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Supersymmetry & Grand Unification: Lecture 7

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just come back from the HMOs arms from moment and on foot and not interested now an emotion we've talked about this before the guns and HBO's arms arrest their described by creation annihilation operators you create a party preyed corporation operating entirely with annihilation opera not interested in moment the theory that they are all of them out there we can describe them in terms of you know the story creation annihilation operators if we're talking about both on then we introduce creation and annihilation operators a plus my guess we multiply them together we multiplied them back expression by the mass of the particle and just in a plus a minus the man's times the number of quanta a plus times in mind this is the number of quanta and we can just say that the energy the amassed and the number of our bowlers on a court that the energy of the Mosaic system if we also a firmly on reaffirming in by a similar pattern creation and annihilation operators from Miller's we've called C and if the firmly under the bars on exactly the same energy or mastering caught master energy then religious and the number of both and also the number of firmly on steep losses she minus melody thing recall the energy or the Hamiltonian our religious word 2 0 about the Lagrangian formulation of balls on infirmary again we're concentrating on different guns and bows without worrying about their motion three-dimensional space I love for HBO's arms we invent the field the field is called were also arms but Phobos on Michael feel far-right usually qualifying and we give a Granzien incidentally just to remind you this is nothing but the mathematics of a harmonic oscillator of a single harmonic oscillator so Tulua described the field theory it is very very simplify context the field theory is just the field theory of the balls answers just a harmonic oscillator winter those a variable flight before the coordinator of the oscillator and we give it a Lagrangian if I got squared off tool plus the frequency squared which is just and square by square or tool the
gets spatial gradients not thinking about things varying space moving in space the simplest situations simplest set up the only a time direction most space directions to worry about and we work out the equations of motion Lagrangian equations motion that followed the derivative of Lagrangian with respect the fight got what they are the orange equations motion partial of L with respect 5 . be by is equal to partial of El with respect the fight nite by deal but if I got as justified got this would give us the 2nd derivative so mistress becomes a fine double got 2 time derivatives and that's equal all I'm sorry but I do wrong here yet he might face right energy as he possibly regards used humor and that's equal so the derivative of grungy With respect the fight Betsy quota and squared minus M squared minus and squared fight and the solutions of this there is differential equations 2nd order differential equation it has oscillating solutions and the oscillations oscillate with a frequency and my square but I am not so not surprisingly the mathematics of all of its the quantum mechanics of it is just a quantum mechanics of a single harmonic oscillator 1 creation and annihilation operators as a brown state with no particles particles with a dagger taken away with a with a minus a Ed particles plastic taken away with a mind so for now let's come now to the Fermi arms the freely on would Gandhi info firmly answers some cents simpler they both on news it only involves a 1st power of a of our time derivative it doesn't them long fight that squared it's more like just 5 . 4 and 5 this is think about that for for a moment had a Lagrangian which was 5 . 4 times 5 relapse suppose for the moment that boss on ozone mean conventional fires are you just think of them classical Lagrangian the Pfizer numbers they're not grasp the numbers their ordinary numbers they commute with each other and so forth deposits was a Lagrangian this is an interesting Lagrangian those given interesting equation of motion if Tillis would get the equation emotions partial Lovell with respect to fight not that's a court fight nowhere depicted the river a that of Beebe by IDT of this it is just equal to what I got that should equal BL by defying which is just equal the fight that and that much of an equation emotion that puts no restriction whatever on anything it is as find articles by about 4 biology every possible trajectory satisfies the other thing you might wanna say he is well maybe maybe the trajectories ought to be differentia Bolsover fight means something but other than that no restriction it is not a useful interesting Lagrangian this is connected with the fact that this is a total time derivative it's equal to 1 half of the time derivative of fi squared whenever a Lagrangian is AT kind derivative of something it's always trivial never gives an interesting equation emotion you could draw away Total time derivatives grungy and it doesn't matter what time derivative of it just doesn't matter it doesn't make any interesting equation emotion but now let's suppose if I was above 80 fermionic variable but it's a firm fermionic variable it means it has to be treated like a grass and number of tanks parcel of both respect the fight got we differentiate with respect the fight back and that indeed gives us fight and we take the time derivative but what about partial Lovell with respect the fight if I invite not our grasp variables and you differentiate with respect of fire you have push it testified about which is also a breast variable you get a minus sign another words if you wanted to write the other order so that you could directly differentiate with respect the fight you have to put a minus sign and sell you would get partial Abell was that the fight is minus 5 . what's the equation of motion now it is 5 . 0 is equal to minus 5 . that seemed equally spilled but not is not equally stupid bring Attila science twice fi . equals 0 accuses the mission does you 5 . 0 that's that's highly restrictive means solutions are high above the highly restrictive and this Lagrangian has some punch to it may be too much punch for our for our tastes and maybe too much let's head something of this Lagrangian here but also a yacht if all we had was fired up square incidentally that are equation of motion would be fight double but equal to 0 and analogous tool but not the same as 5 articles and that now that he's affirming arms we should give the different I should rename everything on the blackboard called just so we don't mix up firm eons on HBO's artist from on sewers course site set why is it
taking wanted to just this side side side radicals but now what put In order to get this site top oscillate I had to put a restoring forces was the rich restoring force him square 5 square that's with this was doing here without it we would just have fight double radical 0 is there a possibility of making sign a oscillate where we don't mix I ask you to make sign oscillate we edit the Lagrangian plus In size squared supposing we had passed ends squared squared but in time size square inch similar to theirs but with only a single power of and then what happens well when we will get side Yukos twice about no I want to make them think that just might want to put is to win here at the world want to put in moment so I got goes minus side got that's what I got from this here I get from this term we are what it sorry pleurothallis go back over again partial of with respect to site got that's just sigh and by IDT of it just like that that has equal derivative overrun with respect to side got outside our respect site outside not plus him twice sigh against brighter son or were said well on but no actually actually sigh square 2 0 40 really should do what we really should be doing is having 2 different size he really get it right but still if there are more consistently but still more consistently let's put it to a different size sigh 1 and the site of a could also be the real imaginary part of a complex side but secular grunge our society 1 . side 2 0 plus side to die applause to adopt sigh it is balls on this list this Lagrangian would be completely trivial let's see why its 1st similar kind of reason billboards arms supposing weak took the equation of motion for site to equation motion foresight to all our office I won because I won partial of with respect to sigh 1 . is Eagle decide to and if we differentiate that we get sigh 2 . 2 this is the equation for sigh 1 and that is equal to go from here sigh adopt Eurydice again trivial if these will bows ionic variables just have signed up to equal side to side too it also sigh about is equal to sigh warned that was saying nothing but because when we differentiate with respect to sigh 1 over here as a sign change the equation emotion becomes more interesting now yahoo's aside a constant that's the solution the acquire differential equation has a solution a solution striker be any constant now we entered its plus and sign 1 strike to let a head in sigh 1 site to do what we get so let's again take the equation of motion for sigh warned that gives us site to 0 . right and from he'll earlier from here bars source 1st of this 1 gives us partial that equals decide to adopt differentiating with Specter sigh 1 and from here we get plus and no site to I think I do that right there do that right differentiating with respect the sigh 1 Rodrigo respect the sigh 1 . site 2 0 and then you differentiate that was at the time the site 2 . Healy differentiate again with respect to sigh 1 but now we get a minus sign because this I 1 is in the wrong place we have push it through he deferred differentiated that gives us a mine site got skewed or sex if
