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Supersymmetry & Grand Unification: Lecture 8

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get back into our favorite subject supersymmetry I started it and I'm going to finish today but I really showing you the mathematical structure of supersymmetry in 4 space-time dimensions we did a little bit with just kind and then I'm going to show you How quantum field theories are constructed just give you an example of the construction of a supersymmetric quantum field theory and show you what the supersymmetry gives the findings diagrams and how the fireman diagrams of our work a little bit this is for a simple model supersymmetric few period is not the most general supersymmetric few example our more general examples of a follower very similar patterns but take this simple example showed you only had 1 direction of space time and was just kind a completely ignored space end for example we wrote their own relationships between the Super generators the Q and a Hamiltonian the energy but they energy belongs together with the momentum and a single for vector how could it be that the dollar's only a relationship which involves the energy of the time component of the momentum on the right-hand side must they be generalization of that With more Lorenzen variants where what's on the right-hand side of me and a commutation relations involves not only energy but also momentum the answer is yes it must be the equations have to transform a sensible way under Lorentz transformations 2 tools get a bit of a feel for it or not In their great detail it's good to go back to the Dirac equation but the Dirac equation in a particularly simple situation if they thought a Dirac particle wore masks lawyers it would have to components which would be uncoupled namely left moving the left of the left and the right spinning our on left them right spinning means relative to the direction of motion the great right hands grew left and do would be to let them and ELECTRONIC course does have 2 components for 2 0 left him right a partner buzzer particles on matters you could have a purely left-handed particles that did not have the have a right the article associated with it no 1 that's choose interesting reason why massless particles can have the property Of only spinning in 1 direction but a massive particle has to be able to have to spin states and the reason is kind of simple on let's suppose we had a massive particle which could only spin the right the right means right enrolled sort has momentum and angular momentum twisting right now and moving it's moving along but now I decide to move faster limit its spinning this way but it's moving that way original was spinning right hand and moving along with some of enemy moving faster than it it's now still spinning the same way the has left him it as left him so a massive particle because it was slower than the speed of light you could always out run it and look at it and I left him particle will be become a right-hand particles to you've got have you can't just have left him in the particles but of a particle is messed Lewis and truly Mooloo speed of light then there is no way to turn a right-handed twist and 1 of the left-handed twisted you could try a low-end strong rotation which rotates at always around but that takes they write hindered won a fight just by virtue of rotating my arm around the doesn't change my right and my left hand OK so particles which more with the speed of light can be Munno dexterous over what the right word actually do know what the right word is the right word is Kiah world Cairo's means spin is only in 1 direction of work Cairo's CHI RAL but prefers that having a handedness and I think what the Greek word for hand Cairo welcomes becomes the Greek word hand which is Kira Kira so firearm error yacht polls mood did is with from article is not quite was to be for a particle is held is just what kind Rallidae all share a steady hand the ain't it he came off got helicity and car rally a little bit different kind rally has to do with gamma 5 helicity has to do with about P unwarranted explain what's in about years in a minute and is only for massless particles that gamma fibers equal would not be not quite the same thing but on the case of massless particle where are Dokic this is particle has Hey half as many states at speeds states as a mass as a massive particle a Dirac particle describes the massive electronic for example and the Dirac Field has 4 components the massless particle romance list Kiraly particle
let's say left-handed candidates a standard the code of the think about left-handed particles both particles or the fields to wave fields for most of the Dirac type fields have only 2 components and the courts I want him to components spinner sign 1 site to which we can also write side Alpha and they satisfy Dirac type equation the Dirac alike them Dirac equation also recalled how we equation and have no Massenet and it looks like I these IDT boats which appear 1st as a Hamiltonian for the particle and Hamiltonian is signal about Peter letters at me it means that its bidding in the same direction as its the momentum its energies positive there it's spinning in the opposite direction of its momentum its energy is negative but don't negative Potter negative energy particles that are negative energy particles would replace them by wholesome fill up the sea of negative energy particles or what stuff sold on left-handed particles have seemed about B positive architect now you can convert this into a wave equation H is of course by T. P is minors I be by X there are 3 is about simplified miners I'd be body X equals put an equal sign here and there are 3 components services X Y and Z and segment are a set of 