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New Revolutions in Particle Physics: Basic Concepts  Lecture 10
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time I started tell you something about how quantum field theory arms gives rise the theory of a particle interactions rule and object called the label on June I think that was not terribly clears I go back to it a little bit before starting to discuss the details of details of particle physics was still thinking about quantum field theory basic technique that I was a Jew that I was alluding to last time is called the path integral method of quantum mechanics arms and it is these most direct route defined diagrams of the theory of a particle interactions that are based on finding diagrams so we just a trial briefly go through the basic ideas which we started to do less time but I don't think there was sufficiently clear anyway the path integral formalism or method climate tremors stood defined it is a generalization as the quantum mechanical version of the principle of least action so I just wanna remind you with the principle of least action is 5 were talking about a particle now I don't need a particle from particle physics quantum field theory procedures to classical Newtonian particle the motion of the classical the particle From 1 point the spacetime until another just very quickly is determined by a Lagrangian around and as a function of the recorded that so the particles time derivatives the particle hand from the Lagrangian where Iran's when constructs action action as the girl along the trajectory of a particle of on June 4 every trajectory whether it's the true trajectory or not the true trajectory from year year attitude trajectory I mean the solution of a lot of Newton's equations from 1 spacetime point to another spacetime point whether or not the trajectories solution it has an action integral of grungy and along we all the classical principle of least action is the trajectory followed by the particles through space time minimizes the action From there you could derive differential equations of differential equations are called Newton's equations and I will assume that you know a little bit about this idea of action those same idea applies to classical field theory and the way the idea works is for a particle from a particle motion the Lagrangian is a function along audit function of an orbit for a field theory and of course it's an orbit which connects some initial configuration of final configuration for a field theory the equations of motion of a field theory classical equations of motion are also determined from olive branch in the Lagrangian but as a function of wherever stop colon generically just find aimed not just that kind derivatives of the field but space derivatives and the time derivatives derivative of flying with respect to x meal at justice stadium shorthand way of saying that tho a grungy depends on fire and derivatives of 1 thing which is required by the theory of relativity is the grungy and B a scalar transform under Lorentz transformations as but that being said Villa grungy committee anything to build better far use derivatives of fraud was the action imagine we have a region of space time time as usual flows upward spaces horizontal now we have a region of space time between an initial kind and a final action in this region the action the action in this region just as the action for particle trajectories the action from 1 end point of the trajectory of another here is the action an original space between the initial configuration final configuration the action is equal to the integral all that region tragedy for X which means he time and the X and Y of this Lagrangian so it's a totally integrated value of the grungy and all the space and time words for the particle Justin integral over time now or do watch the analog of a starting point for the particle of value will of of the field Monday initial surface here give not just 1 field goal of fields and the problem prescribing the values of the field on initial surface here and prescribing the value of the field on the final surfers you could ask is there is a solution of the equations of all will be go back step up yet Is there a solution of whatever the equations of the theory on which starts with a given the initial value of the
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fields at time t equals 0 let's say and ends of the field being something different at some later the answer is the principle of least action the be equations of the theory of formulated by saying yes there always doesn't exist or something in between which is the correct solution of the theory which hasn't given initial value the fields here and a given the value of the field to the end point and be correct solution of the fields in here is the 1 which minimizes this action when which minimizes the action of principle of least action gets extended to based kind principally stature with the degrees of freedom in here are fields OK now very quickly remind you what the quantum mechanical use of action Howell exactly the right of the classical principle of least action from the quantum mechanical version of vitamins quantum mechanical version we won't we don't really need to get into here you could go look up and need anything about path integrals but for a particle let's go back to the particle motion for a moment according defined for according to quantum mechanics of thing that you might want to calculate it is In the board through his ears anything on the value he turned itself not now no more than the possibility of the position of the parties had to go from 1 faced by others faced with what was based on the values of fueled fears boat get rid the trajectory is replaced by a history a history of the field from the beginning to the end a history means the values of the fields all along here everywhere India and the idea of the trajectory becomes trajectory field space which means we have specified actually eased the 4th quarter as well as the field to get it initial here lower here you specifying the initial boundary condition as feel field values all long from initial surface to pick beautiful surface but Georgia picked kind as that's a whole field along some cross sectional got right which is another way of saying the end of the whole field at a given instant of time and the final configuration the whole field at some later are time question is there a solution of the theory whatever the areas which interpolate in between knees in the same way that a trajectory interpolate community the role this minimizes the action that in that if carried out what Toledo we did a number of classes ago but if that's carried out would leave requirement of minimizing the action leads the partial differential equations for the field of partial differential equations on things like Maxwell's equations things like Einstein's equations and so for so let's come back to the quantum mechanical idea the caller mechanical idea has to do with amplitudes amplitude Our things out of which you construct probabilities so for example you could ask the question What is the amplitude Jennifer particle is injected until the world however its objective has faced time point and CD what is amplitude they look through later current X and try and see what's the amplitude that if I look at a later time Woods quality Prime that I will find it at position ex prime prime so see amplitude that I start a particle at this point close my eyes for a while and then the turn on the detector which looks for the park or a certain place the amplitude for finding that amplitude is a complex number that depends on the initial point on the final point it's a complex number always magnitude squared is the probability Russia square the amplitude multiplied by its complex conjugate and that is the that is the probability for the particle to go from here to here fighters rule is a foreign visitors take the action for any trajectory thank arbitrary trajectory and write down the expression he each said body minus ART the action of the trajectory we could just write about Inc the girl now we're talking about particles now interval of the Lagrangian Vt From 1 point to or another pouring it's a function of what functional Of the trajectory and now a true factor of age foreign here