no not menorahs equation PLO by view that core ex parte physical today is equal to deal by the X number assigned from work the site 0 . physical might sign 2 . plus year differentiating with suspected sigh 1 so that just give me side to bigger were being consistent with standard notation I would probably put a half year and then my equation would come out and a half years a half year were shift the soil with the left hand side I would just have outside food doctor is equal and decide to know what kind of Bush's savvy we need a nice somewheres don't wait for Iowa here because of my does belong over here and we're here Taser have 2 . be cause and Scioto Thames
loose as ever oscillating so is now a fermionic variable which oscillate with pride it has I turned me but I did see energy I debugging is energy and so each quarter as energy him now whereabouts I want not what happens what happens so I wonder whether he will go through with Iowa what you are With I 1 wood Barnes saying they old replied British at 1 or if there were does apply 1st got here Thursday with respect Taiwan I get equation of motion for site now I want the equation of motion for I 1 itself respect the site tool and on it or get a minus sign here and I'm not sure I think I should get a minus sign here for do right but each case it's an oscillation at 2 . be interesting point is that the law grungy and furry firmly field instead of having aims squared time size squared has just In time sorry with members the frequency and Connecticut term only has 1 time derivative you've seen it before in a Dirac equation Andracki equations of first-order core equation in time for the Klein Gordon equation which is like this 2nd order it seems to me needed that this last year because rapid variable centered but like slight site too I don't think the physics issues but I think would be a final physics probably doesn't depend on it but for me but I just wanted to go through the Lagrangian formulation our this system consists of a Lagrangian which contains if I got squared off with 2 miners and squared fi square over to an end something like I think there's a half again side start of society sigh 1 or we configured besides start Bob sorry With an heII plus society Dr. Tsai star plus an outsized our stride work done here for Our replaced I won but site to buy sigh 1 plus decided to try might to find that I can be right like real imaginary parts of society just being where courts I once tried to here I think this is this is basically the Dirac equation in a very very simple context with only 1 direction of space time namely kind just ignoring space altogether after from unlike other bows on as a harmonic oscillator and the formula is just a year component of the complex site OK now we wanna come to supersymmetry this Lagrangian does have supersymmetry let's just a review that prickly has as a symmetry between from Iran's involves islands and obviously does it take out a family and you put back in the boat on or vise versa and you don't change the energy so the some kind of operation for our which takes a Fermi on Barzani or a remote was and replaced by Barzani and those are the super generators also accused of right those our cue dagger is a dagger a plus Co a plus C minus and they were just the convenience and notation but for the square root of twice the mass here that's we haven't we haven't defined to dagger yet the free to put the the number like it was a joke
it removes a bottle and remote a and put in a boat conjugate other Q is just C C minus a C plus Times Square to again as a severe square there as a young Convention the Convention having been established summer context OK these are they kill operators and they satisfy an algebra they have 8 in die commutator algebra commutator algebra like angular momentum button and die commutator algebra well it has some pieces which of commuting some pieces were returned commuting so if I can can't die commute Q dagger with killed with done that before your we get we get Emma and then a dagger 80 plus Steve Agassi for a dead miners my dagger and plus 0 OK another words we get twice Hamiltonian we could have made it simple and made a one-time the the Hamiltonian if I would put the 2 of them but then I would have disagreed with somebody else's notation and since by now the notations are completely standardized I put the here Our so that we would agree with Her everybody else's notation This is 1 factor What about me and I commutator of dagger with dagger well that will have for example the product of 2 steep losses the BOC plus from here a C plus from here at every turn and step back from it it will have to to see the title C minus death and C minus 0 service 0 as Is is and die commutator of Q with Q 1 less thing to do since summer ranking inside of this algebra we warmed up with than a somehow humble forms with Q. in a common algebra common algebra but age is a commuting thing it's not an ordinary number it's not a grass the number and when you commute or when you commute or do whatever you do with Caresse men's times ordinary numbers the rulers commute not infect I so we could check 1 last thing we could check what's the commutator of Hewlett and kill Dagger out strike dagger but seemed Q with rich and this is easy to work out leaders a homework assignment to work out the algebra of it the out above it is a couple lines where CSS the still a hero 5 0 0 if it was just an 8 by itself than the commutator would would terminate if sea by itself the commutator would have a city but with both of them have the commutator Issaquah check check that yourselves Net I did spirits on board all we recover commutatum yeah OK well the commutator is 0 and that has an implication that has implications as obscure as 1 of the few waves Q was an obscure thing it it sooner grass variable or got got a fermionic character to rate renew measure it now you don't measure but what does it mean that commuter the Hamiltonian well 1st of all means a symmetry but also means it means that the gross Observatory in the system and this particular case taking our bows are putting your for on board also it means that cues are conserved quantities conserved quantities they don't change with turn and sorrow you can label state by their value rebels as they waited to label him with energies charter of things physical super charges incident political super charges because they're concerned quantities Super from a white but charge like electric charge conserve that's important thing about others this algebra commentators and back from favors of courses 1 more which is the stupid 1 that's just the commutator of H with nature's also called 0 everything commute itself and sorrow we have a closed algebra set of variables picture now was another way of thinking about supersymmetry and spy a dimensions the space time grasp environment so let's go through that again I remember that things like Hamiltonian is an meant that now Manning back in position for a moment just temporarily from moment spatial position Hamiltonian is represented in quantum mechanics as ii times the derivative with respect the time so for example the Schroedinger equation says on sigh is equal to something our it becomes equation for the kind derivative of the wave function of H is Heidi by momentum might decide by
