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Supersymmetry & Grand Unification: Lecture 5

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Stecher University by will
want supersymmetry now I have struggled to find a way to explain supersymmetry In the end I took it as a challenge a my own opinion from looking to lay outside is that I have failed challenge but I can only do my best supersymmetry is probably the most abstract construction or but it is incredibly abstract soft Uzi among other things these bizarre grass the numbers which our numbers of all which objects which do some things back numbers can't do the other hand supersymmetry is full of them and we need to to understand them Overbeck I can't about grasp numbers less time just review very quickly and that I won't tell you about supersymmetry itself supersymmetry is a form of symmetry similar in some respects the other signatories other the symmetries might be rotations symmetry translations symmetry the symmetries of spacetime supersymmetry is a symmetry of space time but I'm very very bizarre I want to make it a superset symmetry of space time 1 has to expand the notion all four-dimensional space-time something radically new will really strange we know is that strange new lists of this new kind of symmetry but I won't talk about today OK let's talk about Glassman numbers grass the numbers are mathematical constructions they can be represented by matrices but doesn't help much so I will indicate sometimes by epsilon sometimes by data usually by Greek numbers and all it may be several of them problem so it's label epsilon some times when isn't even number of them we might be good for you personally could multiply that by ordinary members when I speak about ordinary numbers I mean complex numbers real or complex so you could multiply that by Heidi square root of minus 1 over here square minus 1 you could multiply that by something which is not imaginary and fact Yuki combine them together in the complex combination was if is an even number of them often the case that you tube Group them in pairs and make complex car combinations ever so for example if it tool in your problem if only to last the numbers Ukiah Cohen epsilon 1 in Epsilon tool or you could call them Epsilon and epsilon bar in my notation tonight the bar stands for complex conjugate that's a common notation for complex conjugate is to put a bar friends Samoa Her star so that these Glassman numbers calmer times through the
grass numbers which have the odd property debate In commute the squalor of any 1 of them 0 it is not a 0 per square anyway 0 and what times Epsilon To what is my epsilon too firms absorb more will come back in a Glassman algebra and that commuting whereas the numbers closer things are they certainly on our part but there are the which is mathematical also are vs. even the other kind of numbers in such a set up our own ordinary members let's just call it asian number is a glass with numbers the numbers like 7 3 minus 4 x 4 Expos I wiII and have could be complex the ordinary numbers end and those accord even even elements of the grass algebra and art elements of regrets around the rule is In order times and all are Ruiz a product of tool breast numbers Is it even element that means a commute to about things and 5 times and even tho it said I aren't times I was even part job even odd times even is on an even times even is even more so for example without Quebec means fiberglass numbers always and by
commute epsilon with epsilon bizarre the same figure epsilon I with epsilon Janey as always they can't this means epsilon i Epsilon J. perceptual j epsilon I is equal to 0 last summer means that every epsilon squared is equal to 0 0 because they had
die commutator of something with itself was just the square Price of these earlier on elements we even elements Camilo but said in a letter to numbers equals 0 0 this means any kind In the equals and a new head the convention is that even elements an order elements also commute so epsilon times in his end times epsilon only end by commuters commuters on a pair of epsilon so let's be obvious in even years of grass numbers yard radius usually dressed here as members of a book that would be the elements of the rest and algebra autographs numbers right some are making distinctions between your young the other things which you would act on progress numbers the rules are the Alberta OK now grasp another boat which we've stated over here there the theory of functions congressman variables was a function of aggressive variable but say a
function of 1 Glassman variable well what way to express that functions as a power series of epsilon so be some 80 which is independent of epsilon incidentally 80 could either be grasped the number or not rest remember but doesn't depend on epsilon with thinking about functions are particular grasp of element epsilon 80 Busby times Epsilon but nothing after that because epsilon squared is equal to 0 that is more than 1 variable you might have a function of several of them is at stake there are either epsilon than epsilon tool or epsilon epsilon bar in that case you have kept function of epsilon wanted epsilon who would begin with something which is independent of the epsilon so that there could be a score B 1 and epsilon 1 1st beat 0 epsilon tool and then some steep Epsilon 1 Epsilon To nothing after or not because if you multiply it by another 1 of the epsilon to get 0 again it only to absorb the problems you could also like this In terms of you if you made a complex combination epsilon epsilon bother you correct this reform plus I guess we can cause bebop times epsilon plus um plus epsilon kind movie buffs epsilon bar Epsilon it doesn't matter what use the complex the complex are combinations of real combinations either way with some terminates after because of other factors Epsilon will was 0 that's the only songs and a problem if there are more depending on the number of epsilon variables are all part of the series will stop after some number of terms to view them all up From speak out