3 matrices the 3 Polly matrices so we could put them in he minors ii stigma and this really means cigna XP by DX but signal YD baby wipes it was Edie by Desi we could even rise just got Beebe IDX and all this acts on society side and that's the analog are these simple version of the Dirac equation shift this that the other side and write that on a peace IDT and Coke C X nor for the time component of a 4th right beside by BX not plus Cigna . decided by D X X now means X 1 X 2 X 3 sigma has 3 components this sex over here has 3 components that is equal to 0 and delicate even make it a little bit simpler and by defining a 4th Pauly matrix the orderly Filippelli matrices they are 1 minus 1 0 0 0 1 1 0 his disease busy component as was the x component the last 1 is 0 minus IIE that's the Y component we matrix let's end 1 what the kind component of the probably make bricks at that time component is really simple it's just 1 1 the unit matrix there are 4 of them these are not matrices write any matrix as a linear expansion and now other 1 multiplying guests visitors 1 multiplying beside the X so I could put here but for what this says X not the current component I could but he ate Sigma sup sorry cigna not and this whole thing becomes just i.e. Signora mule where there now are for vectors worth of sickness times be signed by DX equals 0 that's be Cairo role Dirac equation and it's perfectly Lorentz invariant it takes a little bit of work to truly prove that its Lorentz invariant but a really is Lorenzen variants and transforming the truck the signs here transformed in a particular way on under Lorentz transformation is we don't need to write it down all we need to do is to know that 2 components bidders like have a distinct and simple transformation property on Lieurance transformation I that suggests that suggested give right suggestion that Lee grass and parameters grass parameters which are also the signs 1st of all our aided by commuting objects recently in society by armed since sigh they a quantum version of grass number and that commuting field that suggest that progressed the algebra considers not just on a leader but a fade out Alpha just like also called for complex conjugate faded bar Alpha which is state of bar warned bar I'm really independent components are the altogether of these before we can either think of show real behalf real until
imaginary component Oleg think of the state is in the state bars as separate objects that's OK it's it's legal to just think of them as separate objects but if you wanna be more precise you could think before Real components of data being go right therefore a component of data now supposing we have a general functional later for example a super field a super field which depends on the court that's a space as well as these new winter commuting coordinates we can exist we can again expanded powers of the it will start out for example the 2 were just a year of HBO's object an ordinary number it would start out with some ordinary thing which was independent of the is might depend on the exodus and then it would be something proportional but say that for example when course data there might be something with the fate of Barnett data a bar shining how keep going previously we did it at the next term why the only to thinkers but now more and it's going to go all the way is to the 4th power why because the Reforma after the 4th day it is the start repeating and will be the end of it so we can go toe while called a either like quote sometimes sidebars away size squared the start date of our squared later squared some coefficient is often called the arms or I can just call it just dated to the floor and make it simpler and but they at the 4th Byzantine the 4 of the power of anything innings they what it means a product of the 4 distinct later this so we use simplified the many cases tonight I'm going to use simplified notation because next biology gets off on so I don't want to do it them either want us to befriend taste now we can ask what is the super algebra what is analog out and I commutation relations and the commutation relations of kilos with others with themselves and with other thing it's alright them down for you now you're seeing things like it before but now the newly headed kick that things have indices out i it's as good a up you Alpha Q who is also a spinner that belongs to the same family are objects as state and it begins with Debye ID data all I wanna to that I'll do that later I'll go that later but took a pass from Q. was an object which belongs in the same class of things as there was just a figure that right and they write them now what the commutation relations are want directly affect commutation relations and die commutator Q dead out With Q Beta Beta and often don't have to be the same let's try the figure out what that it or that could be what possibly could be remembered the previous case when I have the right hand side them twice the Hamiltonian right salt whatever those on the right hand side is something like a Hamiltonian cannot only means energy and energy is a component of a 4 vector that there is a tool of the editor was a purely conventional us there was no significant to it you could normalize it differently on the right-hand side we should expect that theory is penial the previous cases peeing you just meant P not the energy but it can't have a Lorentz invariant theory where you have some structure here with just a kind component if you do a Lorentz transformation that time component will turn into space components so you better account from the beginning propene Yulia but that doesn't look there's no on this side is enough and they are on the side there a new 1 aside and allow for that serve means we have a sensible expression we need