so let's discreet as anybody remember what the allowable values of R To apply in them over L Wright particulars L. gets bigger and bigger the distance between neighboring values of k gets smaller and smaller and eventually axil gets infinitely big these discrete intervals shrink 0 OK so what's see what's the Pistons but the cable 0 OK can be positive or negative incident but but cable 0 right over here His cake was 0 at what's the interval between neighboring values Kerry 2 pie over L at the distance from cable 0 after equals 0 2 technique was 1 for example to apply now let's take this function this function is a function of K of course we entered granter integrated over X is only a function of Carey was equal to what plot on here it's equal to a where K is equal to 0 0 so right at K equal 0 here as heII Tennessee quota L but said a giver but flight with just for the purpose of drawing it does for the purpose of drawing it right at cable 0 here it has a high equal to L but I tape not equal to 0 0 that's 0 we concluded catered for the L yes if case is equal to 0 and 0 0 F K
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is not equal to 0 how high it is did a tight L. how wide is it well 1 ain't all you can think of anything smaller than 1 and the 1 here Our it has with 2 pipe over what the area if I imagine giving it that much aware of what area armory pot so that means that this function is 2 party crimes the delta function function here to party kinds of about our case why don't of had equals 0 it's not equal to 0 cable anything else it is equal to 0 now where the meaning of this this is the meaning of this is in limit our very very large scale this becomes a very very narrow very very high function was an area it is equal to 2 part will keep that in mind as we go along this is called a Dirac delta function and we can now say but snow the limit of very large L if we got to the limit of very large other ways of making these intervals progressively smaller and smaller were really approaching situation that really does define the Dirac delta function then this and they goes all the ways from minus infinity for bless infinity so I just think of this as a formal prescription for an integral of eagerly I. kayaks and the rule is it gives Delta of K 2 pipelines Doctor of case is is equal to 2 5 kind Delta of that's something important that when you the great over a function of Ito IKX like this DX you get a chronic our Dirac delta function of cake but say guests supposing instead of building the integration over X I didn't I didn't and the goal the case exactly the same into exhibits now overall segment eatery IKX but instead of integrating integrate became what must that be at a function of X but exactly the same structure here except just interchanged X in case you're right XEK appears symmetrically here so fight interchange Exon Katie this doesn't change but Cheney interval of X to an integral Lovett case so this is Nelson function of X for what is always the fortunes of the buy and Delta attacks so we looked a little own observations that we will see happening will see their utility occurring in number places products that dealt a function of suffers of the mathematics tonight was that little bit of mathematics I talked is the kept as a symbolic notation for quantum state I also think I told you I didn't want to use a day I did and I thought it was half of bracket not retained see other hair firmer but now the rural about the back over those visitors notational devices and therefore are purposes is simply notational devices and the question is when they use the by Monday using cat drop purposes we will think I can see as initial states of a process of a process starting with some initial state is described by some quantum state will describe the initial state as a cat backed initial in final states we will describe by Robert this winter put abroad recognized so it can be America bracket this is simply a bunch of symbolic manipulation but In what comes out of the symbolic manipulations on numbers case numbers numbers experimental numbers are teaching some symbolic a 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 and the number of excitations of a harmonic story the number of times a year number of units of energy the increment of a harmonic oscillator Wicker also described it etiquette effect as the cat description of a particular quantum state we get also Wright In terms of above act that stuff I told you anything just to waste right the same thing he said What the difference between and not much Bhutto Italy's Prada club of our use of this notation it will help us do some bookkeeping that's interesting right now war the notion of a new product between a brawl vector and now all of this it is not passed the quantum mechanics classes and I refer you back With the book to work like him like tonight but I have a kept back the and and a brawl that at I compose next to each other and political that connects above it this way always gives the number Bequette sectors are abstract things about sectors are abstract things but the product of 2 of them backtoback are far right front front module which in that form is a number this number now it stands for some quantum state and quantum state of the oscillator and ended stands for the instead of the car we are sort this number this definition now is equal to 0 0 if and is not a quota and that's equal don't 1 and equals so the bottom and the kid vector for the same value of have a product which is 1 that call the inner product inner product between these 2 bits 1 if any calls and it 0 is not
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equal to end or derided and unified formal we can ride it as dealt and and whose definition is 0 I'm less and equals and and if there is not a quota and it's kind of like a Dirac delta function at a discreet former director of the by this is notational visitors notational tricks of things now let's come the creation and annihilation operators were 1 again that before we go for that is how creation and annihilation operators act on brought sectors I've told you how the act can't vectors and I have not told you how they act on brought actors not you could say Of course I haven't followed you and I haven't even told you the rules which will allow you to deduce how act but I'm going to show you how they act in this show you why this rule was particularly nite but so let's take creation operators what does it creation operated due where funny and quantum state of an oscillator multiplies it by square root plus 1 times end plus 1st or about annihilation the annihilation his you squared and pens and minus 1 so 1 of them raises the quantum state 1 of them was a quantum state and teach case multiplied by the appropriate square event value can ask death Delaware asking for 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 brought back so how shall we take 8 plus back on and brought so you know what the rulers rulers that however anatomy and it gives a number of economic project of course the circle that brought back then and then I think it's in a product with me and can affect color of this acts whatever it gives it gives a mother brought us a rule operator acts on a kid victory gives a cat that corrects on Prozac the riches of Bravo and the notation is such that when an operator acts on a brawl vector you put the operated a right and hit the left acts on a cat that Toto need a the rule is that you get exactly the same answer as if you did what you can allow a plus tho act on and then taking a product would have to do it distinct operation however plus should act on it and find out how rich Back from and with the find it so that you get to see me and so far was brought kept as if you were allowed a plus stacked on while giving you the rule for plots acts on and then take the product with so let's see if we can figure out From this from this set of abstract principles or abstract definitions Howell a plus acts on it see but 1st take a bottom line here for bottom artworks calculated end and now a plus acts on what is active square them plus 1 that's a number we take it outside squared plus 1 and then what and plus 1 what does is scarce this this factory give Zero on West End is equal to plus 1 if it is the same as in plus 1 then you get the number square root and plus 1 which also happens because squared and of course so that's 1 way of calculating what sets this over here but notice it only gives an answer Annunzio ulAnsar if this it is 1 unified