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factory each foreign he with quantum mechanics comes through it solid or integrated although all possible trajectory well In nonrelativistic quantum mechanics you don't let them go backward in time but all possible trajectories which never looked back in there with themselves time some that although what all the possible trajectories beach trajectory it's a complex number they had them all out there how you end up such a thing as some complicated kind of integral but we we needn't worry too much about the details How you actually calculated that accorded the defined the amplitude for going from year to year is sum of all possible classical roots classical wrote does not mean a solution of the equations adjustment of possible route some or all possible routes whether or not the solutions of the equations Newton's equations lover of each of them minors I can't see action measured in Clark's constant units incidentally points constant has units of action so this is simply the action in units of the plants can't be right they yes right exactly like that that's exactly right and the other thing here is the Arnie the idea used part of quantum mechanics with now that but some is the thing squares of probability by Square I mean times complex this sum could hardly be a probability because in general it's complex you multiplied by complex conduit that is the probability This is the probability amplitude through the complex thing so this is finds formulation of quantum mechanics and now I can be extended the quantum field theory in quantum field theory the corresponding question now and that actually formulated persuade because so that this is a correctly perform them with fudge over bit and make our way through it without without any return but the same idea you start with arbitrary configuration of the field so that's some initial 5 of X X now being just X not initial configuration of the field and a look at some final configuration of the field which qualified prime of X and visit to a different functions and you ask what is the probability amplitude that start the feel that some value and let the system go and indeed half the field at a later time imagine you had some weighty of starting an initial condition with the field being prescribed at every point of space of course it's an idealization you don't have a chance in the world doing really but you have some sort of apparatus which allow argued 2 0 start the field in an arbitrary configuration let it go and then you had another apparatus which detects the values of the field of every point then you could ask what is the probability that you start the feel a given away that earlier time the fuel have some value again that probability is determined in terms of probability amplitude and the probability amplitude is again this summer over all possible ways of interpolating the field between the initial and final all possible wait exactly where it's all possible ways above filling in between the initial state the final state with a field value every point of each other miners I kind of feel that action at stake in the Gobi hectic XPY Desi before acts of the Lagrangian which is a function of fly and the derivative of 2 evaluate the Lagrangian every point integrated up that gives you the action for a particular let's call it a trajectory it's not really a trajectory in ordinary cents but the history this caught history video for each history each possible history that you could imagine it doesn't have to be a true history of real history for every possible history you there is an action that actually suffered in the world but then you could see some this of although possible histories and that is the input to alluded to start with a field value find an end with the field value 5 times between these 2 trajectory that's the notion of a path integral if they have a a a a a a a final a a house by a home no Mini Mini integrals well OK you're asking them and let me rephrase the gloomy phrases more strongly ending up an incident number of numbers pipes that's your special I think what pointing out that each 1 of these numbers is not only finite but has amended before someone suffer a truth it has amended to equal the 1 it said expert enchilada white time something right so any number like this is a number which complex plane not minimizing you're summer normal Davis is a faulty gave a few these things do could do converge stayed dumped them they converge because because
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of the oscillate all I Indians Monday unit circle around circle on the unit circle and yet the intervals do converge our for example did a that's 1 example but there are many examples where intervals converge on the current thing they can converge well if you have an infinite number of numbers all of which are positive and you let up positive and bounded away from 0 and the number of numbers all of them positive and bounded away from 0 of course that's going to beverage 5 in this case you have numbers which canceled all here as 1 trajectory of a here's another trajectory over here is another trajectory amplitude different trajectories and this 1 cancels this 1 so 0 it's not hard for an integral Z studio around in a circle as Iraq a lot still that's right they tend to cancel a lot the only trajectories which tend not to cancel the ones near the classical trajectories do a donor to discuss that Abbott that is awaiting you go from quantum mechanics classical theory at least in this a formulation if you look for the particular trajectories where you have the least cancelation and those are the trajectories of stationary action Mr. the trajectories where the Action News minimum but that the that's that's a restored This is the quantum mechanical path integral formulation that much of modern field theory basically all of modern field theory quantum field theories based on I know I'm not going to to ride the next day and going to state the next step but I wanted to at least explained to you what this quantity years before I show you how in practice it's you know what I say in practice this it a bit an oversimplification but not too bad the last time we talked a little bit about how the Lagrangian is used they calculate process particles moving from 1 place to another particles interacting when I say calculate process you what that really means I really meant calculate the the probability our and initial state Goto final state but there are tool distinct ways the think about relative as well above fuel cornfield 1 is in terms fields and the other is in terms of particles we know that there is this duality between particles and fields we could ask a totally different apparently totally different questions instead of asking suppose you had a given field value here and you want to know what the input to issue a final field value because they ask supposing you'll have some incoming particles quanta of the field you