something else here we need something to soak up this index and give us In index often beta what could that be said no new Alpha Beta Sigma is are maker seems but they also have an index new stock we can make a nice formula out of this said the new house fire they now both sides match In all in this new world of course this means some meal Einstein convention so we don't have a potential possible low rents invariant hit by commutation relations for 2 0 Super charges Q dagger and Q. We can also write this as to what I said no new Alpha Beta DE by D X mule amusing the connection between components of momentum and derivatives derivative operators are both loser a correct next year by commentators Q. dagger with kill bag Till components in the previous case it was 0 it remains 0 say the S Q Q and the other thing that we have the right 0 minutes 0 recalls if any of them made any offer made a Ali we had that b supply generators commuted with the Hamiltonian that made them kind independent that made them conserved quantities here the generalization is they with all 4 components of the momentum of the strike P I won't bother writing which component any Q With NEPA and it's also an EQ dagger with any paid all of ours Oh sorry when you have a grass with variable and ordinary things you use commutator come if Ada or
Kermit data what about the commutation relations between different components of momentum Baker why commute because they represented by derivative operators removes protects removes but Hawaii the order of differentiation matter so all components of momentum commute with all components of momentum Inc. the kind compartments this is the thing which generalizes to a low rents invariant theory the things that we called the super algebra West OK any question is he said Alpha Beta for the signal that victory the stock look often embedded or the Matrix index and is based on barracks and is doing he added the formula because we have the soak up the index news somehow we have put back in the next often data None as a concept again of sober fields tried out a component of a super field of super field as a function of coordinates failures and later bars and if the role of the queue and also the peas could differentiate fight just shifted to make will shifts fight with respect the X and respect the later for example the action of pee on Friday Mr. multiplied by I times and derivative with respect the X reaction of Q in the same way that he says but Our hands on which upper a lower component of what's just right as ID by DX and the same sense that P is ID by X Q Alpha it is all in all it's just they almost Debye ID data out took Q was like a component of momentum in the rated by election day the direction is this in the past small direction what it means that our momentum in that direction is not something you could conceivably not quite finished the sustained deal as last time and plus I normally have less time data bar Debye T is what we have played a bar code body by 18 now becomes the bar data the bullion indices but after that under a simplified love Deeb IDX meal we're Mark Taylor bar b by IDX New misses the analog of fate a bar and be by duties but this doesn't look so good enough on the side of the beta on this side that were look good and as a mule Haywood nobody socket up Winnie the stigma items same will be said no new data How from that queue or to rewrite that I'm a simplified and just write and data plus I played a bar stigma Dean Neo just to save myself writing these indices Over and over again the signal is here forgive the finger the right Alpha Beta structure but also to soak up property the should be a mule but simplified get rid of all indices in the expression being means derivative with respect acts is different this curly d means partial derivative with respect to X which acts as the 1 that matches the component of Cigna here 0 critter just visitors written shorthand Q Bar equal to be Argus and the right to Barbados just for a variety of Debye data bar Bader plus again signal new Beta Alpha they that Alpha IBX Neo but end forget now that of this was later if this would save us from state bar officials borrows from Slater their conjugates of each other more simply d by data bar plus I said much later d . for short and those are the killers don't were the killers due to kill operate on Super fields and they deal at a small change in the Super field under a supersymmetric transformation in other words the variation of 5 8 it's her mission if they are this will there also metric except for this 1 the railroad arrived job data on the left and think I have a right Cook said if From with may offer the data already they alfalfa meal it but they have a right OK the supersymmetry transformations really just means a small change in the Super field which is given by the Q Thunder Soper field now was killed times you need it simply means you hit the super few of these various derivatives or Q. bar Times but a calmer here you Conover
supersymmetry transformations were killed you with Q Bar think of Q. 1 fine