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next sorry yacht sorry not only gives an answer yes is 1 unit lowered its and as 1 unit lower than M and a pass comes along increases in by 1 unit and then we get a nonzero and again on 0 and is it game is 1 unit lower than well let's look at this over here surpassing that a plus acted on in to really and then really would only give Mansour if M was 1 unit bigger than ever but according to this world we only get a 0 answered it is 1 that I think the program is said Tuesday it accurately I get my from this form we see that and has to to be 1 unit begin and end together nonzero aunts and must be 1 unit and if on the other hand a plus increased the value of an over then we only get a nonzero inserted in the opposite situation where In the was 1 unit what less than so this can't be the right ruled that when a class act of the increases the index he work must do is decreasing index here the fact that the definition of the correct definition where but in a plus acts are that it doesn't increase and it decreases and minus 1 hour about the numerical factor they have the correct numerical factors just square if you want a cure Yukon workers out I wonder if it what about M a might was at wealthy histories and giving you enough rules that you could turn yourself work hours what it does is said increases and in plus 1 times Square and plus 1 another was to make the story short when a blast on kids vectors it increases and and multiplies by square event force 1 a minus decreases In the multiplies brisk word of when a Pozsony minors I am a brass sectors they just interchange a past decreases in and multiply by squared event a minus increases and a multiplies by squares and plus 1 that's the rule this that's over all of which leads to look really lovely calculus that usefully Kwan character calculus meaning now tricks for computing simple things that I give you example let's calculated 2 distinct ways the following quantity but at stake a plus a marine what that was where therefore for yet stands for its quantum mechanical operated that stands for the occupation number the number of court in the state and that's calculator this quantity here they might assumption let's calculated in 2 different ways for those who studied some quantum mechanics you know that this expression stand for the average value of whatever the quantities In the quantum state and put them on the list calculated as an abstract exercise but let's calculated to ways in the 1st wave let's allow this operating attacked the right 1st 80 miners acts the right was a minus give back it gives square root and kind and minus 1 we still have a Plus which we have used up yet now numbers come money outside numbers like squared event may come outside squaring no was a plus to acts minus 1 the Bush is a backdrop and multiplies by 1 integer higher than what appears here so that means again squirrel event it is To square Savannah which means and crimes and what if you AP plus arms in mind this brings us back in in in what is at as 1 so just gives us in acts a plus minus 1 which within the quantum state in just gives us the numerical number what happen if we did the office not the opposite order but by acting the vote left on the brought with Stewart what happens when a plus acts are in it gives us an minus 1 Times Square remain but we still have Rackerby minus what does a minus Nova Heffernan minus 1 it raises you back up raises you back up and gives your mother squared event you see with this definition of the creation and annihilation or the raising and lowering it into changed when you go to their action on broad sectors and kept it doesn't matter which way you a magic these operators operate with did the same answer at a useful I use former patient and so when you see a thing like this you don't have to ask should your happily with this right and then take the inner product with left inside should you operate the left and then take prodigal right inside you get the same that's the beauty of that particular definition that's good now Correa deliver orders because of relative study quantum fields which objects built out of these creation around Irish operators are Gabbard Yelets Let's go right there with store the quantum field Quebec part simple quantum field only discussed already and may come to the description of quantum atrocities scattering across the seas creation of process he's annihilation processes and so forth how the described by quantum field How the described in terms of mathematical expressions of border quantum field while interested in process is like collisions in creation and annihilation processes particles figures in the microscopic world that's about all we could do if we want to do experiments the only real handles we have experiments is colliding particles together and see you see what comes out and describing those process sees how the initial state of a particle of 2 particles for example morphs into some of the state involving 5 7 9 particles for particles it is love tools that is quantum field theory and was setting up some simple examples but so let's go back to the definition of 1 of the simplest quantum field the simplest quantum field we took it To be a function of only 1 coordinate namely X is no reason why we can't think of X Y and Z year then make position into a threedimensional then if we do so momentum also has to be 0 incidentally will work units in which each bar is equal to 1 hour victim right the speed of light will comment anything but what's the connection between momentum and Katie page farmers equaled 1 that sank
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normally you have an age far over here it Katie but a virus said equal the 1 Katie and momentum of the same thing right OK so we concocted thing which we call the quantum field associated with the point of space X said X could be a threedimensional point of space there is then Casey has to be threedimensional if we're living in 3 dimensions and the momentum was threedimensional meaning say has 3 components and then cable also have 3 components but owner and they have the formulas are going to write down are pretty much the same in 3 dimensions where a number of the doing sigh of X was a somewhat all all the allowed values of momentum they we're talking about the universe Thomas circle periodic universe than these are the cases which are to pile L times an integer we're talking about infinitely big universe then convene in East anything near a number some OK summation the Katie farm the creation operator from particle of momentum k times each of them minors I K X that now has become a quantum field there's an a conjugate quantum field and now we pretty much of wherever quantum mechanics people Cork about talk about that permission conjugate if were classically oriented people