DX so they often very very often in fact almost always the case that symmetry generators are related to derivatives with respect to some kind Our variables whereby angular momentum and angular momentum which rotates space path is related to various combinations exit Deeb IPX the related to differential operators namely the differential operators which is rotate you will be sold when we look at these kilos you might ask are they related to some sort of differential operators but not derivatives with respect the tea or X but the rivers with respect the the White is addressed the numbers after all so grass the numbers and they have to do with symmetries they should have to do with symmetries of coordinates which arrest members themselves so I think we wrote Obama last time be our representation we head complex dimensions the time and figure freighter or we can also add status of complex grass variable we conclude fate of also the
complex conjugate Albert I would differ interchangeably use the notation bar and complex conjugate means the same thing so are degrees of freedom depend on TV and these grass in court on very very obscure an odd thing to to invented but were still at sea with I will write 1st of all dagger it is equal to derivatives with respect the fate of bar as possible and changing notation wrote the last time I don't remember debugging Paderborn minus ii made a Debye D. end Q will sell here I know that I had a mistake the other day because somebody pointed out to me is see by these data and I think the last time around I wrote plus I fate a bar Dubai duty but I was wrong the the more I want somebody a but it worked out them whatever you but what they really boss yacht will also work gold rings and miners to back yard who also work reminds the correct they usual not take part of it is convention part of it is convention in chair part of his convention but the fact that these 2 signed should be the same that dictated by these commutation relations and that commutation relations besides should be the same that looks funny this is a complex conduit of this you might think you should have changed the sign of alright but go back step remember that in quantum mechanics it is a chore real houseplant the things which are permission and are observable remember that fact that momentum is equal to minors ii Debye by DX that's a little bit peculiar is it because momentum is a real quantity and so the right hand side had better be permission operator and the answer is that Debye DX it is not a condition operated on 8 times debate DX permission work that out we done that in the past I'm going work a mail when you see a differential operator like Beebe IDX think of it as pure a measure Mary it's pure imaginary it is not real thing which is the real differential operator is minus ID bike TX I'm so be by she is sort of Israel and when you complex conjugate you don't change it that's a that's a the thing that I got wrong last time both assad should be the same now if few intact commute them with busy that commutator comes from the me and I can't say comes when you enter communities you'll have an impact Camille commutator of Debye data With faded with me but if they the bar with a bar what Saturday country His warm so give you a RID by Beauty and the other way also would might be but fee now refer do we really need because of he is that a for about a year get juices alone 1st there was not enough to reach our Olympic knew would be Gerber these in commute with chairman and these here not I think it really is minus yeah so master class now but see the plaster as the enough I worked it out at home when I got my got minus nonetheless you from me I by decree is each young prepared I knew I guess blasts my mistake and any case you will get to age limit to where To come from the tow comes from the 2 terms in that kind data of they borrowed paid a bar the commutator of B buddy favor with favor so enter commute fees you get exactly twice twice on i.e. Debye DTD dispute by Kentucky and they're on the right OK and ID by deep-sea is H that's the that's 3 basic algebra Of this very very simple supersymmetric system by the next is to introduce fields now in the world that we're considering Narrows is only 1 dimensional space-time only kind or their come back and forth in space will make complexity and make it more complicated than this but until we put in space is only 1 dimension of time physicist at constantly thinking of multi-dimensional spaces they think of four-dimensional space-time which has 4 one-dimensional time 3 dimensions of space they think of 10 dimensional space-time which has 1 dimension of time 9 dimensions of space we sometimes think of 5 dimensions the call applying theory for dimensions a space one-dimensional we even think about 1 dimensional time which has 1 dimension of time and no dimensions of space and afterward going here is just simplicity just to see how the machinery works are right now wish to introduce a field and that field should depend on trying to end the favors whatsoever is is time and the Raiders so let's call the field capital 5 was not this fly over here at 5 small fire this fine big 5 is a super super field which means it depends on and the they tried out all possible expressions that you're going to it willing to assume that fighting is it bows in other words that it's a ordinary number that means the 1st term which contains no favors is an ordinary field and could be this field of India folks at a time reduce Rican house things proportional a bath later barred time site is a fermionic variable they that is a grass unbearable fate bars grasp and then we can have Our sidebar times and the last thing we can have it is well called bath data what comes at the Dana Barros they don't like played a square times they the Barbara that's 0 made at times they Tabora square that 0 you can't have more powers into and freedom are so terminates at 6 at sea that's the general dependent on Super field in this context is very simple quantum mechanics context for the very simple one-dimensional world a function of time course also functions of time and D is also a function of time because the general expansion that but what will you see supersymmetric transformation was due walk when they act on 5 what does an ordinary transformation like he goes to a Michael shifted he do the shift the tea just 6 5 5 plus as shifted value of to what does supersymmetry transformations it's a coordinated effort to coordinate transformation on key data Cedar bar set of axioms
settlers course transformation but on him write down what is in the house show you that it really does agree with these things here it is a transformation he got was that tea that are willing to introduce a new award on grass and number new grass number will be called Tsai also has the conjugate get the what I'm doing another rumor form former Majid in doing that you ordinarily might do in space really space you might shift the coordinate acts by writing Expos X plus a small ship was most shift epsilon that accorded transformation might do which would shift to shift x-axis epsilon would be up perimeter of the transformation now will would doing what we're doing now is where shifting data because the data plus grass and number sigh a bar goes the state of bar plus the conjugate grass number plate bar so they shift their translations of the FedEx of sort translations of space but at the same time you translate time but how much did translate and by prime goes to plus or marine tea mariners on Tse bath later Myers REA see Figure bar I will not swear signed because I work amount and they seem to change this guy was magical way of changing between earned might drive from home forget signs seemed to change on the paper so they must beat hope when Ike commute to work assigned chain How Hall 1 with the at the don't of the firm crew chief for is the year they be these other variable it's just the power series it's just coefficient defined as a function of p they then made a bar we can X ended in a power series expansion and powers of patent Barbican leave the colossal expanded powers of tea but that would not be useful thing to do which is leave it as a function of tea but expanded powers of bar about data and that's it came away from and that that's the most general functions and played a bar with a key dependents in each 1 of the places nor it mean that means that the system containers a From above bows variable firmly on variables in the complex conjugate an appeal was like another bows on variable it appears that the mother boards are the Pizzaro to bows on variables but in fact who redefined that's not the case we defined as something I think the reason is called these