there are very many functions usually when you write a function of grass and variables you assume that although the terms are either odd or even for example if f isn't even function or on loaded commuting function that a would be even the EU would have to be archives so that beat times epsilon even this be would be odd and see would be even the other possibility is that efforts suffers a crime element of the grasp of algebra the aII would have to be on element B would be even would be are so you could have functions choose which are but nevertheless what they are if their functions about a finite number of grass and variables the number of functions is not very big but polynomials and In the importer broker because that's the algebra of their breast variables and a functional analysis of them well forget perform tour Malissa's just functions what you could write this 19 and epsilon assigned function is just an expansion in powers in just ended the 1st the sight of epsilon epsilon now which is now a structure that make interesting functions other than a couple upon meals now is even a Capcure so grass variables both differential and integral the notion of a indefinite integral doesn't really exist the notion of a definite while you have announced that it exist so just the in the glow of a function of epsilon maybe I'll go through it the next time I don't know go through it be integral calculus of but it's very easy to remember derivatives and integrals are exactly the same thing don't differ but will come not doing to goes about just how you about the differential calculus progressed variables delivered with
respect for beta sorry days rest variable around but faded Newcrest variable analyzed article you made in terms of instead of epsilon yes I actually do know why but for the moment no difference derivative with respect to a grasp variable of functional breast and rarely very simple 1st that grasp of the function doesn't depend at all on data for example of only consists of a constant and answer Iraq for prospective fade of 80 plausible beat website fade at times beat that's just tickled debate just exactly as if you were just differentiating with respect we respect the late at nite did not to be a little bit careful let's suppose that this function he was an odd lot on our even element and even element of grass algebra that means a isn't even OK no big deal but it also means that it is an odd element that means you have to distinguish between beef data and faded be indifferent and their opposite in sign so if B is Alana elements be fate and faded be a different if you want to differentiate with respect data 1st bringing the dependence of the left and then hit it and the derivative disk is that's just 1 of the rules Our fiddling around with these grasp numbers so is much to the calculus of a photograph variables leader of Gaza Conestee what he'll have a function of let's let's let's take a derivative with respect data judge on top of a foe to know this is a complex for a favor other functions which is in a class B say that class so it's court BUT Busby Ba Rb Barclays Beaver data Bar B plus see may have often act as generic situation if you have to status the problem and I've arbitrarily organize them into real toe to toe complex pairs use they want who also but see what says the delivered over respect 8 0 sets a 1st class now we have to be careful Is this odd or even function here so let's say it's even function just be specific 80 is even was beat arm so that is why I hit it with respect but the river respect I had better bring the they the left weapons would bring a bit a left my best electron would be minors we borrow now the derivative of data Of the baht with respect the data as you but was a different variable data we could use data once they get to it doesn't matter but the derivative Our 1 variable With respect to other variables 2 0 so you 0 from he now about the remove spectra fade out of this term he yes plus seed earlier yet they above all my airborne my later Bob because a the hit with a derivative we 1st have to bring the city through to the other side OK and that cost us a sign of miners seat bar that the example of a derivative of a derivative operation on grass and variables you're the only thing you ever have to remember it is real order things so that when you differentiate your always differentiating something that's right next to the Red a pick the April theorem which is important to the because I had our yacht yes both these yes but
because you had baby the fate of odd element on here give your minus sign and you had to bring the data they the bottle over here all sorry I'm in mistake no no right with ASEAN on but it's right Soon the Idaho horror even element Red see then see must be either right now the whole thing could be an odd element but suppose the whole thing the whole function here was odd element in that case in that case they would be fined would be even would be even now and see what the art but get the different chair again differentiating a gives us nothing but so we can bring the thing that has to be barred because the B bars is even element and then differentiated so this time we will get Class B Bar not miners and over here we have to bring fate of fruit to what odd elements to get left that will not cost us assigned each time we bring it through we get a signed bring it through toward elements no more signs of that all plus see them marked I did a step to do the bookkeeping carefully as amazing that these things a useful we do not know little yacht this is not see say threat functions here's a sea change that red lips astride functions they were born were 2 point that a sign that might be less if it be a bar now so tough but a significant that diseases reversed was a significant they did that as a bookkeeper a bookkeeping device for keeping track of friendly answers were the most significant other than the significance we will look at as we go on In which case is on like a function of our members well now we're sorry if teams in order of Baden and Baden bar that I should write penis Steve faded faded see data by data it is my see made a bar orders by us it's a real it's a mathematical will look number and the kind of number when you switch toward the art elements you change the time if is also in order element in you bring something past seat you also change the size of a shady area is a case . at threat as any of these manipulations do not change ought to even but they can change the sign of the object of a busy he mine the complex they could be complex numbers they had that if they happen to be even if there is even it could be a complex number but it could also be something else it could be the product of an even number of epsilon or favors but can still get a card if this thing if we have a 5 we have 2 of which if grass field as well their only 2 elements will always include the numbers around With