we simply talk about complex conjugate call sigh Star side dagger of Rex complex conjugate which is a similar things involving annihilation operators can veto plus side his that is definition Rice's definition but do not definitions the question it's always a pub when somebody on our gives you a definition say Why is that the definition and usually answers Wait wait till you see how we use it and you'll see that it's a useful definition sold aren't afraid that's a situation here why is this the definition because this is a useful definition I could put something else here and it would be useless definition solos it's it's premature to ask why this is the definition but this is a nice simple expression of very complicated some although not allowable values of momentum creation operated antique might decide IKX or the complex conjugate which has annihilation operated times Ito the plus side KX not going to do something even a little bit fancier I'm going to give you sigh of axes some time independence so far they don't depend on time the definition of those involved was not involved but I'm going to introduce some time in the game and I introduced time Whitling introduced time but a number that a connection between the momentum of the that Casey and the frequency from oscillation these are Fields Beaver fields fields are story as written doesn't ocellated doesn't have proindependence if this has anything to do with a real field waves and so forth with that introduce some kind into it well it's not so hard because we remember that each value of Katie there is a frequency frequency tells us how things change with time surpassing may have an object a mathematical object which has of certain frequency omega there are like function that goes that isolates with frequency omega Peter the Ohio maker eagerly I Omega which also happens of course to be sign on be achieved plus I signed maybe this is a function with a definite frequency the frequency is all major I now past lectures I point that out to you that various kinds away waves each Katie Katie is a wave vector its inversely related to the Waverly 3 HK there is a frequency so we can say that there is a each case there is no maker of K she and each oscillations each time we see in each of the miners I KNX the we can't put in front of it and oscillation from behind it and e I home again Kanye times now this thing has not only space dependents but has kind the Pentagon's and Moeller it's kind would do the same thing here of course the miners Omega FKE T let's sir let's erased sigh beggar for moment will come back to win a more room on the blackboard an object which has both space dependents and kind the time dependence has been arranged on such a waiver each the value of the wavelength of Katie it oscillate with a timedependent which is just the right time dependent for that wavelengths now functional spaces and climb it truly is a quantum field now varies in space and focus I wanna take an example show you that any example is an equation of Bowie equation for sigh of X and Celia figure out what the wave equation it ceiling and find the wave equation for sigh of ex ante knowing the connection between Omega and cake I haven't told you what a connection between Omega K. years but let's suppose that we're talking about a nonrelativistic particle other words I'm not talking about a photon I'm talking about a species of particles which moves With much less a speed light that uh who were working in the approximation in which everything is non relativistic OK when a spot equal to 1 of our year age far equals 1 Omega has other name whatsoever name from a guy energy remember energy of a photon energy of a quantum of a single quantum widow a single quantum now the energies each borrower maker they hours acquittal 1 Omega and energy of the same momentum physical page Barre Times k ee so when I'm when farcical the 1 energy is frequencies and momentum as cares wave number what's the connection for a slowly moving particles slowly means only slowly moving compared to the speed of light what's the connection between energy and momentum right energy of a slowly moving particle is the square of the momentum provided by twice the mail yes it Of course the same thing as writing energy physical toll onehalf and squared and is is equal pay and a fouryear if we solve and plug abandoned here we get this relationship here OK that also tells us now the relation between Omega and Kate K square Over 2 but it so well we really do
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not know the precise form of sigh of ex ante some over the allowed values of momentum and Omega of Katie is not just some arbitrary Omega of but it is case Cuero over to where With that provides old of of ex ante solves satisfies the differential equations a wave equation but savings you with a wave equation is 1st of all In wave equations derivatives of space and time I enter and their equal to each other in some form a wave equation is equations warehouse space variation is related or have time variations related space variation so let's consider 1st be time derivative of size what the time derivative of Siam cause I thought Standard Notation prime derivative side that we get we obtained by just differentiating the time dependence with respect at equal some OK a plus a K deeper minus side K X times what kind I Omega inside the summation I Omega of K E we Idaho made of chaotic I say every time a storm of summation here every time I differentiate with respect to that polls down a factor of wine maker that's I got now let's look at based derivatives what is this space return decide IDX while we do exactly the same thing every time we differentiate with respect to exit polls there on a factor of minus are you OK so this is Eagle cursing kind of thing summation of minus Katie a positive Katie times he drew my side X are all maker of K T differentiating with respect to x always just pulls factor of IKB differentiating with respect to par imposed by a factor of our maker now there's no simple relationship between this and that's why not because all media is related case square if was simply related to AKD for example a maker I had to be equal to carry them just say so I got physical IDX resemble imagine a world are kind of particle with a frequency is in fact equal OK as a simple situation in that situation I Omega K is the same as mine years apart from a minus sign is the same as I K M to adjust say that such a field satisfies equation excite dot is equal I think the minus decide by DX is clear but that's not the case here we have have all made equals case squaredtoe squared take another derivative each time we differentiated brings them back there Of allied K solids different you again that brings us minors I case square what's mines IKB squared forgets miners case square right minus a squared my years where the miners but now case squares Mr. end times Omega K squared is to end Funds Omega so won't let us love right back to and from Jamaica and let's divides the lefthand side by 2 where 2 and is just a number of issues be thought let's see let's check when I differentiator respect time it given major so that's all maker known a differentiator with to Katie gave him from writing this I Katie and I did it twice selectors minors I shaky square was minus Chase my minus case square mightiest K square so yes as I can tell a minus sign there maybe it was was on my disk a squared and then I substituted for mine this case where money is to more maker and that's what this came from OK I'm more familiar with this 400 which we divide both sides by to win this doesn't matter of course but divided by 2 and we get 1 or 2 