express things in terms of field quanta rather than in terms of classical field configurations supposing I told you the field initially was In a State which was described by a particular collection of incoming particles incidentally when I say the field I mean art collection of field I tell you not the value of the field along here but rather the particle content coming here and I tell you what the particle content is going out and I ask you what's the probability that you win From 1 particle content to another part it's a question of a similar kind of question but expressed in terms of the particle representation of quantum field theory rather than the classical field configuration entered the interior not surprisingly involves exactly the same object we talked about a little bit I think what I told you last time not too correct but was small piece of it said would you do if you took 1 hour does remind you of something I said and will say it again but more correct for you take 1 plus the Lagrangian remember what we did we said Let's divide up in space time in lots of ourselves it's hot to give me the objects like this like this have to grow our directly the waiter give meaning to it is to divide up space and lots of little sales instead of thinking about continuous functions are even this continuous functions think about value of the field in each 1 of these cells that makes it more concrete and then need and you divide you allow you let the size of the sales goes 0 so let's not put the size of the cells go to 0 but are keep them finite like this what I told you is that you take the quantity 1 plus the Lagrange in each cell Shiite itself but also by the value of the Lagrange in each set out a new multiplier all ourselves remember I said that if you don't doesn't matter because we in the right way this time by and they need be we did something with this to try the calculate on I try to share their pieces in here which described the propagation of particles of collisions of particles will back over again really a central bite a 1st of all I missed the entire Adair was not terribly year career it's really 1 mind is 5 times the Lagrangian no 1 on 1 1 plus a small this is a moment of magic This is a small quantity I should actually it's not 1 plus the grungy and
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it's 1 plus action and each sales no reaction each cell it is 0 grungy and Karlheinz spaced at times arm times the space time volume in each cell based on volume was dealt text of the wife of disease times don't the thief rich so call at a small number but score space time volume of course off way is a small number retool sellers small so the action in each cell is itself small because the sewers more now 1 minus 0 1 plus a small quantity of score 1 plus abso aren't is an approximation who eatery Epsilon eater the epsilon is approximately equal for small epsilon but the exact formula is a power series of epsilon it's 1 plus epsilon plus Absalom squared or 2 plus epsilon over what comes next 3 factorial 2 6 we kind 1 and so forth and so on and so have 1 plus epsilon is approximately 4 small things Eagle Beta Epsilon benefactors more efficient and correct the really write something different than what I rode over here name wait this sector but let's think about workers there there's if a if that would make a a if I'd probably windup sipping stage for equal toward his usual broke right OK was the subject line arms but this exponential here what's in here we're or matters in replacing not by interval but bias on the interval the space and time he just imagine that we've replaced it by this summer whether some of its some of the action in every little self right each 1 I replaced some of the integral by some Robin he replaced interval by some in here where are really doing is just heading up the action and always the that's the thing about 80 exponential it is that exponential of a summer is the product of exponential eatery the Cosby there's eatery eh kind that could be so this could also be written as not when another way forget the summation he affirmed that something over pay before we do that summing over histories for this subject is it's an exponential of Osama so it's also the product of each other miners over age are aII in the 1st cell each of the miners Iowa each foreign action the 2nd and is actions EPA's minors I each bar 80 in the 3rd cell end it's exactly this kind of product product or what all the cells not of 1 mind is Iike the action but of Ito minor site terms action that close to each other with actually wouldn't matter which we used the time practice but OK so let's now the Crusaders Christian having this form here let's go back to war get particles from and think about fields Henry express beyond repair grow ID so Healy have a regional space time that's been chopped up into tiny themselves that was the idea of an initial condition did not the idea of an initial conditions to start on the 1st row of cells here and give the value of the fields at every point in initial conditions of ASEAN along initial condition of final condition is specified the fields in the last World so the question then is what is the probability amplitude of the field is specified a certain way and the 1st World and midfielder specified unless row what's the probability to go from 1 another or the importer would be amplitude if you started a certain way and let the system run you later find it in the final state final state now means the field in each 1 of these cells he you you take Ito miners are I some of the action cells but we now realize that this is simply the product of each of the miners of the action in each set out multiply together the action in all of these cells forgiven history forgiven history for a given history which means a given value of the fields in each cell and the multiple I'm altogether we compute the exponential of the action and then that we summit of all always populating the cells with fields all possible histories means all possible values the fields that could be in every cell with the exception of the 1st loans last 4 of the 1st goal in the last row of fixed by the initial and final conditions and in other words by be over initial final conditions we don't look at the field in the interior we simply
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stopped assistant and then detected in the role is some or all possible our field configurations that could exist in between but that reduced this idea toward discreet for they which we see we have here is a product worthy of his product over our ration might know much try the formulate some ideas about particles instead of asking the question What we start with a given field configurations 1 of which start with a given particle configuration and end with a given particle configuration What is the amplitude go more to the other so I'll tell you how you think about it that right we've talked about a good deal we talked about the idea is a good deal we talked about the fact that fields of made up of creation and annihilation operators are and will multiplying together he is functions of the fields functions of the fields and their therefore fullfunction cave somewhere want are which disagree that are not sure nurtural who would more context we're now natural who said it with the context was in power you are will give me a little more to go on question was somebody said something my grandmother probably set off a quantum field theory on high seas if I can fire came as a shock don't think I could think of was loose talking about quarks and of course in some sense quirks of articles about some of the since never