the order elements so typically we might have some finite number of odd elements but also you have a complex of real numbers that you could define more sophisticated exhibit defying things which only have a finite number of ordinary numbers also are just the integers modulo 5 and that were also make grasp the ones we're really interested in the ones that have complex numbers and some finite number of grass elements that will call From time to time McCormick from Poland From a suit the secretary there was a lot of these groups other than that he for 1 of us saw it as biased and days that version was not a career but all is that he period that said that if are the the roles of simple whenever a Glassman element goes past over Glassman element it changes the sign now in fact you could also think of derivative with respect to what data as a kind arrest warmer but permission to remote into resting on the fact start with ordinary functions of ordinary variables but such as a function of a single variable and consider the differential operators derivative with respect to X and multiplication by X Lozal 2 operators and have the following properties the product which way is the the yacht with smoke but by IDX times X minus X B by exodus ever seen this thing before that's the commutator of X would be by Beebe IDX with acts of because closing quantum mechanics of a cousin quantum mechanics because the momentum of a particle is represented by an operator which is miners ID by DX What is this commutator b by extends X minus XT by DXY Orlando above the anyway you take it you apply to a function any function and you see what you get lots let's but begin with the 1st day of the 1st and gives us derivative of Inspector X of X times FOX and that's the sum of 2 terms for the 1st time the derivative hits X News 1 of his just gives FOX and the other time derivative hits Soto plus X fund got f climb of acts where
Prime indicates river the move committed acts and we have already been here Louise hit we've X know about the opposite the opposite order we have access times derivative with respect to x times FOX that's just ex parte aims at prime now mixes on the left a derivative of X and that's just an next times of South the difference between a user is just a difference between this that's the X times if Prime cancel or we find out is that this is just people Everex that all welcome be rewritten in a simple package the commutator of derivative with respect X with the X is just 1 while not because it does nothing to it just acts on a function and his back the same function so this is a simple feel about the commutator derivatives with multiplication by rejects what about derivatives with respect to grasp the variables you might guessing you might then you would be right that exactly the same thing is true 0 except for 1 thing for and by let's check that out force just check out from a function may force be fair Chu Our even case you could work at odd case Our where aid isn't even element and b B is alive that's the case although let's 1st hit it with Our later Dubai data and they will have to hit With the opposite order but nobody but if they don't had say gives what 0 0 about by ID data can't be fair he is an not elements so we have to bring this around to the other side this will give us our forget this bomb with this were just give us a might best be led arrive sorry might be letters be free but never really had a look labor Gary paint solids might just be be data they all left but this is not so we gotta be freighter B is not elementary echoed done this right now it's fine it just give me back be the racy wait for pig Brett so that's the 1st hurricane erase now about the opposite order d by did say that if a death times it aII plus beet Beta Theta times they just gives us timetable was about stayed at times B times there as well I could bring the fate so here and then I would have the product of Tuesday's is product of 2 phases 0 so this is just equal a derivative with respect fade out of at times aII right which is 8 so us oppose elect a head leaders I had to use that I get began by commuters Our fate would be by 4 acting on an arbitrary function a plus figure goes back a people's be there's an 8 they so this is 1 of a glass derivatives with respect to grasp and variables be hit by commutate out of favor with bifurcated is 1 West orderly numbers or ordinary functions of ordinary variables it's the commutator a derivative with respect to coordinate is equal war so I shall do this for a variety of reasons but 1 of them was just to show you how to manipulate these objects and said the only Maliki have to keep track of his past 1 on element pass mother changes signed otherwise it's just like ordinary algebra much simpler because only a few functions and not very many rules the because there is very much a fab rules about a does you couldn't makes functions but they're really really it there really not normally you don't allow makes functions are an expression should either be on river our nuclear you can investigate stop playing yourself what happens if you have mixed functions but normally pay an object is the real sorry is Auberry if what is that that that but this is ordinary function ordinary functions even function I was reminding you of something here this is kind of the commutator of X with PU right this is this Kabuki or from multiplied and minus or than this would be the commutator X with Pete is equal to or like a guy job but that number like 1 or 2 is a limit to what company could have on Germany's number if number but there's even a really number of cracked France abut that is what a relief Glassman algebra Grasmick calculus the only thing that I didn't tell you as I said was How the Bulls work with value right now I maybe a work out for you but not right now are being integral be I will function as data the same thing as derivative with respect data as ever say that I won't tell you why right now if we ever happens that we need derivatives I will show your stroll it's the only rule that allows you to integrate by pipes you insist that the integration by parts formula be true then 3 it necessarily true that they said derivatives and the rules are the same