and times a 2nd over those sigh with respect the space square is all media terms all the stuff but that's clearly proportional the site not what's the what's the relationship here we have yet we better let's let's yacht is good in formula we can divide by all I which is as multiplied by my side OK to make a long story short the right equation should be minors I saw I got that kind derivative of society of violated is equal to 1 over To win the current the 2nd derivative of siding with respect to x squared now have a sign right or not but of course the the right that was a writer the by nostril the well to and not 1 of which 2 and 1 or 2 are OK are let's say is a minus sign not minus year and members of plus here so it looks like is that looks like it's plus size 4 Lecont the plus side now now well of course I will do that will do the trick updates was as anybody know the name of head equation as a Schroedinger equation but it's not the Schroedinger equation for for the same thing as an elementary quantum mechanics and elementary quantum mechanics society is just a function of position that's not an operator it doesn't that do things which just a thing whose Square is a probability here it is a operators when it acts on states it creates particles annihilate particles this is an instance of the relationship between particles In quantum field quantum fields are operators they happen they have the same equations as these Schroedinger equation of elementary quantum mechanics are very closely related but their quantum mechanical operators observables you can observe them under what circumstances they behave like classical fields are you can measure them and the same way you measure the electromagnetic field is so it is that they behave like classical fields when the number of quanta it is very large number of Qantas a large then the magnitude of the feel of a large and the quantum fluctuations of small quantum uncertainties small exactly like a harmonic oscillator a harmonic oscillator efforts got a big motion behaves classically there
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is only got 1 quantum unit of excitation behaves require can't so this is an example of a set of quantum field and has creation and annihilation operators and it nevertheless it's a thing which satisfies an equation and the equations of the Schroedinger equation our this is obviously a more advanced notion of the Schroedinger wave function they're just saying it's a thing Square is a probability for a given particle it's something are a little bit different OK and as I said it is a quantum field with the simplest version of cornfield How use it Oh that describe processing giving and we really talked about this a little bit of what I didn't talk about which I want come now is energy and momentum conservation and how energy and momentum conservation are codified codified the right word is in the various dependences of society and the way that we describe various passes that 1 I want Reginald describing very very simple as process in which a particle scatters off target with target in this it is anything which is so heavy that doesn't require pedicle comes in here reflects offered scatters offered in 3 dimensions it could have its trajectory changed nor coming in from the left and go off straight ahead but what can you say about particle scattering often ordinary target target to stop it doesn't move stays there forever ever which of the conservation laws would you expect to be true for that article about momentum he more work yes there has been a very attractive but last will absorption for the moment the particles comes and goes out so it's not absorb but yes it could be reflected being banged it could be refracted in which case direction emotions changed in any case the momentum of that particle generally it is not the same comedians going out of course secretly what happens is momentum is really concerned but what happens is of course target absorb some of the momentum but let's just fix the target of cemented nailing down the target the target is not part of our mathematical description for the motion of the target is not part of our mathematical description then in the mathematical description of a particle scattering offer target momentum of the particle was not conserved and what about the energy down the energy of the particle is kosher today a tennis ball what was a bad example because a heats up when you throw against the wall but an idealized cans wall will you could ignore heat euphonious wall bounces off what can you say about the way bounces off you can say the momentum is considered the momentum is gone from that direction that direction but the energy is conserved so how could we see can we see our from simple model of the scattering process that momentum is not conserved and energy is conserved using these wave feels so we talked about this last time a little bit of how you could describe AT process of scattering creation and annihilation and so forth using these quantum field so here is is marble here is a spaces horizontal time vertical end the axis over here represents a target fixed in space at nite goes on forever the past it just sits there not particle comes it it hit the target at some point any that bounces off goes forward scatters into some of the direction so with this guy me initial state by saying his want particle With a momentum OK let's call came in for initial initial or incoming and then a particle rose off having scattered off the target and let's use language now it's absorbed by the target and suddenly really emitted by the target think of it instead of just thinking of it as bouncing off the product that's in the back of our minds have a picture in which it is absorbed or and annihilated by the target and instantaneously recreated by the target but possibly with different moment that's call at final momentum final purchase court case by and KTF Michelin 5 and won only interested in or interested In probability amplitude the quantum mechanical probability have plateau but the particle scattered From the moment the case chauffeur formal now 1 will ingredients Gore who assumed that the scattering can happen with equal strength equal at 8 times always at the same place or the place the target at x equals 0 but we're going to assume that a scattering process can occur at any time whether that kind that particle gets there it will scatter or not gather or do whatever but with no special dependence on the tongue in which the process takes place that's gonna be are basic assumption had only describe is vindicate of a particle being absorbed apposition x equals 0 but we describe that by means of a side that there should I think should be started dagger daggers or goal with plus thanks the of mine yeah the big so and then this will be part of our Member bucket we describe it in the following by a man can and bookkeeping it's all bookkeeping but it's useful bookkeeping you absorb particle this is a peace this is a field which creates particles let's write their own we are there conjugates field which annihilate particles complex conjugate or sigh of ex ante which is made up out of annihilation operators need plus IKX of minus on a Omega of K just every place everything is conjugated he minus IKX becomes plus IKX himself for and these harder because of this complex conjugate retail permission conjured but let's imagine 1st of all a absorbing the particle at the origin and how we do that we that operate will we start with the initial state the initial state has no particles with and particularly moment them except 1 particle With momentum KI so if we describe it terms our occupation numbers only the occupation number associated with momentum I would be 0 could also just labor lap by saying is 1 particle with momentum KID and be done with it we don't need to specify all the