detectable particles because they can never escape from 1 of the only thing I can think of quirks of the quantity of the work field but they never directly detected as separate quirks well separated from other quirks assume there could think of nite well I guess jury June unitary Lucia facility will stand and movie I had pools that identified few years states worker on if you accept the idea quarters of particle it's true that does not want a war correspondents necessarily between a R B fields of theory and the particles that's an idea which is truly as long as the coupling constant the furious more so armed is not necessarily a onetoone correspondence the I mean it is true there a field theories that for 1 reason or another you wouldn't describe their quanta as conventional particles but do not some level it's just the difference between words Our Pereira if you define particle to mean quanta the news of difference between their indivisible Bob very very carry energy and so forth or store were from my money I recall the particles but not if a toe toe No no no in fact they don't have to be continuous are 1st of all of course the idea of continuity of discrete space like this doesn't quite mean much the field in the neighboring cell is just gonna be a different value of the field the rule is the role is divide the space the cells and mn actual practices it is the way quantum field theory is defined you divide fearing the cells used some of what all possible ways Our armed of populating the cells with values of the fields all possible worries no restriction things which are approximately continuous and in fact quantum field I'm not approximately continuously jiggle a great deal and you some of all the possible values the heroes and go your principal Richfield related to sign sign it all classes that online where you you don't assigned the derivatives separately from the field the derivatives of costs related to differences of the fields in neighboring so you populate with your values and derivatives are replaced but it is not only a new could use a different down rules that you might stop after you might define a derivative he had to be attributed difference of the field hearing he rack yup that's about that's a minor detail in a minute then you might take realign writ small and you will still have some of them not if that is what we feel is doing over by defeated by the field here or your would you give value to each possible fuel in the period this fall since these are complex complex functions and function of further from the of the relationships that feel that I have to fly half half to conform is complex very complex variables about you with analytic function analytic functions and extremely smooth the smoothest functions you could think of these functions do not have to be more there at the complex valued but did not analytic function the field as functions a position definitely not money advertise
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discontinuous a there heard solid it is time to give sign up so of course this this by now a major industry called lattice gage theory of lattice quantum field theory lattice because it divides the world into a lattice and this is the study that power were how do these integrals and practice on it by now it is a very effective tool but it took you know 30 years Mr. terror developed a computing technology I was involved in the very very beginning the very beginning I wasn't involved in the computer technology but setting up the rules for lattice gage theory of history the 1st part of it was setting up the rules for it and that took about 3 weeks and then some 30 years to develop the technology to compute and you're right I mean you know how do you know when you know when you've sampled enough of the space peso so there are a lot of the wisdom of it came from statistical mechanics were you do very much the same thing you calculate partition functions of probability distributions of Florida this the checkerboard not I don't feel theory it could be a real crystal lattice and you might be summoned all the configurations of whether there is a reason the electron nature each site so allow the methodology came from the quantitative study of statistical mechanics as we are the same question how do you know when you when sample enough and this is not the subject the mites lectures so the answer is the experts know perfectly maybe putting us good doesn't agree with experiments or circuits it seems work OK so let's let's go back and remember that quantum fields are a shorthand for creation and annihilation operators and if we're talking about let's say the product of some fields in this box the fields in this box that can represent the annihilation of a particle in this box and the creation of a particle and they box so we might just represented by a particle moving from 1 box to another what kind of things In the action he actually do correspond to a particle moving from 1 box to another there are things in the action he which strictly speaking on not associated with 1 box but with a pair of boxes namely the derivative derivative terms I was a little bit hasty when I said you multiply always things together 1 3 box as a little bit of a cheat because there returns in Lagrange in which are associated with him of boxes so really you could think of it as something over boxes and summing the neighboring pairs of boxes but that's that's a detail which turns involved pairs of Fox's terror which involved pairs of boxes of things like the derivative of fire with respect Phillips with respect he squared the derailment of flight on a lattice becomes the becomes the difference suffice in neighboring box so Indiana derivative of fly with respect he might really be written as fired at point X fire point T U and X minus 5 at . neighboring kind an ex that no 2nd Aziz a never OK never bad idea was real damage loss there for about derivatives with respect the special court saying things that corresponds to do it to be different to neighboring point of space and then instruction Mr. take 1 half of the square of derivatives for 1 half of the square of the derivatives will have In a destroyer focus now on the terms which multiplied by times fires neighboring point we just focus on is also turns in here which multiply by 5 the same the plan of the value of the same point but in particular there are terms In this Lagrangian which multiply fire 1 point com fire even port every time you lay down I love a term like this it represents the motion of a particle from you year you could think of it that way OK let's let's take what's concentrate on these terms let's forget the interactions and more grand jury and these are simply he's a call the comedic turns the quadratic terms and were grungy and various things which are easy to deal with and they do correspond to motion of a particle from Aug . 2 a let's look at best and that's included only those but we don't need right it was almost just write it for the the martinis I tires solemnly all pairs of boxes of all neighbouring Paris of boxers are set of all boxes but it's clear that quite right we were vigorous sums over neighboring pairs of boxes of things like flying in 1 box times flying in a neighboring box or use the notation next primed to represent neighbors on alive that's what goes into the action
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articles into the exponential of the action coefficients in front of