so that makes chapter was very very easy right now let review for a few seconds after a couple of minutes the idea of Symmetry Groups and then we're going to come through supersymmetry symmetry grew upset just quickly review we have to look elements I'm interested in continuous groups now and let's call group elements GE G survivors group element I could and I hear just in indexes you want to index as struggle last time the whole structure of the group is encoded in algebra of commutator steel algebra globally algebra Hooley algebra of commentators on these genes are the authorities jeez are not group elements fees jeez other generators of the group generators of the group infinitesimal elements unitary group for it element very close to the identity is 1 plus ii becomes a small number this epsilon is not at the moment progressed the number of of times G and they collection of independent linearly independent generators the case of the rotating group of generators are just the angular components of the angular momentum the different components of the angular momentum the elves LXL wife for more complicated group still more generators but the collection generators and be important thing about the generators is the commutator so that tells you basically an infinitesimal version of the table of multiplication of group elements GIG commutator of 2 group elements will have to do with it has to do with it group operations about different acts sees speak and then only doing what you get leftover
would talk about that might be a general rule is that the commutator of too good element is lot 2 group generators is another generator the standard formula is always an idea and there's a set of constants but columns see you might JKE times element GKE that can't the obvious I and J element misses some overcame course repeated indices that dictator of 1 generator with another is a linear combination of generators that the structure or all continuous groups and these constants senior his numbers they have some symmetry properties recalled the structure constants of world where are they for the case of a new if they get limitations epsilon for example but in general but just as they have some symmetry properties for example left-hand side the commutator is always an isometric I J. GIJ GI complicated GJU changes Simon if weapons so and by symmetric under interchange right Jay and the fact you who discovered by a few manipulations that sent by symmetric on the interchange of 2 of them but that's not important important thing is the algebra closes that means that the country caterers Our generators give back generators get that's it fundamental to the character of what a good there's no jeans typically sensible symmetry groups are always permission operators the Commission matrices that means they have real Eigen value that does not prevent you from putting them together and the complex and are complex combinations but we give you an example of of what we studied in the past take the angular momentum components L X RO Y N elves both represent angular momentum there observable quantities consumers in in the real world and of course there permission because observable quantities are mission that does not prevent you from putting together combinations like Al X plus or minus I L light given these things were raising and lowering operators whether they do they acted on stage that good when they given the value of LCD we called it in and either load in by 1 unit so soap I just point out to you that it's often the case that we combine together at the complex combinations form of various algebraic purposes that the values will toss In putting them together and the complex combinations that doesn't change the fact that the original objects were permission 1 last point before we get to supersymmetry is that we have a real symmetry real symmetry our of a physical system is also applies something about the connection between the generators and the Hamiltonian In particular it says of Colleen of the system commutes were the generators commutator with the generators is equal to 0 at another Of saying that when the generator acts on state it doesn't change the energy we prove that last time that when you have a cemetery it means the generators when they act on state do not change its energy that's another way of saying that G commutes with age but also says something else it says that the kind derivative of the generator it your as a car on mechanical equation of motion it says the G is a conserved quantities so I put those things together is reminding you of them because without a play that with generators all the ordinary things of physics always represent physical quantities which are even elements of grass or anything new measures for you measure it can get a number the number is the number it's not element of goods on autoimmune progressed algebra so firmly on fields ot but you don't measure for me on fuels for themselves you can't measure the fuel for meals you could only measure the square of the field before meals or something like that the things you measure nature are things which you should think of as even elements that there's any reason at all to be thinking about arrest algebra theories the symmetry elements rotation symmetries all the normal symmetries of nature and commutator the above commentators and not a not infect come sober symmetry is a cemetery where the generators are 5 elements arrest which means everything you would say about ordinary symmetries is exactly the same thing except to put in by comes at a price on only give you know an example but butts event 2 generators G Q Q and few dagger what you bar does matter complex conjugate as I said it's sometimes profitable combined generators of groups and complex combination supersymmetry is nicely expressed in terms of complex combination is intended to be odd elements on elements means that the entire economy made of Q with itself 0 accommodate Q Bob with itself is 0 that commutator painter of Q with Q Bar might be something else just the same way that when you have any other kind of group when you commute to generators you get a 3rd generator you may find out they when you and die commute Q with Cuba are you get something else would be an even Oren on element of a the out of it and so that's on the limit construct for your example it's constructed out of fermionic fields or basically something simpler than that I think we maybe even talk about it but both of the details because it is the basic the basic super similar missed the most primitive version of the supersymmetry is the most primitive version of it related to the real supersymmetry the same way that quantum mechanics and quantum field theory is related to the harmonic oscillator that subprime but if you know all about harmonic oscillator as you know a lot about quantum field theory because quantum field theory is just a collection of harmonic oscillators same thing here very similar but now we have to introduce tool kinds of creation and annihilation operators we can think of them as can even be even are just harmonic oscillator