1:01:17
particles of orange there but nevertheless it useful to keep in mind that when we specify the initial state was specifying all the occupation numbers of all the various quanta in only 1 of them according to assumption here is president of 1 year Williams a 0 0 label it by saying forms 1 particle the then that particle is either it is annihilated at the origin of its it's at the origin of described body sigh of ex ante but X is equal to 0 0 and T wise x equals 0 I'm assuming that if the process happens it happens at the target and the target use x equals 0 but it can happen at any time so that the process of absorbing the particle at time t that's what this says but they particle is immediately recreated that a White said why doesn't have to beat it doesn't have to be that the marble that I'm making a particle absorbed in immediately readmit that's a mathematical model it's not necessarily a lot nature about any particular kind of particles and back in the real world a fulltime can be absorbed by anatomy and he needed a leader or earlier so we're talking about here is a simple mathematical model in which the process happens at all and an instant Parker absorbed the reunite and related what described by its described by the creation of a particle also at . 0 that's side dagger of 0 and the same kind but what time what time should have put their the same time but would start you will matter but so why 1 0 0 Obama say can happen at any time any time of the particle gets whatever particle gets that save integrators I'm possible talk another words there is a process in which the particle absorbers absorb this time at this time at this time and so forth so that's not prejudice what time it happens but just averaging integrating global possible contact as a mathematical expression which we can work out I will show you as we go along with the implications of this In the goal over time but the basic physics of it is that the process could happen at any I'm no special kind of no particular time singled out a special only give a of the specialness other value of time by just integrating a that is kind of What does is give this he is the final status here's an initial state these are operators which operate on the initial state give some final state how do we calculate the probability that the final state consists of 1 and only 1 particle moving with some of momentum Monday rules of quantum mechanics weak take me in a product of this state With the final state so let's say 1 particle with momentum k final This is a number remember can't affected operator Bravo that give us some kind of number is number the probability for the scattering not quite we have to square we have to take the absolute value of it and square 9 left out 1 important ingredient in this expression with evaluators will go through it however see what it says key because I I'm purposefully um purposefully I would rather be able to explain all of this and complete logical order but given an hour to do all of quantum field theory we have to draw drawn on things that we've done in the past of the past explained that probabilities are squares of amplitudes an amplitude in product between initial state final states so here's the thing we want to calculate but left 1 thing out I've left out the strength of the gatherer the strength of the scattered it is a measure of strength of interaction between the scattered and the and the scatter read Sony give you an example you could have a charged particle at the origin which is scattering awful a photon comes in is absorbed by the scatter and readmitted what is in that case what's corresponds to the strength of the scattered the answers electric charge of the charge the bigger the electric charge more probable that is the Electra the photon will get scattered there is a measure of the strength of the interaction between scattered and scattered particle the strength of the coupling between our which which has to be codified somehow end it is simply but in the medical number of pushy it's called the coupling constant g g for 1 would use for its a measure of the strength of the coupling of the strength of the interaction between scattered and between scatter or the target and the particle itself by your example some examples of the scattering of Amazon from Proton is a strong process leading to say that if the measures on is absorbed and as on arrives at the location of the proton has a high probability of scattering that indicated by a large coupling constant a photon in the vicinity of a charged also could be a proton of photon is also absorbed and readmitted by proton the probability that scattering the probability that the 4 current gives redirected is much smaller then the probability of Amazon and that's indicated by the coupling constant being much smaller for the interaction of charged particle with you think something even smaller yup neutrino interacting with a proton has an even smaller constant for scattering that's what made after the target being an electron beam were right now we just right now we're just making a model in fact electron or something like that electron scattering of target sold the of his being valid for any real think this is a simplified model from number of different situation what about the scattering of gravel conned by AT by nucleus Gravatai Louisiana analog for gravity photons while it can happen but it's an extremely small constant dialog and try to tell you how much income minus some large number by comparison so the strength of the of the interaction the probability that this actually happens the particle redirected his indicated or or described by the coefficient GU which appears at just the number of its cold the coupling constant called 8 coupling constant many coupling constant I so let's see everything calculators China calculate with trying to calculate the probability we're going from Katie initial Tecate finals at your best is there he was maybe a little reason to resume is no
1:10:41
object to it no logic to but and there are situations where it's a function of Omega Our will also use of situations as I said this is a new model of a particular kind of simple scatter there more complicated Kansas scattering a more complicated kind would be in this would really be for the case of photo Adam for example photon comes in Adam gets excited Is and then be excited by amending folk art he is a little gap time gap between the time the photons absorbed pundits admitted this happens to be equivalent to saying the coupling constant is only dependent OK so as I said we're talking about a simple model and just trying to find the consequences at prayer with consequences are was the only 1 of Powhatan consequence and is very general the rest is not so general the 1 important consequence Over Warner get too so let's just won't go away a just coast straight ahead being in society its value written in terms of creation and annihilation operators and they use those creation annihilation operators to annihilate the initial particle and recreate the final particle and calculated be expression that's up there the square of it will be really probabilities do it OK so young what is that the juice quarter Warner young are that's a good question were you could as well have residues much bigger than 1 doesn't mean the probability is much bigger than 1 not it just means Red you have to go ahead in tow more complicated calculations this calculation I'm doing his only correct for very small G correct errors