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course but that's that's of secondary importance and now we can explain about this exponential but see what's there is 1 Martinez the sum of flight X Friday of X Prize and then things like but see what's the next turn on icons minus 1 were referred minus summation of fire a 6 5 x prime squared times another factor of the sum of 5 of X 5 X Prize others could deprive X Prize echo X double prime text triple prime X double prime troubled times on 1 pair of neighbors another pair of neighbors and I think I left out a factor that should be 2 factorial downstairs from simply expanding our the exponential here what's the next 1 well and has 3 powers of this sum for powers of the summons or for let's look at each turn here the 1st turn has something which involves a somewhat over the latter's I love annihilation particle 1 point and the creation of a mom particle of mobile poured their neighbors or the next Turner Our incident rate the rule that I would tell you again I'm in the business of telling you some rules for v amplitude for going from 1 thing to another the rulers you've got close off you must not have dangling end point dangling in pouring could only dangle like that if you've got an operation to put a particle into the system dangling in point for example have diagrams which have particles going from 1 point to another if they like this illegal on Leicester's instruction to put a particle in at this point pick a particle our sold endpoints like this we will simply rule our those things on lest they correspond to were of a year specific instructions for a particle and at this point they get out of only putting particles in the initial state taking them out final state the only end points would be allowed on the top and the bottom OK so let's let's take a particular attorney at 5 x 5 x prime is neighboring points on a lattice may correspond promotional particle from 1 place to another is a sum of terms the could be a time for particle from year to year turned for particle from here to here the term for particle from here Bahia
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but none of these will contribute unless of course it would be 1 situation in which they would contribute that would be of course yes this is our layered structure here was only 2 layers piety and we put in a particle Olivia and took 1 hour of the neighboring point saying it would be a contribution to the input to of that coming from this trend the proper 5 x times 5 neighboring acts that would contribute and would contribute to the amplitude basically the factor miners I tell you that the amplitude to go from 1 point to another was remind us privacy coefficient here OK but that's not very good if this entirely different yet the here they from year to year How we get a particle from year to year and calculate the amplitude but if we put in a particle that was detected over here for that we have defined in some of products here we have that terms which will leave no dangling in filet steak let's just take this here Ms. has a particle moving from next X prime and then another particle moving from
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X double prime X triple Crown the 2nd only only only when the initial and final state it is always Jeter mister now but this notation means neighboring particles neighboring boxes and healing have 1 neighboring box in Healy have 2 other neighboring boxes yet life 5 but you're there by that the I just means pick from neighboring parents some over all possibilities some offices some of them all neighboring pair no no no no general not but it can be be effect is could even be the same but it doesn't have to be somewhat more but so what does this 1 to move particle from XX prime this 1 was a prodigal from X double prime extra so 1 of them was a particle from here to here this is X prime tax Double X X 2 X Prize and the other 1 was a particle from here but he yeah but X double private extra crime but obviously it is good to be dangling scenes on less well that's our words take this case he only have 3 layers if we only have 3 layers the only way to avoid dangling his the 1st particle connected a 2nd particle sorry for this box connected to neighboring box and then this box connected to this part of the any dangling and except for the dangling unions which correspond to the initial particles and the final particles so if this who was only 3 layers that we would find the contribution here namely where x Prime is the same as X double prime that transports a particle From the initial position for the intermediate position and from the intermediate position for a final position on what would be the amplitude associated with the particle Goran from year at least country corresponding to his turned it would be a 102 factorial minus I would be you read off the coefficients coefficients tell you you're the amplitude thus following we don't have a way to get from here here now with 2 steps anyway steps here however if this was only 2 layers that we would have no way to get from here to here across the bad had a legal across that no we go again this 2nd order 10 year term which has 2 factory it and we find that turned which takes us from X X Prize and Informatics prime From XX Prime Minster next time of from year to year and here to hear but what's at there are there it the path for you jump from here 1st to hear out of it as any of them now noted all knowledge us to track right OK but now there was no reason to stop deal with only 2 terms let's goalline we can have returned 3 terms would correspond to going from 1 box adjacent box well would correspond with 3 distinct steps of those 3 distinct steps would connected together into a chain other words in this some of the products we found the terms which took us here if you do come right back again that would contribute together with the direct jump from here to here would contribute to the amplitude call from his point of this point Tuesday built think that this prescription it's something like the original particle path integral idea amplitude to get from here here it is the sum of the amplitude to go from here a year and the amplitude to go through another route also was 1 back here and eventually if you spend this out of arbitrary they will be a trillion every possible route that you could take the go from any initial configuration any final configuration of that article so don't 1st of all just thinking about a single particle moving in space time missed fuel grungy and contains information about these amplitudes the goal from state arbitrary the row from a point at any other point would you do is you end up all of the possible ways of going there and what is the coefficient each way of going there you read it off the coefficient that multiplies that particular turn in stock 102 factorial whatever 3 factorial some of them I eased some of them don't have used by squares minus 1 so of some vise and some of them don't have eyes and so in general amplitude to go from 1 point will be a complex number that's what's new that's 1 of the things noon here that they can propagate up and down and that's a feature relative said to thing but not raw