creation and annihilation operators a plus or minus houses remind you what the algebra of these things is the commutator of any class with a plus is the same as the commutator a minus with a minors is equal to 0 and the commutator of 18 minus with any plus his eagled 1 that's of creation annihilation operators and all these or are perfectly conventional harmonic oscillator raising long ago that also introduced a collection of fermionic creation of annihilation operators say exact rules except that coming come a tale of 2 creation operators is equal to the comic tale of 2 and apparatus and commentator excused me physical 2 0 and the in dichotomy aII creation operated with annihilation operated Fernandez is again equal toward incidentally this is a C plus squared is equal to 0 Miss as the like squared sequel 0 only 1 creation operator for friendly on only 1 creation operated from both so I'm not talking about anything as complicated as a quantum field we could just be talking about up for me on or bows on a West it see that there are it's not what it's about is on near 1 of the 2 of them with 3 of them were not worried about motion just a question of bombs on an arrest how many of them are that's control but we could create annihilate them or fermionic rest my word about their momentum OK now I will construct the U B Armed generators of a Super Bowl or of the supersymmetry let's suppose these particles have amassed but suppose they have amassed any energy is just the sum of all the masses of oil particles MC squared off interest the energy it's just the total number of firm eons times the mass of early on but the total number bows on the Massimo if unless firmly on the buzzer happens to be the same as just total number articles from eons or bows on time the may of the particle so let's call the article and Q is equal to the square root of Banda Aceh convention of course putting a square and there's a convention times at the creation operator of buzz on time annihilation operators offer me aren't so what do we talked about this last time of this remind you it remove the fermionic is when there and it replaces a prior bows sockets a thing which takes a fermionic Somoza on likewise for this 1 accept my quite likewise squarely in Thailand miners seat plots but I forgot to write down something we didn't discuss the crime figures Of these days with the C B a these are even the seas are the rule is any 80 whether plus or minus commutes with any NEC the commutable themselves what that means in particular doesn't matter which what're you could take us odd element to art elements and bring it passed redevelopment was likewise here these this these are the generators Of the supersymmetry algebra do I would like to know what the the can't be impact commutator of Q with Q Bar as the analog of knowing what the commutator are group elements they tell you the structure of the group so that ceiling and figure out the analog year please supersymmetry operation I said what they do what they think is space they don't translate things in space may simply take of Leland and replaced by a bows on all of our users must look at me and I can't repay its work out mathematics and of southern hard the OK and that kind of thing is not hard but I tried doing it in real time will surely make mistakes stand here puzzled so I will use my note OK hereby commutator of Q with Q dagger what is that where officer as in front of it Q. times acute gaggle haven't queuing outside QEQ Bolivar kid that P Q borrowers the different QNQ bomb in and that it will have Q 1st book you where which is a plus minus Q and a Q Bar Q Boris should be a minus plus that Q. times to barter but now I have said Q. and killed the opposite cell and here last nite miners because we're talking about in that can't be figures the opposite 18 miners See Fast Times 8 plus C minus report that ASEAN by commutate this times this plus the opposite order again at work on a root beer 1st of all you could take days past the seat right Yukon and on Eller and even element right past cost nothing so recur write him and Tripoli aides together overseas together at a plus a minus at times seem like a C plus period over here it's plus a minus a plus see the path C minus face so far nothing very interesting about it but now only and subtract something where he is right that we care for each rats it said it is either bucket you cyber bracket Ross scare Rite Aid what 18 miners see classes C minus but not edited about what is going on getting things are trapped miners a plus a minus steep losses seen OK now would be the 1st of all this plus hair but over the contain a plus and minus factors off but there was a multiplier multiplies seem like a C plus plus C miners with that as a commutator received bus with C minus right below that 1 song the policy might assume I plus the puffy together making and saying 1st saw contained in time is so that's got twice 8 plus a miners strike rate as the
1st about the stand this has see plus C minus policy minus again but it has a minor say plus among it's a plus they might have it has the commutator of 80 miners with a was at 1 all this comes down an and plaster glossy might well-timed twice somehow Syria out not got to does not too is there now ever get not to but not to not too good because I hated that too OK was a plus they might have been hallmark oscillator language but it's the number of those on where well-received passing minus it's the number furry arms so we can arise that this is equal to the masses times is the number of bows on plus the number of Romania no I have assumed that the mass of the Fermi arms in the bows on the scene and assumed that we'll explore what would happen if I didn't assume that I did I would have got a got worried about what is this expression here we're talking about part of what we're not thinking about promotional particles were talking about particles arrest offer particle arrest the masses equal to its energy and so this is the a particle and total number of particles that energy of the system which can is quality energy or the Hamiltonian so here is always discovered we discovered mad didn't buy commutator