warned you what you have to do and is lodger is extremely interesting and will be very important to us and followers the answer is you have to do higherorder perturbation theory OK but we're not doing that now where we just doing the simplest thing with the simplest thing and just plebeian society its value signed the 1st sign up aside dagger but the precise it is a summation although K not Katie initial purchase case summation index a minus to absorb the initial particle of Kerry eagerly IKX each of our RIA all makers of K the facts Ali initial state which happens to have 1 particle of Mormon job and Michael thanks a 0 actor scattered soullessly bailout heated 0 is just 1 which turns in this summer will contribute only when Katie is equal to will again and right white because the annihilation operators gives you roll when they act on states where there are no particles known particle around the momentum until only when Katie is equal to we get anything but nevertheless leave it in this form just for moment now what about the other side that that's also a sum over momentum and that is different index because I don't want to use the same summation index twice a bit confused that scored Eyal and now we put a class of that create particle momentum at all times each could be well Exs still equal to 0 Exs to equal to 0 so we don't do anything from here but then we have need to be Hari Omega sub L times what about the final state of final state his state where particle momentum k final OK which can at 1st the term in the sum of a K which contributes is only link a fullscale direct what about the sum was l have equal a final remember that
1:16:05
when a 8 plants acts of the left it and 908 has defined the particle torn irate paper only gives you something when l is equal to cave final so really both of these sums collapse only K equals Katie initial and L. Eagle's K final I about JD but have also left that something else where it In the Robey take a in there will be which is the thing that I'm really interested in life a selloff the only contributor here is Kate was KID the hot but what does a minus of KI daughter acts are stay with momentum KRU it creates a stay with no particles right it violates the particle momentum chaos what about this summer this summer is only 9 0 when a L is equal to K. final so we complicate final here and when this on a state with a final sorry yacht OK right OK the skirt what happens when a fax on stay with Katie initial it just gives us they would know particles right such as not particles what happens when this act on state with particle momentum k final but has given no particles again what is this number it warned that much is 1 all this operated of creating annihilating particles logos away and all we have is a number GE that's the probability for the scattering of St amplitude for the scattering Jean appeals except that we have this integral 0 9 this expression I think Durell thinking about a relativistic problem so no no no there was only 1 particle here because it was only 1 particle Yao who would only 1 particles that OK but did plight in fact real Ali thing that really and illustrating here you see it will over time and what is it in the Gobi bt each I Omega let's just call a final minus all made their initial case initial times that's the whole upshot the kind integration gives us an interval of it eatery I will make a final minus Omega initial although they operate and nonsense just all 1 nothing very interesting the coupling constant gives us a G and now we have to evaluate this integral was assigned the role Delta function this is an example of the delta function in a dual DT of eatery on something kind but see was it to Pike and built the function of Fernando and here was to Part GE crimes Delta of all made final minus Omega initial is a string about Delta will make a final minus Omega initial it's only none 0 0 when Omega final is equal to Wamego initial on May the final is the final energy Omega initial is the the initial energy so somehow magically notice that if we had not integrated over time we would not have got in this dealt the function of energy so somehow there's a connection between the fact that is a conservation of energy and the factor does it is not preference for any specified time we could have made a model it which the scattering only happens if the kind within some boundaries then this interval would've only gone over some limited amount of time it would not have made delta function so we ingredient here which is closely connected with with energy conservation is kind translations every time was like every other all of the square of the delta function the idea of the probability of the probability we have to square this is probability and the probability itself will have the a square of this but who cares about the the square of delta function and other square the functions also 0 if Omega finals radicals with the worry about squares of delta functions but in any case this will be 0 unless the initial energy is the same as the final energy with only initial energy is the same as the final energy then this is the operative important part of the scattering amplitude caused by its name that's a scattering amplitude and the probability of the scattering is proportional for op I squared G squared the Strait of the scattered or the probability fritters gather contains G square they always want the containing parties but the important thing is G square in which which which final moment that can happen to well I specify anything special about the final momentum here is final momentum could be anything as long as the energy was saying so this gather has a property that takes a particle nor the probability proportional to G. squared change momentum with equal probability any momentum with the same energy services a scatter which could scattering indeed directions with equal probability any direction with equal probability that of course is not sure the wall scatters is very simple model which gathers revolt with equal probability in all directions and coefficient G squared for pi Square D squared in this case is the probability so this is a good show that many things so far the nite in particular the definition of a coupling constant the fact that the integration over timeline is think which insurers energy conservation which is just another way of saying the problem has time translations symmetry was nothing specify special picked out about 1 time or another time and a and Everest raided the idea of a scattering amplitude or the amplitude the thing which becomes square in order to calculate a probability this column as of February generic set of ideas purpose but the details of all where they have always been requesters body yes yes this particular case yes the question of what would happen had you not put x equals 0 here right that's car requested OK it what you would have found is aware that another
1:25:07