beef a dummy a dummy variable are I didn't invent terminology called be OK so now let's say this this it is field where we need have field equations we need firm action we need election From pellionella how you build actions you build actions by building up various kinds of products derivatives and so forth the fields and you make a Lagrangian and then integrate the Lagrange Ian Over position or time to make an action you integrate the accident an ordinary field theory went to great action over time and that he will integrate the action over tea and data and face the bar just think of the bar I would never coordinates finished here end right next to the action as contain various kinds of expressions involving capital 5 years and its derivatives just as we would do it if they complete a bar world ordinary coordinates press Overman action actually reaction was Caribbean it will be tea and dual these data and Inter will be paid a bar on some let's call it land idle workload an ordinary Lagrangian super Lagrangian was quoted lambda instead of El normally would use L Lagrangian turn Lagrangian would not there would not be used the court capital super Lagrangian enter the super Burundian is itself a super feels like this but it may be for example my contains square of Super fields of might contain all kinds of derivatives of Super fields and so forth sold whatever it is due a grungy in itself might have a similar expansion in powers of no later later a bar they don't say the boss so it can have grunge runs in here as a function of T data available are now we have to integrate all the data keeping in mind that this land contains various terms what's called a thorough the 1 plus well it'll have various turns of exactly the same kind of 1st terms will be bows on the 2nd turn will have bar times a firm beyond the 3rd term Ohio the referring on a 4th term will have bit Bob Bullock Garcia may be more complicated won't just be a simple field contained time derivatives quadratic things all sorts of stuff y happens when you integrate a function of the bar when you integrate of function with respect the fate and that of bar but I tell you less time integrate over faded out with what happened it is the same as differentiation yet you get the last turn the constant you get the last 1 which contains the maximum number of bidders and that a bars and that's all you get sold whatever grungy and is this part of it just takes out the last term which is proportional paid bar made its call the detail of the Lagrangian although it is not necessarily just be turned in this expression here it's the last terms it's called determined by without naming it it's just the last term in fact Wetzel we can call it land made bar rest turned the expansion of land in the world detail another words whatever the last German land is it's just orderly Lagrangian the ordinary Lagrangian as saying that we integrate with having never heard of favors and later bars this last terms is here it's just going to be what we could redefined the ordinary around here so supersymmetric theories have super Lagrangian which don't do an expression like this but they also have ordinary Le Grand Union's which just last turned this expansion here the trick is to keep system invariant under the supersymmetry transformation words to make sure that the symmetries preserve absentee steps that's the trick there word 4 discussed OK so we need to know how to build various kinds of expression out of the Super four-year-old we meet the norm below expressions which involved derivatives which involve higher powers of the field exactly the things that we might do
if we had an ordinary field and we wanted to build a Lagrangian we would differentiate finds where the derivatives we would multiply higher powers of all the usual things we do to build an interesting grandeur we tried Dylan we're going to succeed where tried for 5 years on before we do that yet before we do that or ignored we arms yacht what I didn't show you is that these transformations here are the things which is generated by these cues didn't tell you what the connection between this transformation and this would attend the but we did arrive it was just go back again arms if we have a super feel like Paris and we shift the arguments over what happened to the Super feel what's the change in the Super field but said the change in the Super field if we make these changes the change in the Super trio 1st of all the says shift they got a little bit that's a shift data differentiate the sympathy over with respect that data and then multiply that by excited like that we shift did they by America Tsai a change in the field is a derivative with respect faded and right now we also shifted later bar plus the derivative of fog with respect to fate of bar times see borrowed yacht and last wish shifting the time variable little bit what door-to-door shift time variables but they declined to write their time derivative plus fight times the amount by which as shift flight on the budget the amount by which I shift key so that that'll give me a 5 . times my on a two-day strike later minors on see bar data I believe that's what happens that's a small change in fiery wearing I make such a transformation simultaneous transmission of data they the bar and he bats I'll fight shifts notes collect together the pieces of multiplied XIV that defied baby data and also a will flight . com exceed current data but the pieces which multiplied see very serious succeed was see prices d by ID data the Culebra fire out here and there have you we 1st differentiate what is that the data and then they were 0 apiece but can you miss remind multiple multiple now OK debating data now what the other thing multiply scsi it is minus I see border it is a terrific see outside funds Debye Paul all-time books not there's a favor they not far from that nuisance will sign for some reason or another I seem to have Aug it is just gathering data will be written by the entities that desire raw numbers I off brokerages but that does not think that the do have any tho White there have to their want your order with the story preferred that is another turns mentally the terms were complex conjured you would see Bonn Clemson was tough times are you will soon have a 5 . 4 reais where plus I continue to think that I really thought I was not right not yes CUG his house is die he says changes he Karlsruhe I'm not saying goes team what some might get some rain right miners and that pressure these the functional you just differentiate with respect to thing is shifting and multiply it by the amount shifted the the argument by right now is a very simple curative there's the a reason that I wrote my own this year where's because I was tracking some signing conventions that I have inconsistent for years so aware someplace over here at signed convention consistent yeah I'm a fuzzy about the size of all with that's my problem with this subject but could never keep track of the signs are that they come and they go can keep track of are in any case a consistent set of signs would probably have have plus here parents members must be passed over here OK so what is this thing here inside the bracket What is that differential operator inside the bracket there's just killed services seem times a differential operator Q and then from here plus C Mark times Q. bar acting on flight number what these things are these derivatives and various kinds of differential operators is simply work it through you simply take these derivatives backbone fight fight you want differentiate with respect the state is straightforward you just going only thing you have to keep track of the signs and that's what these cues are they're like momentum like momentum but not shifts of ax but for a shift of these data simultaneously shifting partner in this way that's the basic symmetry is called supersymmetry it's kind of shift symmetry in these grasp court but not just shifts it was just this shift cemetery in the corner in the grass court and they will only be these Debye ID status in the by data bars it would be no I data by GTE himself serving you would just have the Easter Bowl a baby and commutator virtue with acute dagger was only Debye favors 0 hour 0 so it would not satisfy this algebra where was Of the world lost now would not have the right him inside contained energy bill would not represent the algebra that we started or not the we started with very simple making something fairly simple and something fairly complicated and complicated story but it could see the stone is kind of interesting it saying that the basic symmetry commute thought of as a geometric symmetry as a geometric symmetry of shifting and translating