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indeed gap yacht time let's talk about going back signed with cigars works his form goes backward in time you could think of this as he therefore allowing the new rules where particles go backward in time or you can follow time word and say what really happened here is a particle move from here here and the a particle particle particle in fact were created the particle half of it going to here and the entire particle combined together with original particle go annihilate figured you away but the role on now allow was trajectories which go backwards last come back to this from a toe the crew the insect certainly help the convergence but then I'm not make it converge but if jet yup the others yet no no no no no some Nova configurations doesn't have to converge Yap August this bottom and this is a finite number of full radiation it OK arm lips Our now let's go on to a slightly different problems we studied the particle moving from 1 point to another nut suppose we put into particles and want to know what the probability for the 2 particles devoted to other particles another words we won start and will take the case where there are only 2 layers from moment there a case from 2 players from all and I want to know the probability that I start a particle the here it will get to hear no I stuck to particles swarm over here in 1 over here that they'll get here well come back to this turned them all at this time differently before before it allowed me a half 3 units but that same terms allows us to particles each to go 1 you look at it this create a particle this annihilated this can create a different particles and annihilate this corresponds to a 0 graph when you start a particle him he goes to hear that this fact that many other factors instead of taking the particles that sorry they're moving over here it just takes a totally new particle removes it here so this seemed factory the same Turner contributed both for the motion of the singleparticle 3 boxes and contributes to to particles each move in 1 box right now Morris 3 boxes 1 who worked at about so there's a line here besides just most particles from 1 place to another it has information about any number of starting particles going to any other number of final particles now in fact particle number doesn't change as long as we just take this into account why not because every starting every year dangling near end point here well on Jasper quality start here was sent obviously something it has for mean they have the bid dangling in some words Our supposing I wanted and I can how how would you get from 1 particle to the further positron like is far likely that would be something that would be something that would just be something like this the particles are a 1st for another layer in him now yes but that strong that we color Spartanburg electron and positron for example annihilating each other other the problem with dozens of energy John facto you a couple is amplitude you would get 0 but uh but their yacht but it the offer but under certain circumstances if was electromagnetic field to soak up the energy our yes you're right with would you would have information about electron positrons annihilating right so that just a lot him here it was walking here on remote everything not everything to find out the or other things that implicit in the field theory there are other terms In now what you put in the Lagrangian is determined by experiments and it's really just a way of codifying the results of experiments but what else could you have and Lagrangian our we have the derivative terms say viaduct squared the miners a derivative of fire with respect X squared why Square square electric stuff that was a staff that I wrote down fire at 1 point punch line another point neighboring point they possibly other things to just involve foggy and not derivatives so for example you could have been 5 square and or the fish there's usually a 1 have here at convention remember what the
1:05:27
coefficient of defies couldn't the square of the mass of the particle provider should derived by 2 of the Convention and if we included this term he what that would do it doesn't particle from 1 place to another if order particle 1 point and emitted from the same point from the same point so it correspond to graphical constructions were would add a new rule if a particle and enters a region hiya from 1 box or another worry that terms that can act users ask where it would simply take the the particle and do nothing it would leave it in the same spot enter another term could come and move it but would weigh that particular path with a factor of them squared every time I missed turn Mac it weight is the path with another factor and squared the amplitude so they would be terms with no and squares act were only these derivative terms at the words were expanding product about have a derivative territory that would be the fury of massless particles the theory of massive particles would have no role that but 1 of the possible things can happen at every step is you may only not just wave at that point with square that I've so the man's His another rules for how you evaluate pale the please half the world's Nova what's we we don't have to a given to detail the main point is about the missile article as codified by another terminal a grungy and witches and squared 5 square which kuwaitis pair In such related knows about them maps problem more interesting is the more complicated turns things like flies square with could G scifi killed fueled can do a number of things it can annihilate 1 particle and create 2 particles can annihilate 2 particles and create 1 particle that can create 3 particles can annihilate 3 particles so that a new kind of thing that can happen on this checkerboard here in expanding our the action and expanding our the exponential there might be additional terms here for example G 5 killed when we expand our we can have not only these products which take a particle from 1 point or another but also Prosser sees where particles are created annihilated in free use of the to win wild toll out 1 in 4 3 in 3 out that looks like I'm here it is for example supposing we have several these quadratic terrorist which move particles and warned cubic what cannot look like well what a condo physical take a particle molding from him and suddenly in that box with butterfly queue in that box it can absorb 1 particle and 2 more are now the next thing that could happen is to particles could move 2 other boxes but basically what it does is it takes a particle jobs it creates tool announced a particles could be on their way and this would then contribute to a process where 1 particle came in to particles out In fact you could make a very very complicated things you could make things of Abu let's go back to the 1 particle going from 1 place to another 1 particle going from 1 place to another Ireland is drawn checkerboard such just imagine the checkerboard 5 1 particle from 1 place to another thing go directly just but to small stops is a bunch over these hot but also something else can happen on away at some point the Cu turned back Gilbert and connect it could take that 1 particle coming in and Splendid Splinter 2 0 and then a both particles will along on till another cubic terror something quadratic in the Cu terms something with 2 cubic terms in them and create an extra particle reabsorb so in fact the actual amplitude to go from 1 point to another by 1 particle is not just the sound I simple trajectories going from 1 point or another through so the space time but it is more complicated here has infinitely more complicated has process sees with the particle splits and we don't know the words where the particle emitter 2nd particle which is then reabsorb an example of this would be an electron moving along and photon ribs on the part of the amplitude from electron to go from 1 place to another is still process of emission and absorption but think he'll wildly more complicated of course we have they had all these up the rulers in up the amplitude for every possible way to go from the initial state of the final step you could have several of these planning enjoying sir you could have things jumping across the year big gap between our rules and this genial arbitrarily complicated shot from the point of view of quantum of electrons and photons like this could happen or rose through photon emitters another photon emitted but even worse the photon can break up into an electron positron but then electronpositron exchange of photon between them all these processes he's calm hour of expanding the action like this and finding the individual terms which do all of these things you want the amplitude friend 1 of these processes sees you just go back what appears unifying maternity looking for and find the coefficient in front of a mail may not have an ID depending on whether it's an even power and art power on an autumn factory was so poor that will give you the amplitude for a specific process but then the full amplitude go from 1 thing to another Is the horrendously complicated some privacy and things standing multiple multiple as at which then will have the to that process for action more Yao from all different