factor the other factors in here would have been the airport tour Ito the RIA initial might escape a final times the position of the scatter called position of discovery Echter at just busses Claude X Our the scattered X of the target excellent target number exit target is just the position of the target when the position of the target was EUR missed in the car if the target was moved all over the position next time you get an extra factor but what happened to the probability no change because when you score where 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 and multiplying by its own complex conjugate the scars away service does not appear in the probability that the same probability no matter where the targeted is but in intermediate stage of calculation calculating the young amplitude the position of the target area also came 2 years the man let us see what they we said is were can have that did not get used them because of where then is worth more well of course and In the real world of setting for example the interaction of 2 particles we take everything into account both particles and so forth the scattering can occur at any time at any place that ah but you could and Nagin you could certainly imagined and approximate situation where you my OK if you're asking me is a ever a situation where I hear the approximation was we just ignored the recoil of the target therefore the momentum was not concerned is a ever situation where energy is not conserved and that so the situation is what happens when you have kind depending on the coupling car of time dependent on the strength of interaction now yes we are situations where that can be a good approximation to say something something makes a sudden change in the system from the outside 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 could do we could try cut cocked Warner can track down before us but yes settler the concoct a situation where some sudden happens in such a way that makes the scattering possible at the time and energy can it here said what I newly Illini when this time timedependent the coupling comes that usually means something that you've ignored in the system which is movie nor timedependent something always dynamics and degrees of freedom you're sort of ignoring in the same way that you ignore the possible recoil repositioning of the target he sold all our think of a situation where there is an interesting go violation of energy conservation all where bookkeeping was such but you through some of the energy away but the looked tour they are getting a moving target heading a moving target right thing moving target there was an example where the target where there were again the motion of the target is thought completely heads and I am I say they worry about the size type person we sadly yourself that favors here in this model time target was 0 yeah in this right you're absolutely right something interesting happens target has a certain finite size is weak and we can discuss but done it is combative the question of removing the moving target if you think about 4 minute of course if you have a moving target energy is not conservatories sample supposing that wall was moving toward me and I had a tennis ball is the wall hits the ball on the ball moves off the energy of the Of the tennis ball where we're not accounting for the energy of the moon when I tried to worry about that so the what moving that corresponds to what kind dependents in this case a timedependent of the coupling constant time dependence of the location of the scatter that would be enough to make energy not serve no we could see it in EU in the mathematical formalism OK reiterate these creation and annihilation operators are the tools which allow you to discuss transitions particles from 1 state to another state From we have not talked about but who we could talk about creation and annihilation armed now let's a let's discuss another situation always supposing society was the field operator from electron each kind of particle as only separate field so electrons have a different feel the photons when you put aside here he better specify what particle you talking about armed let's think of the electron side describes electrons here was a process in which an electron came in electron came out what happened to the total charge the the total charge change not 1 particle came in a part of that now let's imagine different situation let's imagine a situation where 1 electron comes in 2 electrons go out a crazy it can happen but works from moment nevertheless trial imagine how might we described by the same kind of mathematics 1 electron comes in 2 electrons go out however modified then you my simple model is electrons go out from exactly the same point yet we might put another Mike square there's another words sigh of 0 TCI that 0 please side there's a 0 0 3 5 0 T. props distributed Europe but what about now this is this of course can't happen nature because electric charges considers this corresponds tho the annihilation of 1 electron the creation of 2 bad idea but nevertheless Roach what about electrons in electrons out is OK but we doesn't violate electric charge conservation how we describe that another side so I signed beggars better about 2 electrons in free electrons out sigh society sigh dagger aside daggers dagger OK which warns of these are allowed a what's the rule not the right now is is still the same number of Asia's today the same number
1:34:12
of size as side daggers is not allowed Rice's allowed us not allowed is same number of size site that Carlos man with it is not a question of the Order of the operators it's the number of size vs. No. side occurs when the number of times when the number of size is different than the number of size daggers versus when the number of size is the same as the number of such occurs How could you diagnose these combinations because it's it's really easy just to count the number of size inside that of course but mainly due on mathematical diagnostic God there were really does have a deeper meaning let imagined a transformation now dagger does mean the complex conjugate of upside it is permission conjugate Navy language Yukon quantum mechanics but the Commission conjugate easy analog require more complex conjugate than we imagined that I take an expression of just a mathematical expression like theirs and ii trans forms society by multiplying it by a phase phase means and Ito the RIA time some number of scored Alpha time I just unjust doing now blindly playing a game where I see sigh I multiplied by the hour by hour 4 times what happens so dagger From multiply 8 complex can't quantity but Peter the eye out for what happens with complex conjugate you like his eye out for side that what happens objects that have an equal number of inside there the they say the same weapon objects which don't have an equal number of science and they change so it could characterize the allow would process seized by the ones which is described by operators which are in unchanged by the operation are changing the face of the opera now that he indeed but there for the moment let's give a simpler name and variants under changing under redefining the field so that you change its phase almost all everybody change by constant phase factor yes we description if the interaction expression here is In invariant been charges conserve if it is not invariant charges not why happen disposing had sigh plus I dagger what kind of things that correspond to Our leave that you think about change I think this is a real part of flight and if you multiply by a phase you change the real part of so that's not a quantity which is very good that led to the military you that many the early but if you have worked target target of electrons well sure you could have guessed yes yes the bright by you see what happens if Adam and you hit the atom electron earned 75 electrons goal by judges not not violative of course what happened the has changed my life this is situation end which you can get away with ignoring the dynamics of the target in the particular important dynamics of the target is the change of charge of the target you can't in this case treated target is was a completely passive thing you've got to remember and I love but it whatsoever think yes that means the target itself has to be described by a quantum field we we can discuss that I think Oh we've gone far enough for tonight so if we found that that