in this space which Malcolm things not only for but also the loose ball and on does that might cost of 4 . 5 4 . 0 or it really just it's just a bookie forward point of view it just a bookkeeping devices that keeps track all of Of this simple algebra here the reason that it wasn't the reason that it is isn't port to go this route he is more complicated route here is because we don't want to make Lagrangian which I and systems which on what complicated then just harmonic oscillators interacting with uh which side or not interacting with size is very simple system up there harmonic oscillator and the SEC dagger and nothing else is no military particles on interact with each other they scatter off each other they don't do anything I have something interesting happen we have to make more complicated Lagrangian and the question is how do you make grind rat at every stage preserve the supersymmetry which preserve at every stage of HBO's arms in the Premier undergoing have exactly the same man that's not obvious how do how we do it in a relativistic physics how we make sure that our dynamics is relativistically invariant or translation invariant or whatever wealth but we write oranges and we make sure that the grungy and have the symmetries that we want what we're doing good Lorenzen variants we invent vectors and tend to resign all these objects we have a whole cancer algebra and we build our all of the objects of here build scalar Granzien has to be a scale if still scalar it's made up made up of vectors cancers all kinds of things derivatives of vectors derivatives of cancers of the Lagrangian as scalar the theory will be invariant under Lorentz transformation with don't we're trying to do and now we're trying to do we will succeed in doing it but much more abstract bath basically constructing the methods producing invariant Lagrangian invariant under what variant under these new kinds of transformation which contained shifts of later and translations of that's the Bulls the font haven't paid calculus of constructing invariant quantities in particular invariant Lagrangian Nova managers may be much much more complicated than that simple free field that revealed no interaction between the premier arms embargoes on how would you construct something with firm HBO's arms interacted that you need this machinery for like this there were no I would do that OK so let's talk about that machinery of a bit what can you do with a sober field to make another Super field 1 of the things you can do is square Yukon square super field but so got through the exercise of squaring the super field and that makes another Super field Joel let's just take we have fight fight is equal to wearers so here I have to go far right us data sorry plus sidebar data plus beef tires but right want square at their best when square Mr. right down from another version of a parent they're OK but by square fight times 5 which is just by square roots times by despite square but see what it contains will contain the bars ionic variable 5 squared ordinary number squared Miller contain 5 0 contained a term with data bar class the fate Barre Times fi society where will contain plus faded turned fight Siad Barre trader now from this tournament contained fire these data but they that's a 1st term I get 1 multiply this the next 1 multiply this by this world we had yet to forget bar fight site members are to step aside this is the same as the front that 1 of gap that were again now borrowed it this times this that will give me later bath data that would give me the food over here later barred society sidebar played at this time this what about Western over here 0 because it contains about and that a body found next we have this times this that looks exactly like this year plus far-right Siad Barre later on this 1 then we have a sidebar data for us later later Bora side this one-time this 1 this product Give 0 2 status and this would also do 0 is only 1 more turned pro after plus D Not yet but yet so we 1 Western it is these firms From state of operator that's it all the other terms of 0 was nothing else besides that now I can a them up this combines with Mr. give a factor of 2 is combined with this a factor of 2 and then we have all these last terms here with fate of 8 0 at there is a fight time is D and deep times 5 fight Indy commute with each other so it was just a factor of 2 fighting be paid about data service and here we have a mother Cedarbaum let's bring this state North Trujillo passed here and there past here for actually have the signs right I can tell I do not know if I can tell if it's worked out this 1 should have the same signed a sworn so were work brings Botheroo through if you bring made 3 2 variables it doesn't change signs of B plus they had a bath polling or Myanmar might announce I always like food reasons that had nothing to do with anything I was like to keep the bars on the left and the and it's just a way of keeping a mnemonic for me so I was like the bars a raft of it but the bar on the left here that will change sign that works library that mellow about the 1 over here won't do it bring the rule over here but they shift state and made a bar that's gotta give another sign Mrs. Getty get twice this yes 5
square don't vilify squared it might b a thing that I would want to put in a Lagrangian why not presently good support field by square and I told you before the within the buildup liberal June or super Lagrangian out of How would suffice derivatives of fine and so forth I'm so this might be a fitting in a Lagrangian what I do with within the Corsica part of land that maybe not the whole thing but part of land but
farmland whirling do with it integrated integrated they that they the bar defined the ordinary Lagrangian In other words I pick out the last Turner and integrating over fate and face bar With just picking up the last turn of Western which contains data and bar all this work just to pick out for Western here so if I put this into Lagrangian where would give me would just twice Fahy V my against twice sidebar that would go in Lagrangian that would be guaranteed to be supersymmetric because it was built up out of Super field which had 80 characteristic distinct transformation under our under supersymmetry transformations but this is the way you build up the ground you that this is not a very interesting will grant yeah there was half fight times we don't would be years and sidebar side has no time derivatives as 70 time we like to put in something like a candidate turned to get some kind derivatives so we have to know is how amid build derivative of Super fields out of derivatives super fields you might think all just kind differentiate them that's not the right way because when you kind differentiate a super few you don't get another Super field our so you have to be more careful address are you getting exhausted not really RAM is did that OK so we have to know now how we figured out how to build powers of Super fields incidentally would be nothing to stop us from writing fight queue . 5 fortified anything else too little exercise take flight and pick out Lassiter she would you get what you going to to get it various things were 5 squared d you very get things with sidebar side times body but that's an interesting exercise the trier of fact pick fight to in power if could in power and see what the last areas resume the rules to try to work it out the and part work of the halls of power or 1 those pull out the last turn of fate above faded and I'll be attorney interesting candidates return branches OK now on what about derivatives the revenue a socalled bombed Our won't touch before falling