1:13:34
histories going from initial the final so there 2 ways to think about field theory 1 must think about fields and think about histories as initial values the fields final value of the field and use some or all the values of the fields and between the other is the particle way to think about where the fuel was replaced by distribution there then again amplitude for initial configurations which means of initial particular set of particles and a final set of articles these basic processing the basic underlying processes which happened are governed by this Lagrange that is the most important thing 8 sigh we'll have more 1 into fields interacting you have an interesting interaction process were particles would enjoy and so forth what happens if you have fight for the 4th year is a perfectly good interaction that would be someplace where particle for example could come in and break up at 3 other particles 1 to absorb the initial state 3 to emit final state or it could be a scattering in which 2 particles come along call I had to go off anything that looks like a vertex for particles altogether To win until out or to win 3 in 1 hour or so for or just 4 hours for foreign there'll governed by this and so so forth and so on procedure Wiederin spontaneous car chase yacht that once again you would violate energy conservation so if you're really work that our land they ended up being amplitude from every position where this could happen every space position what could have been you would find out that a present at the captain and the the residual add up to 0 so process you would violate energy typically um all aid up 2 0 and your head of sheep quickly if that all if the they can we did some examples are so you honey integrating the vertex all kind conserves energy how integrating a wall space conserves moment the same thing is true integrating the position of these vertices Indian points of these things of all possible positions in space and time will simply ate up 2 0 on less energy and momentum is by apart from the conservation laws everything that that you could drawdown will happen with some prop with some probability now how could possibly be that you end up from among this thing and he answered doesn't come out infinite well however the answer is that there are coefficients he recall efficiency it is called coupling constants particularly the coefficient for the higher powers here we could call is G 3 you we could call G. form and the rule is the diagram defined in Bagram this every time you have a vertex stay with 3 particles coming together the amplitude contains a factor of G G 3 in this case so the mall complicated the diagram gets the more powers of GE each vertex gives you a power G G is a small number In each successive power of G gives you a smaller and smaller contributions and you have a chance that the thing might be able to convert the technical question whether really converges on my was a difficult 1 but you could see that you have a chance if G is a small numbers if these coupling constant small numbers then be simple of process In other words if you would number of vertices be bigger contribution and the larger the number of vertices more complicated more of a contribution another chance at any rate at some sort of of convergence career if the the coupling constants a large it means Vseries do not converge and if they don't converge it doesn't mean that the theory is wrong just means you've chosen to unite he earned prescription for trying to work Quebec but in practice most of the quantum field theory most of the things we do with quantum field theory of based on the assumption of small coupling constants and those circumstances possible towed to evaluate is fine and that evaluate them but a had them up and find out that no 1 with a with 4 vertices is much smaller than the 1 with 2 over so that's the name of the game yes No rhetorical and now the next step is to name is the right down the list of all particles centerright down either in the form of Le Grand Union's or in the form of finding roles the finely roles and Milliman genes have the same information and freely identical particles coming together but 3 bows on fight particles going from 1 point to another those other Connecticut terms a quadratic corrode terms here so be it finding tight rules and we will run June on the same day you can't even describe the world by specifying all the vertices incidentally the line between 1 . 0 my mother of call propagate whose need a place where more than 1 particle model to particles come together those Accor vertices vertices propagated but there as I said just simply another short he in the way we are describing a Lagrangian Lagrangian short way of describing graphs us so we will have stated all of particle physics is kind of dumb way by simply writing down the list of all particles writing constructing or even just expressing quantum field from each particle and then writing manner Lagrangian Close particles which contains all of the interactions all over we can write down all the particles and simply specify what all possible vertices are you the way they come to the same thing they're kind of a complete list of everything that can happen when she know everything they can happen you can start thinking about calculating amplitudes for particles to go from 1 place to another particles to collide particles to collide scatter
1:21:39
years particles colliding skeptical electrons colliding scattering by a photon jumping from 1 place to another again grungy and tells you how to calculate the input to reach views and firm as acidic contains only information including symmetries including conservation laws but it is probably too we write down a list of all elementary particles that write down by the direct mail a city fathers the initial situation talked right that a bomb was the initial configuration of particles Pappas a final configuration of particles and as I said it is kind of complementary way to think about field theory on field very complementary to the field description where you would start with a given field configuration and begin kit field configuration is some armed sends the complementarity between particles Fields is very much like the complementarity between momentum and position of 2 different ways to describe the same reality and the uncertainty relations between them if you know with precision number of particles and then you know what that is uncertainty in the values of fields if you know the values of fields and uncertainty in number of particles so of user really too were armed complementary ways to describe the same thing and Her brewer Grand is useful Bush said was theirs sometimes under certain