of Q with due Bora simple system is Hamiltonian 1st of all the Hamiltonian even element Is the Hamiltonian and generator of the symmetry group yet generator of tying translations prime translations are a symmetry of systems our work time translation variants and the Hamiltonian is the generator of time translation let's finish up let's finish up the goat ratifying out we now hear what we've discovered is accused by themselves are not enough performed closed algebra when you came when you hit that commute accuser themselves you get back the energy of the Hamiltonian what happens if you and that commuter with acuity Bentley that's just square if that's just Queiroz to go up Top Hughes squared will have a seat to see through C minus Q squared will have a plus 8 plus C minus seem minor but seem minor squared is 0 so the impact coming later Q. with itself is 0 I'll be die commutator of queued dagger with itself is 0 0 0 was OK on the right-hand side commutation relations program right what about what's missing what happened which pieces of EU algebra I considered 822 right we've by is that commuting QFQ with we found out that it's got to to contain the Hamiltonian now this is very odd that these symmetry group not only contains these operations which take Leland Ceballos on board also contains the Hamiltonian which means kind translation a very interesting OK well but she with age I'm not going to do on the blackboard normally to you is an exercise to do all they have to do is age commute with fuel just show you raise you wake think about answered Q. What a 2 0 and commutator of Q which is 0 not because it's an on or sorry about him but you don't it by commuters are would even you commuters are reviewed collated of it which of happened and it is even ages even ages the energy you wouldn't that you would like a system whose energy was a grass memorable but what happens this is 0 now it would not be 0 show you worry it would not be 0 if we masses other big bull bars the and not the same so share our conceded 0 yes and actually only if the massive both arms the mass of the formula that this is we've done this manipulation many times you pick you wage minus age you and your act on any state which has a definite energy at a state which has do their duty look where on what this Q do Hugh either removed I can remember this Q removed from Iran replace a bows or the other way whatever it does it takes 1 particle out inputs another particles in exactly the same may ask if the necessary so for the moment that the masses of HBO's on in the Fermi under equal than what's the energy of the state that you get when you act with Q on a stated that energy synergy because you've just taken 1 particle out and put another 1 back in but particle of the same maps pink so this must eat h acts are QED registers me easy when the EU times White because skew times isn't eigenvectors the energy on the other hand what about Q. would each on you well a tax on ease forgive just be eat their words it just gives us a few times times each visit was saying careful but always true with Eddie have an operator which acts and doesn't change the energy of the state always commute of the Hamiltonian things which commute with a Hamiltonian on things which when your act on states don't change the energy this object as long as the bows arms and the firm eons have the same Mets then Q does not change the energy of the state and if that's the case then Pew commute of the Hamiltonian well 2 particles didn't have the same man not just wouldn't Camille and and you wouldn't you wouldn't have a nice closed algebra you get something else on the other hand side here Our which toured eventually make a mess you go that similar to erase that the chief right out of if U.S. a question or your Quebec perfect the that that you may
start of Haiti was like that age got any said do that because of that she would also like to say that the dead alike flag was a complete abruptly argued but the again each to acting on daily eigenvectors energy any eigenvectors near why those Kewaunee have the same status as place places for a say exactly the same right OK the that that so cute times he is eigenvectors 2 as well as a plan must must say it again it is William vector means it has a definite energy so I guess that Q. times he has exactly the same at but they have an indefinite energy so some eigenvectors each with the Eigen value eat so we can write that a here could just be replaced by heat times QE over here this time it's even easier at H acts on EDP each time you write girls say they do cuomo E times you wealth it the way I would time it does once he decided they did not find the right of a fight that he placed operators his area not the budget that's right that's right I just was being a little bit of that debt ah but you see quest eases the numbers so it can go past a Q because now and this is equal to 0 now is what other element of the argument he noted in make which will make now this is true for every eigenvectors of the energy and if the eigenvectors of the energy are complete basis of states then if something in it the object which is called KUK for object if it acts on a complete set of states every state in the complete set of states to give 0 there's only 1 conclusion Katie is equal to 0 so anything which gives you roll on complete set of states 0 so if eigenvectors of the energy are complete which they normally are David follows rat commutator of Q with H must be equal 0 that also says that he was a conserved quantity secure was a conserved quantity but odd element of the algebra has an odd number of on 0 it's a thing it's a thing we haven't seen before and the work it it was also true that we haven't seen before it is a group of symmetries which takes what kind of particle pull another when you commute Don you get something wedge in this case the Hamiltonian has to do with symmetries of space time preparing symmetries a space-time mix is up internal symmetries which changed particles 1 until another With rheology which Is the generator of time translations so this is something entirely new of course not hiring goes back 1972 was 304 like bad but but you use rather than usual that's close supersymmetry not as another way to think about which is even more exotic and more exotic misses the exotic gets more exotic the more exotic way is to say that there are no dimensions of space I know a simple example would doing here there is no space is only kind is only the energy there's no