they but a constrained a constraint on a Super field which preserves the fact that supersymmetric there has supersymmetry up for you I'm thinking about our what supposedly talking about ordinary space when this place we might have to coordinate in white but I interested in I think the myself I really don't have a field which is a function of toe coordinates I really had a field which is a function of only 1 coordinate all 1 combination of the coordinates for example it might be a function only of X where plus Y squared or might be only a function of X or it might only be a function of white would it might satisfy some rules such as the derivative of the function of a field With respect to Hawaii is equal to 0 now that's a kind of stupid thing it says that it is only a function of X Y they introduce a Y dependents in the 1st place will never mind but I might I might be interested In fields which only depend on white now suppose it the theory that I was considering was rotationally invariant what makes sense because they let me only choose those functions of why sorry only those functions of position as possible fields which are independent of wife of course not because of the theory is rotationally invariant a thing which are only depends on acts when you rotate it was suddenly have some white dependent so this would not be a rotationally invariant constrained who would not make sense to say only think of fields which satisfied derivative of f with respect the Why is equal to 0 why not because it can't be rotationally invariant and if the physics with trying to amend rotationally invariant that's 1 way of saying familiar saying it is if I had applying this constraint but given mean there's let's call fields which satisfied this constraint let's call them man Our widow corn Magic Fields worker is a magic fields here now and S is a magic field if therefore if arms if as satisfy this like definition magic field OK now let's rotate corn another Lawrence let's apply a rotation in the form of an infinitesimal generator for example to F we rotate by acting with the angular rotation generators Eyal on just corresponds to our rotation small rotational half whatever is times if still rotating the field still magic field the result of rotating the midfielder was or doing actually is riding a small change in Africa wrote the feel the change in F is equal tho I suppose might Times l firms half way is the angular momentum generator of some kind of derivative just derivative which rotates things Little Bear will Delta F Alsobia magic field Noah was no alarm and the reason is if you take anything which doesn't depend on wiII but does depend on X and rotated it's no longer satisfies Atkinson constrained now Hal what's what's a weighty on drug doing this mathematically would do the let's call this constraint let's call it d thief derivative with respect the officers score White equals 0 this the forum's being magic let's ask whether an closed dealt F is magic server question is what would be it is also a 0 well built the is just proportional beat times at 0 4 definition of magic but what about ID cards dealt the is proportional to L. Times f is that magic but well there would be 1 thing which would assure that times was still matters it would be that bill commutes would be supposing L. commuted would be that would say this was l B. F already assumed that deep times at physical 0 the only problem here is that it's just not true that deep commutes with L as a rotation as a generator of rotation D is a derivative along the y-axis when you rotate calling you come you when you do what you get when you come to a derivative with respect wide with rotation you now do if new lift that tax but remember what these Warren Ellison and Allison infinite possible rotation wouldn't do it rotates the components of vectors right a rotates the x component the vector until why component of back if you commute the rotation generator with a translation generator you just get the rotation generator in the other direction to get these don't commuter routes but that passed the
test for whether a constraint here's a question is a constraint like his consistent with the symmetry as it is and In Paul who constraint Summerfield consistent with a asymmetry it's consistent with the symmetry when the constraint commutes with a symmetry generators and that's a test as the constraint commute with the symmetry generators of erupts now ask those symmetry generators in the case were interested in other molecules accuse go accuser go a blackboard worse From were E. Fiat workers rights we have accused on a blackboard we have the killers for example Q. I think is equal the Debye Bader plastered with I made a body by beating cube are physical could be they bar plus lies favored by it could From let's see if we can build up some kind of derivative amount that tell you the answer is very simple answer for a derivative of the court capital B core capital B but with Bebe ID data that's basically derivative with respect the data but not quite because Decatur those not commute with Q b by dictator faded does not commute with sorry it does not commute with this term he or back and they were interested in asking whether we can find a D which in fact commutes with queuing to block analog of community anger with In the ordinary algebra is filthy which is a kind of derivative with respect data but which has the property that commutes with both fuel into if it it does Phillips a candidate for something which allows us to define a constraint which is invariant under accusing cube OK so the by beta commute with QMQ but now it doesn't commit with Mr. here if anything toward make it Chamero while natural thing to Friday at deliver something like appears he don't strike plus I made a body by DT not work because we know it's just Q and Q dozing commuter and that doesn't enter commuters to bar for hundreds of a minus sign here I would to work it out but you work at a theater is a commuter both of them I think you check that is that would go to a check he that was a problem right I missed identified 1 of these and call the not it's it's a nice Jack the trauma this d and by commuters were both QMQ bar sodas day bar at but missed the big house video but it yes there are there to layers of is of those what does but floor could see remiss commutes with Q was deep commute was QLB but say commuters would be by defender and bike plus a commuter meaning they buy data and by commutes would be by we might be they impact commutes with say the bar paid a bar and commutes and faded barring the use of the Bar of this 1 in commutes service when trivially that's trivial but now I have to check those this and their commute with few Bora and that's a little more nontrivial are you haven't intact commutator from here Phebe by ID data With data but did you also have meant that come from here and they cancel they cancel sole remedy indeed 3 commute with both queuing or was that we that means that the constraints I say let's consider super fields which have the property that deep bar and both both of them was too stronger constraint D bar on Super field is equal to 0 this says that in some way the field is constrained when you differentiated with this kind of derivative here it doesn't change constrained 1st question of policy think we change Body by a supersymmetry transformations Delta Five contains things like Q 5 Q bar fight it contains things like to find Q fiery is a change in 5 was it also satisfied this constraint the answer is yes but see but still Deeb is is
equal to 0 let's try and find that would be barred Q 580 is that equal to 0 question mark we made a little shift and 5 by applying to it and now we want the puts quotes called let's say that it be butterflies equal 0 we could give it magic but actually hasn't seen it's called Kira Nagy super field which satisfies this constraint is called Carl super field see analog of the magic constraint that we talked about before but let's find out who's this magic constraint or Kyodo constraint is something that survives after you make a little supersymmetry transformations if not them we throw away this constraint prior to its would be barred whether few times 5 is magic or Cairo while the answer is easy d bar 5 is equal to 0 and if Q. What can die commutes would be barred which would proved over here we can rewrite this is minus Q D bar fight and that that's equal to 0 equals 0 because we assumed that d fight is equal to 0 salt if fires Cairo than the smallest shifted it when you rotate it but when you shift by the supersymmetry transformations still Cairo well that's a good sign it says what it says Our basically is that this is an invariant to impose its invariant are not on that anything you could think but on the supersymmetry transformation imposing this on a field is an allowable thing to do supersymmetry doesn't get in your way it doesn't a supersymmetric transformation on fuel doesn't violate this constraint OK so that we are that is the notion of a Cairo role super field we are plunging deepened who me the morass grew were riddled with your fault so let's consider then eh cairo super fear that satisfies be barred equal 0 that turns out to say something were worth see Russia easy way to construct 5 days which satisfy this constraint it's kind of neat her work on supposing this was just be by D data bar on Friday because it's not supposing only contain this term he was equal to 0 whoa that's a just say the fight is independent of bar it would say that flying as a former fight plus