circumstances there symmetries which relate coupling constants of this that coupling constant a process might force symmetry reasons related to another couple incomes the trust us some of them are related to each other by tree bark their values in general Laura simply come from experiment Procomm Express on much of this if you really go and learn about quantum field theory has beauty and has both have elegant beauty so now only start writing down the list of what particles 0 are what they masses are what their coupling constants are it's just on Mets it's just very very ugly mess with very little coherence except the coherence that really that comes from symmetries symmetries tell you relationships between different kinds of particles the process but apart from the symmetries which a few from March 5 a large number of random facts about about 80 somewhat on motivated art collection of different kinds of particles thought but it the that because want to measure the measure looking ahead they get absolutely no no I mean fewer OK let's put it this way you have to Laney particles which means tho many fields Khamenei Ferrer any esthetic I'll call you have a lot of them 100 of them or something like that coupling constants which adjust the more less numbers pulled out of the experiment and a lot of messes which range over the matter of a few relationships between Our Barbara wants you know that want you know them that's it you can calculate with great precision anything about those particles are thinking about those particles reading about quarks electrons and photons and so forth means atomic physics particle physics Atomic Physics of Nuclear Physics it means a chemistry and maybe it means biology sold sold the amount of input because it is perhaps bigger than you might like the the amount of output as your letter earlier he 400 are they for electrons and photons elected than want electric charge you have charge you noon well in a sense the electric charge as a kind of G 3 electric charge a kind of G 3 electric charge is the coefficient and the amplitude representing electrons coming in an electron going out the photon being emitted in terms of symbols this diagram would be represented by an electron going out China coming in and the field operator for a time which is a end up writing down the details some polyester damn summer Dirac matrices but then the numerical coefficients in front his electric charge in search of new certain dimensionless either the only process seizing on electoral and dynamics the emission and absorption of photons from electrons but but With like this Over truck with lecture collision with a gap electron collision with a photon just the simplest process would be time absorbed a club protons and usually considered part of the part on October 6 now you could think you'd you'd when you don't want which you are doing atomic physics you think of the proton as a point infinitely master where you don't think was something described by year quantity of fear you just thinkers and nailed down heavy particle which never moved until you will open up the the theory protons and neutrons that this gathering by a photon of photon by electron did shot a almost like is proton yes there were rumours right now rest on the stroke let's just diagrams just to get some feel for what kind of things we have to get enough this is the lowest diagram goes into In 2 scattered photon by electron era to powers of the electric charge isn't he squared but this is the amplitude the amplitude gets squared defying the probability of a crosssection crosssections another way speaking about the probability for the scattering I'm that means that the the whole process the probability is proportional to electric charge to the 4th me electric charge at least in suitable with a suitable definition is a small quantity of the square of the electric charge would begin with a suitable definition is about 1 box a number which is 1 over 137 fine structure constant so this is proposal to the square of the fine
1:29:46
structure constant scattering in quantum elected dynamics is unlikely process of photon comes in and strikes an electron what is the probability that a goes right through it versus the probability that a really scatters it probable really scatters it is governed by the Ito the 4th of its a very small number now we after it had all the processes that we confined in the not perform really amplitude incidentally is another 1 he would like where the final 4 times it is made it before the initial photons absorbed the process after them all but then we start adding more complicated things focused on here and now we can and thought time in many ways this is not the only way we could go from here to hear from here he many photons behind the powers of electric charge Aziz have this has fought powers of electric charge in amplitude and 8 in the probabilities service is even weaker but then we can deliver more complicated things we can cut a electronpositron pairing he each time we complicated we tool more powers of electric charge so Kwan electrodynamics the expansion is typically expansion in powers of the square of the electric charge every time you hear another complicating structures find to powers of electric charge and means it's about a hundred couple hundred times smaller in amplitude than the previous 1 that's why we knew we had the mop it looks at it converges more more complicated they get turned the weaker were media probability from that particular configuration incidentally you don't square this an aide at square of this editor of the square of this find probability you Ed them up and then you square as the role of a head and then square square that muses cross terms between them In calculating the probabilities Court of interference so that that's the structure of quantum field theory but that's what we do with it on to the next yours I said where were now at the stage where we can really start sensibly talk about particle physics we can name all particles we can specify their masses those of the parameters in Grand Union Bookamer and there we can write down or ground genes governing them partly from Serie partly from experiment and discussed their cemeteries the cemeteries could be discussed by looking at the Lagrangian will do that and we can write down the standard model of particle physics and discussed some of the properties so I think wherever they are an off tool graduate courses were set now to more overhead and really discuss the world of genuine real particles and what consider that word he steals not right the 3rd 1st that where ONE XNX prime could stand for 2 different times that's how we managed to move vertically both of them them now does the committed there are various terms were particle to stand still Wendy wrote 5 of X miners folly of ex prime squared that had did have to do it like violates squared private square those attendees of the particle standstill still across the ones where moves of it terms of the particle standstill both from this and from the West if the 5th form OR please visit us at
1:34:48
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Titel  New Revolutions in Particle Physics: Basic Concepts  Lecture 10 
Serientitel  Lecture Collection  Particle Physics: Basic Concepts 
Teil  10 
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  10.5446/15074 
Herausgeber  Stanford University 
Erscheinungsjahr  2010 
Sprache  Englisch 
Produktionsjahr  2009 
Inhaltliche Metadaten
Fachgebiet  Physik 
Abstract  (December 3, 2009) Leonard Susskind gives the tenth 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, including, the diary equation and Higgs Particles. 