position where we put the particles is not translations symmetry theories kind Why is it time with his energy this time ridges remind you what the connection between the Hamiltonian time news was a connection between Hamiltonian past a a a a a a a a a a toe in 2 ways we can write that kind derivative Ava state vector news fight times of time derivative it is the Hamiltonian time In fact that's 1 of the definitions of the Hamiltonian is thing which elect mistake to shift the Times another way to write the same thing 5 This is form of an infinitesimal shift or differential shift of time but if you want to shift the time by a non differential amount you write that each of them I each now shift the time by an amount that dealt with the court he workload dealt because later on more to be a small amount that EDT on society it is equal to try exercise at times he is able decided what people's but the relationship between who is just to expand his just to interview you shift by Delta by differentiating and if you expand out the seat of a minus IHT for small Delta you get something proportional age Salazar really the same statement that whatever you have a Hamiltonian you have ruled up state vectors translating them in time so this is on its the in Bangkok data of 2 would-be supersymmetry generators amounts to a small translations time it's mixing up space time with identity of particles of operations which makes 1 particle toe another we operations that makes 1 particle with her mother for example our these symmetries but of color or I still spin they mixed protons and neutrons for example isospin symmetry takes a proton to neutron Carlos symmetries take red quarks the blue corpses before but nothing you could do would have no combination of those symmetries and a shift kind which space said the bill was based time they just shifted raid the blue to green or are up quarks bound quirks but never anything to do with space time here finding some kind of very unusual symmetry which when you don't die commute cool accuse you get a small shift of Korean there it is a way to think about Paris where the supersymmetry generators are shifts just like the Hamiltonian itself are shifts Of course work but not ordinary coordinates what kind of accordance with you guessed grass court grass court so why
would have thinking about supersymmetry is to increase the number of coordinates of space-time this example was only kind based on this just kind of going through in detail I'm very very simple supersymmetry example our just show you how it works but let's that's go as a protester functions it could be the wave function of something or it could be a field operator of some kind let's call it sigh the just sigh of xxx I would say it was the way functional particles as a function of position but I'm not gaga directs ongoing perhaps a sign of leader and they bar and of course it's also going to be a function of time we cannot also put exit here but we won't from now this is the wave function defined some kind of strange super particle at located at congressmen call would and they bar not because that's the most bizarre thing you ever Bruno and Berra idiots Our very strange let's imagine some symmetries the 1st symmetry is good be gold time translation growth he goes the T plus her that what's the gentle and of course what happens what happens sigh when you shift feel little bit words Webster changing society when you shift feel little bit well of they have been a bar and feed rose To sigh I say that later bar feed last the derivative of sigh with respect to tee times Delta right you shift to you a bit that differentiates it and multiplies it by Delta ordinary stuff Mr. later and fate body and nothing that they a bar argue could remember that derivative outside with respect CD is related to the Hamiltonian acting on society what I would not so this has to do from multiply and abide by all I can write this as the operation of the Hamiltonian outside at this Of course symmetry of any good quantum mechanical systems time translation barriers and I I haven't done anything except tell you what a shift of time does put a function of time but now there are other symmetries if if these were exit and not leaders you might have space translations cemetery they would be associated with derivatives of with respect the space the question comes up is there any kind of symmetry which has to do with shifting the natives Israeli notion any kind of possible mathematical notions of a function which is a function of time favors but where a shift of data has some nice interesting mathematical meaning sold would only to invent cemeteries where a goes they never class Epsilon Data is on variable progress algebra so was epsilon otherwise you wouldn't be allowed to do this hour we might also allow state of barter do something for the moment with just leave a bar out of it they also at the scene kind at the same time we shift we're a shift feel that he is very low tree plus what fight made about our plans Absolon now that is very weird thing to right now this at at least at least this isn't even element fate and epsilon are both our elements at least this is an even element and I was in development what on earth could mean very hard thing to write down but let's pursue a bit let's see if we can find out what they would be generators use which does that what happens to society when you shift played a little bit shift on matters of Barbara Keep it simple just shift played a little bit and shift time by the same amount what happens aside sigh those deciding Class B and know that you shift made a bite of score epsilon delivered of sigh with respect for beta best as the formula when you shift an argument of a function by a small amount of you differentiate with respect to the argument that multiplied by the amount shift and then also shifting shifting time by this amount here's what's Rhode Island it would give us some miners but kept track career on the eve of elements our Epsilon Data but he society by did team I got here I set out a shift time a little bit so that means side must shift because of me shift of time how much shift it shifts by these IDT times the amount shift the but this should be a I hear from rump steak and but this doesn't make any real sense so obviously we can normalize because normal mathematics Butler following were following and seeing what we get is outside is what let's factor Epsilon out what's factor epsilon out of it here's 0 yeah if we would just shifting time we call the coefficient fewer of the small shift to called the Hamiltonian acting on society at current factor sigh out of this and think of this as an operation acting been on site the operation I think I'm sorry let's give it a name that give this operation differentiating with respect the fate of ME shifting this abstract is really very abstract additional and at the same time shifting time by an amount proportional potato bar let's give that operation and to generator of something to generators infinitesimal generators something and they got a suggestion last call Fred about Cuba it's an odd element because it derivative would respect the figure Discover Freedom body what's Kolich your cue ball through I think there's an idea Q bottles its complex conjugate it has derivative with respect the fate of violence because of favor he said and probably a change of sign of