Cedar kinds of played a bathing this time so sigh barred translator but then nothing else because reveals elsewhere faded bars so if they a bar on fire was equal 0 would be an enormous simplification of the Super it would say it contains only 2 terms of 5 and a sidebar terms later females OK but that's not what we do and and that's what we're doing is be bar more complicated but nevertheless this is almost right so Amoco yours right you right that funny it is equal not but let's call towel end data fight it sorry plight it is flight that foreign is a function only Our Powell in data but what is How how was a new thing it's not to regional current Powell is trying plus things plus ii Vega a bath what a weird thing to do shift time by an amount I played a bar data right that a field only depends on how end later this is like saying that the deal only depend on 1 Kwan 1 combination of course it only depends on how which is people so I paid about beta and say that by not made a bar separately only depends on May the bar through this combination lets proved let's approved any field of this type of of far-right our teach cause I played a bath data and data let's prove that it satisfies this relationship could well Dean Barker said this was relationship be bar is on by ID fate of bar where we not to be by the state of honor in the minors are David D. by DT what happens when you differentiate this with respect to play the bar differentiating this with respect the fate of bar gives us the only place we see theta Boris together with team that gives us plus my kids miners are feeder fiber duty when you differentiate with respect for made it close their a bar here it's equivalent to differentiating with respect to time and then multiplying by I made a bus drew district of Amman now what about derivative with respect the time that also gives you a time derivative and also multiplies but but with the opposite Saarinen any function of the form here which only depends on fate of Moran special way here satisfies D bar on it physical 0 it's another way they've be Boris equal 0 is another way of saying that fight only depends on fate of in this very very special think well that that solve our problems we could start with any fire which only depends on track and favor and then later on just remembered powers the shifted value of cotton so as is fired C plus Our side by side of town of town plus sidebar of towel data and nothing more we just have to remember that he is marked her 0 er we have to shift power war will break pursues to toe it is that the old formulation said let's not yet if appeared sober says this is a former or current super fear now let's work it out in detail its work it out this full of ugly glory full of glory remember what towers towel is T I paid of our data start with this 1 over here starts out as foggy TV but then we see if they'd that a borrowed a bit so we have to put in plus ii leader Boris data kinds a fight . we just we're just order Taylor expansion in shift here we shifted by and I made a about status fighting force I made about fate fight got them what would be the next turn next term I guess would be 1 half I squared which is minus 1 minus 1 have played a borrower they say the size New year
spending under fire but yacht rhyme fish resources Her boy but it yeah I'm just substituting guy was tune he foresight about data now they yeah but it all there was no higher returned because they've played a square 0 pretty Botha and about but later 0 half guess such true tree and then we have plus sidebar of tee times data but then we also have to differentiate sidebar plus sidebar got of TV time state a bath data later but uh 0 bitter-bark times greater 0 7 0 that's all things the Cairo super field is just fine and side combined together plus an extra tourney which has paid about data and coming from the ship that saw Cairo super fielders now it seems like a very heavy machinery are to do some fairly simple things this is what the Karl feel this what do we do it well basically it we can square at a multiplied by its complex conjugate and use it In a Lagrangian let's see what kind of things would happen if we squared it and put it In 2 Lagrangian so we want to square multiplied by its complex conjugate words we have been modified equals 0 0 years 5 but multiply by its complex conjugate that will make another Cairo super field multiply them together and then integrate them with bar so I think always have to do what is right if down next its complex conjugate placed on 50 defies complex typically now flies complex plus ii later bar later times Friday start dark and plus Saeed he made a bomb on a middle terrible chip data did or didn't because but they FIRMR I'm really too tired to figure it out there and I'm not recognizing it frankly it doesn't look right arm IBM after stop you're in f does but not enough punch lines but but plans if to do it right and you get beat wide and be barred RE Multiplan together amid the Graham all the data that a bar you'll get the Solingen best punch line young firm but I think I better do it over next time this is this is more than 1 lectures worth were run out of Our steam deployed let's let's would point the point is the build supersymmetric were granted the where the supersymmetric or brandy and it's 1 which has this fantastic symmetry that you could replace both on formulas and that in fact means that guarantees the firm eons and bows on common in match multiple with exactly the same and will do what exactly these things which we required over Granzien which will cancel all our budget comes matched for me on HBO's on pears then you could prove also of things such as the fact that the ball sons don't get any self energy why not because the firmly on don't getting divergent you don't get any beverages and you could prove all sorts of things that are rather magical and mysterious coming from the symmetry between firmly under bows on particular you could prove orders no no no no statement is true for free field theory of all orders and everything perturbation theory non perturbative the whole works you could prove that of programs in the supersymmetric there are exact cancelation all orders which makes you with these divergences don't happen that Steve that's the punch line and I feel I could punch there will be much but a tour I was teaching this morning in an hour and a half of supersymmetry is more likely work I'm sorry they questions we are good cairo Carl C. age I already know your car or your him several bypassing the got a look and arrived finals from non not Duke could you well not know who will you get prime From what Ranjan correct tell you what the final rules are it would be very interesting the entered look the interesting thing is here is that this is a generalization of what interest in here it is that this generalizing our ideas of space getting Coalwood and we want them the fine diagrams for you are going to fun and Bagram calculations you're going up the slack and say I'm here and the door frame diagram calculations you want there are people who could do that they could do a lot better than you or me arm what's really interesting here Is it this is a generalization of space here it's just kind a
generalization of time and awkwardness space it's going to be a new generalization of space time and generalization of the symmetries of space time that would generalized special theory of relativity to theory which makes up the axes with a bunch of beggars that's what's really interesting about not just any old certifying the diagrams it's 1 which condoms wrote of he presumably very very deep space time symmetry generalization surgery so why would I want to but would you do with that there are no rules to Nonah Olympics soccer mom is not so much the Lagrangian as as these as these symmetries 2 symmetries appeared to be really really fundamental the fundamental generalizations are of coordinate shift rotations and rotation of rotate in a funny way trying to get by commuting quarter net no I don't think this finding diagrams themselves would tell you very much they would be very doll I would tell you this coupling constant as coupling constant would be found end looking at the fine and diagrams you would have no guarantee just looking at the defined environs you would see no guarantee reasons why all these perfect cancelations take place you have to work out diagram by diagrams and or contributions himself energy HBO's arms canceled and it would just be a year of accidents just as in exactly the same as if I gave you the fireman rules were well if I gave you a set of find Rosen started calculating find environs and discovered here amazement that a scattering amplitude always Lorenzen variant but you have to go step by step by step by step each fined diagrams each order of perturbation theory you would start they ask command must be some reason why these things are happening why is it that the power that the scattering amplitudes are always the same in any reference frame and so forth you would eventually come to the conclusion that there was some warrants transformations that theory was invariant under Lorentz transformations I did give you the fund that rims of supersymmetry ensure your cancelations and you would say why I would go back and say we call off the theory has this kind of space-time symmetry is generalizations based 3 sold me that's what's interesting arm I'm not going to take this much further show you want to examples of Super Lagrand gyms and that and will tell you the implications for particle physics a little bit more about the implications for particle physics which are not much more than saying that there are always cancelations which take place and they are super partners we should be there in the experiment . 