time translation invariant corresponds to energy conservation these invariants weeding phase invariants corresponds to charge conservation and we have worked it out yet we will work out next time that spatial translation variants corresponds the momentum conservation I'll show you how that works next time but making always slowly In how particle physics process use are described by field that's our goal for the the next 1 more lecture maybe half a lecture and they will move into some of the symmetries some of the more interesting narrow loss it is our society yes yes but it's not hard to really are OK right that I've I've chosen always daggers left and the size of the right reminding next I'm not sure you have a rearrange other expressions of become like this I know what you're asking you're asking what would happen if instead of this we put aside side that terrorists that and I will tell you next time but assume that this is allowed not quite quite quite not quite remind next time to pirate now historic areas going into it but they remind me absolutely next time What is the difference between this and that's show you next time I I love the exit the Contras on questions of your questions are just Ruano steam what well it's it's our
1:41:23
trip if you have any size aside daggers means direction each the same number of electrons is spits out Mike each annihilate some electron each side dagger creates a lot of all I did was point guard that all of the objects with equal numbers of size inside Eckhard will be invariant under phrase operation here if they have different numbers of size inside daggers let's take a case sigh side there's a dagger yet right much the question is if they equal numbers of size inside daggers phase invariant any consider Astra has different numbers of size inside daggers last season variant and it doesn't consider charge so we just observer that haunts phase invariant conserves George Wendt aboutface fly the the charge for a moment we haven't we haven't gone through momentum conservation although was mentioned by somebody asked me if the process can happen at any place instead of any time yes that it considers momentum comparable do that case yacht in classical mechanics North theorem In quantum mechanics it's even simpler but a but here answers shortcircuiting all of that discussion but is showing how works mechanically Lindner in terms of squads here he made a pass it a was less than 1 0 with something for you have for your model is very incomplete that if U.S. Liam awaits incomplete I will tell you go well it's incomplete real scattering process is not described by just a particle coming in getting scattered out as described by something more complicated which is a sort of problem of some of the amplitudes in which the particle comes in scatters off the target comes Ingalls backed out come back instead of the target twice comes back and status of the target 3 times a whole infinite series each 1 each tournament has a factor of Jesus G. from a bit odds only when is small but only the 1st terms import but will come to that this is a basic theme that where the coupling constant small you can get away with a simplest minimal processors on the coupling constant gets a large 1 has well it's just call 1 its last summer Peninsula number of finding graphs basically these pictures drawn or find and the vertices of the findings graphs are described by these operators the vertices of fine diagrams tell you the basic elements that can happen particle committing bounces back out of sight and dagger Topol articles comedian and bounced back out its side that Agace Isiah whatever the will roles give things their proper name next time I'm doing when got going a mail delivery rose ahead of what an arm you want your here on the existence of a beta prop parameters QCD the existence of a feeder parameters QCD I'm sure that means nothing to you it does happen there are complex values of the coupling constants it usually means the kind reversal invariants been broken complex coupling constants usually indication of a violation of time reversal in there and that's really what it comes down to the you can see that from what I've told you October however stiff you can 1 and I could see it but only because I know how it's not obvious but that was the case complex coupling constants mean that time reversal invariance news is broken and other words things don't look good if you run the process backward as a movie going backwards about possible is allowed to sell off all these gatherings of more than 100 number I did say that the rise of regret the but we will None it's a fair question but it's not the time for it now there last they left her brother at
1:47:50
that time I begin to jumping way ahead and trying to go slowly Everybody is personally and OK you really wanna know your Edison this impose Square a Medicaid in the best I could but who we're dubbing it about a year after being cut because they were the U.S. cast him as a knot nonrelativistic process from a relativistic process can end of his nothing in basic quantum mechanics which says the rest change such a combination quantum mechanics and quantum mechanics in classical mechanics together with an invariant principle and the invariance principle is Galilaean invariants which which is an analog the nonrelativistic analog of Lorentz invariant on but the I destroy nonrelativistic go situation total mess can change but it can relativistic scattering so war were arrested I played him as a major of Carmel after all 1 is absolutely true nature the other was only true for very very slow velocities is nothing sacred in physics about the concentration of more natural on electron positron annihilated 2 photons is the homework conservation of mass there is a concentration of electric charge 0 1 of them is a sort of accidental nonrelativistic a fact the others deep underlying symmetry of nature which happens in simple case just corresponds to this season next I will talk Milan's will talk about what happens if you have more than 1 species of particles supposing you have electrons and you want quarks and so forth at describe all of this what's allowed what's not all our kind of prophecies and all the languages you get his arm reporting good point a good point good point yes yes yes yes good point yes this so we have Armed with talking about Hey bows on version of the electron a solid at completely missed that you're right right but there are particles in nature which carry electric charge is a charged pioneers on a slowly moving charged pion would be an example of a bows on which you were described in this way so that that absolutely no rush for more
1:51:38
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1:47:42
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Titel  New Revolutions in Particle Physics: Basic Concepts  Lecture 4 
Serientitel  Lecture Collection  Particle Physics: Basic Concepts 
Teil  4 
Anzahl der Teile  10 
Autor 
Susskind, Leonard

Lizenz 
CCNamensnennung 3.0 Deutschland: Sie dürfen das Werk bzw. den Inhalt zu jedem legalen Zweck nutzen, verändern und in unveränderter oder veränderter Form vervielfältigen, verbreiten und öffentlich zugänglich machen, sofern Sie den Namen des Autors/Rechteinhabers in der von ihm festgelegten Weise nennen. 
DOI  10.5446/15071 
Herausgeber  Stanford University 
Erscheinungsjahr  2010 
Sprache  Englisch 
Produktionsjahr  2009 
Inhaltliche Metadaten
Fachgebiet  Physik 
Abstract  (October 26, 2009) Leonard Susskind gives the fourth lecture of a threequarter sequence of courses that will explore the new revolutions in particle physics. In this lecture he continues on the subject of quantum field theory. 