so the same way you can then Q which is just the complex conjugate guess what happens if you take Q. What in Cuba and hit by commute that could you guess what happens with the right let's let's write it out with steak killing Q Bar and die commute him so we have commutate a bomb Beebe ID data as minor circuit minors I think body the intercom commutator of that will be by Bora I'm I'm not gonna try to be careful about the signs are well screw amount that's just forgets beta bloody but see feral we get the end I can't dictator of deed derivative respect the data and derivative despite the fate of I didn't tell you what that was but can you guess and I come derivatives of failure it is very much like the commutator of ordinary derivatives 0 commutator of ordinary derivatives of 0 commutator of 2 different derivatives with respect to grass variables also 0 so this time he gives a 0 Wednesday and die commutator of thing about with data it's also 0 and die commutator of any on element any on element is also 0 0 and the derivative played a T years that just goes through from arrive it doesn't change anything so they commutator of this 1 with this 1 0 0 but what about the intake commutator all of this with it contains the entire commutator of Beebe ID data With they die was look just Debye Decatur with data we work that out a list blood of what right where not begin by made of fate above artists stable with Bahia and then there is about would be by favorite but of them give 1 Watson net result the net result Allah is debate as I gave IDT ID by the full and by car mutated here looks to me like as a factor of reverse screwed up some wasn't here should ever factor to wear probably the definition from definition of this should be half of Q by skew the report the yet by commutator of this with this is just minors I things might sigh by was minus ID by H here's a representation of fuel and Cuba are now tens of creation and annihilation operators but operations In a fictitious new kind of space time that involves commuting coordinates its the generators of the symmetry transformation which shifts time at the sale but had I a peculiar amount paid about times epsilon while you later by America on the other side of this Is symmetry structure was geometric structure a generalized geometric structure involving spaces which contained not only ordinary quarter like X and but which contain grass coordinates which are very hard to get your head around but must share moving around in this space 1st of all entails moving around in the X isn't he heat the generators of those transformations are just momentum and energy just as you would expect but also transformations which move you around in this latest based in those transformations of supersymmetry generators that's called superspace symmetries of course supersymmetry and based is generalized space where you think about fields of functions and so forth hard in this way is called super Spurs that I will stop you because I think I probably flooded you with information and they say that is a lot of factors to toward miners at ID by duty right it comes from community be Biaggi figure with Bader getting BID by GTE from here and & Co hereby commuting data Bob would be by data ball which gives you a little warm and debug use of the desolate small as he left at what was meant by commuting parameters squares 0 so anything to 2nd order in epsilon was 0 right so you'll get the think of it by commuting a small right right that's a good question but I would stop here because this is a lot of information but they others said is supersymmetry 1st of all is a symmetry which takes from bows on it can be represented at the simply just in terms of operations which replaced firm eons with both arms but exactly the same algebra exactly the same have all the br unusual thing here is a pity operations close only if you include the Hamiltonian that's telling you that somehow the symmetry operations contained not just the mixing up the of HBO's arms but when you want when you do it twice annually and I commute then you get a translation Hi that's me completely manifest here by saying there's extra dimension to expedite mentions include these by commuting dimensions and the supersymmetry generators translations of the latest based together with certain translations of so it's a rather marvelous structure it gets even more marvelous when you make crude spaced out of the curved superspace knowing you make curved superspace general relativity which is called supergravity it's a rather spectacular structure and as I said at but somehow extremely deep generalizations Our would Mary space time on whether it will turn up an experiment that another question I think will quit now because Hugo 1 we were going to go on but we are the ones with the little more time on mathematics of suspicion tree but the 1st time being think but probably not a thing what little traces should not get out of way out it was centered just the bookkeeping device keeping track of and both times In another since it seems to be much much deeper than that for more
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Titel Supersymmetry & Grand Unification: Lecture 5
Serientitel Particle Physics 3: Supersymmetry & Grand Unification
Teil 5
Anzahl der Teile 10
Autor Susskind, Leonard
Lizenz CC-Namensnennung 3.0 Deutschland:
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DOI 10.5446/14976
Herausgeber Stanford University
Erscheinungsjahr 2012
Sprache Englisch

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Fachgebiet Physik

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