1 of the most were aghast series expansion but are here see that the use of similar to that is based well enough to know what it was based coordinate and fill out to to expand a bad things you Fourier Transform that's a useful thing to do it's not particularly useful but powerless series expand and coordinate rich is not a useful thing to do on Fourier transform of more interesting and that their gives you waves and so forth power series expansion in the state is thing to do because a candidates they had the power series expansion from taxes would not terminate purges bees from arbitrary a part of it did not fully couplings and the interactions with the ordinary particles are determined by the soup Giant sold all of the terms of the Lagrangian involve a much smaller particularly interaction terms coupling terms like fueled the 5 sigh sci-fi terms they all governed by a smaller number of coupling constants than what you might have Bozo or very detailed constraints among the parameters Of the particles that have to be respected so decay rates cross sections are all lawyer determined the terms of a smaller number of constant than the general theory that you were right now that's particles they'll be clear it'll be clearer of the work half of this year to say that they have made it clear that to is lot that we have he's breaking of supersymmetry a mail we have to talk about the thing is symmetry breaking the supersymmetry which itself is a hellishly complicated story yes put or about Lubeck on now it's an extremely involves story and it's extremely frustrating for some money will like to teach physics from likes to explain and physics this is incredibly frustrating it's frustrating because just as I said it's just a bunch of formalism and index manipulation and symbol manipulation and In the end you you are in the end of the day he even after you've finished with it you have to break the symmetry that
involves a whole series of its own and then you have to put a get-together with gravity it's enormously complicated enormously complicated and I find it extremely frustrating but a very frustrating something which seems very very simple the idea of a symmetry between from turning bows on becomes so complicated so quickly that it becomes unexplainable even the people who have learned all about Kwan mechanics learned the bit about quantum field theory learned about relativity and still go through this stuff would be really cakes a quarter on quarter of this class but a quarter of a firm twice a week supersymmetry class go through and even then it would be so it's a big subject them to difficult subject technical subject and yet at some level it's probably saying something we don't know what it is these layers of scheduled so formal so this this has reached the point of abstraction where I think it becomes unexplainable except Of those ships to initiate people have initiated themselves by going rule in Ireland the subject you learn subject by going through detail by detail by detail by detail and doing it all over again until it becomes 2nd nature and that everybody I know who was learned it has complained bitterly about having to go through detailing a lot of details like that when you in general relativity ill-gotten details of Index index in keeping track of signs in keeping track of details the way this subject was learned by the people and it was like going through the details Over and over again and it's extremely tedious extremely tedious was saying I think it's I thought it was no guarantee that the rules of nature would be PDS that they wouldn't be tedious having to keep track of detail detail that is just what we impose what we would like but we say the theory should be simple and comprehensible and their composed of pieces that we can understand instead it saying it's not complicated some simple rules but the about follow through on the simple rules the through on the simple rules to use them to construct and is just an exercise in frustration are enormous amount of years the people who did this work really did comment Jimerson solved in a kind detailed tedium until the got good at once they get good at wikibook added do much easier my are for the day there will be more thought if oil earlier in the day out there not that receded theory is still slot that the fares that I thought they were cited they said images of each of the transformation of a fire there is good because it helped me to say that the Birmingham is what you get when you pick a both on and that would be generated Q no 1 does it on In some sense the goal from refer me on the bows on is the process all of them on shift being made our wrote that the shift in favor from 1 of you can't really speak about the values of favor don't take on values in the usual sense by this shift operations which shifts played a little bit at the same time of the that is the operation which takes you from a friend comparable with a little emotion in the superspace a little emotion roughly speaking from 1 point of superspace to another point of superspace picked you from firmly on the bar on it is analogous to the weight rotation of coordinates will take me X component of a vector toll white component of back B X and Y component of vectors Ucon visualizing your head the firmly owners and bows arms could be off also often called components of the field that's the feel of the fine the sign sidebar these are the components Of the super field that called the components of the hero and the army the supersymmetry transformations of mixes 1 for the other takes 1 the other In particular it always takes a flight was signed an aside for combination of fine beat so what we just do it His emotion in the superspace which takes a firm Arab on could you picture can I say what that means about these particles and a language that you could understand know the language is not English the languages Glassman variables and and earned this you don't you could I was gonna say this is why we don't teacher the courses like this but there on order courses like this so I think but this is why I have hesitated and work around the an avoided it but U.S. Chery so stop finished in a string theory is not as bad as sincerely string theory is is is is is much easier to understand for Lula
these days us at stanford .
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Walken <Textilveredelung>
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Manipulator
Parallelschaltung
Stringtheorie
Mechanikerin
Energieniveau
Radioaktiver Zerfall
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Satz <Drucktechnik>
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Gigant <Bagger>
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Elementarteilchenphysik
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Großtransformator
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Mikrowelle
Jahr
Ringgeflecht
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Lineal
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Fliegen
Tag
ERS
Computeranimation

Metadaten

Formale Metadaten

Titel Supersymmetry & Grand Unification: Lecture 7
Serientitel Particle Physics 3: Supersymmetry & Grand Unification
Teil 7
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 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 10.5446/14974
Herausgeber Stanford University
Erscheinungsjahr 2012
Sprache Englisch

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Fachgebiet Physik

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