as a small change in price that's the role of what know we could we could go for a week take super field and expand out of the components components simply need the coefficients of various powers of data on people speak about the components of a super field that talking about the coefficients of the various powers of the Wiccan work out what changes are in each component but I don't do that that's that's not useful for our right now right now I just want to follow the bookkeeping and show you how do you have the manipulation the work OK so that's good bright What is the simplest super field the similar super field you might think is a general functions are they Exs and traders and fate Boris all of them but might there be something simpler which depends on fewer variables which is still a good super feel could there be some constraints could could maybe a possible constraint compose on the way back flight depends on its arguments which simplifies which gives a simplified super field which is still has good supersymmetry transformation properties so let me give you some examples supposedly we're talking about rotation and variants here is a bad idea take a vector field and say it I'm interested only in vector fields which was satisfied V. X. equals 0 that's not variant on the rotation I would discover it by making a rotation of the vector and discovering that AS the rotation What's the change the 2 dimensions what the change in the x component of a vector on rotation as employers total y component so if the Y component is not equal to 0 0 to begin with there after the transformation the x component will be equal to 0 anymore so when a rotationally invariant theory you don't want to polls are constraint like VAX equal to 0 that idea OK how about the constraint these squared equals 0 this means VX square why lots of that's that's a very very strong constraint is I just as these equal to 0 the sets forth its regularly square equal 1 is that a consistent constrained in a rotationally invariant theory yet this is only the V and said equal to 1 now and may not be a useful thing to do in any estate feel theory but it's perfectly invariant on the rotation how check some variant of the rotation rotations you do a small transformation on the rotation and then you see the result being components of V satisfied same relationship if they do you've done something that's consistent with the symmetry let's ask whether there's an interesting constraint that you could apply for a super field decreased the complexity of the superficial not the decrease the complexity of already complicated warm but some constraint which would give you a subset of things which might be simpler than was generals this is simpler in the sense that it was less instead of having to independent components only has 1 independent component in 2 dimensions so if these quizzical but warned there's only 1 independent component that column from garlic gas pressure from Roy to make a guest of courses I guess MOVE answer I didn't work this out of your work about that there is a constraint that you compose the you may impose a super field which has a form of some differential operator which I'm going to call kinds 580 it is equal to 0 and deep bar is a differential constraint a set of derivatives set of a differential operators on superspace What is condition that this be a consistent thing to do which commutes or which our which is consistent with the supersymmetry surpassed I imposed such a could that Toyota would be Boris yeah because would be barred but supposing there was a constraint like then it must be true that if you do if you make a small variations of 5 by hitting it with a Q it must be true at that Woods also another words it must be true that D. bar on 5 plus the a small variation of far-right which is proportional to times 5 must still be equal to 0 otherwise it wouldn't be a consistent thing to do would consist of the supersymmetry well I'm thinking of these a linear operator of some sort of fire we already know was equal to 0 or assuming defies equal to 0 and the question is if these 5 equal to 0 0 would be bar of fire is equal to 0 will be be barred of cute times fight was I told you it too will be but his abstract general questions about what kinds of constraints could be consistent with a particular symmetry but now this is bar and she was a group of 5 Tulsa would true that fuel which is after all of which is a grass the number had variable supposing Q either come with or can't die commuted with be it is the case with the writers as plus or minus Q d times 5 always already
agreed that we fight is equal to 0 so therefore and this would also be equal to 0 so it's a sufficient condition for a constraint to be supersymmetric but the constraint operator of differential operator of constraint commutes I can't die commutes with the symmetry generator that we're talking about this case killed so what is our raise and raises the question either differential operators which come you with both queuing Q Bar if there is such an operator we can impose a constraint and possibly make the Super field simpler crave constraint like this would say that the super fueled the depends on variables In a particularly special way and they have few would dependent variables and you expect that this is really not complicated I'll tell you what be borrowers and you could check it pledged Monday the but With less but yet that cut through where Q. fight I just switch the order of be barred under the assumption that she wouldn't be barred either commute on APEC knew if the commuter rail commute Ucon switch the order and then use the original constraint on fire the sale will be true both before a trend before variation and have after variation but is a very simple answer for a D bar and the which both it by commute with both accused Austria What is the new contract the little bit your D Alpha they Barbados Excuse me is exactly the same as Cuba bar with 1 exception nobody Jerry guest 3rd start I Sigma new data but well let's just to simplify at minus Sigma Theta climbs derivative say exact expression that I have here except With changing this time if you go through it and check whether it would see what the whether it by commutes with With these things by fate bar and die commutes with barter give basically a 1 so and that commute this with be I'll get this piece junk over on the other hand when and commute Bader would be by here or get the same thing a minus sign deal cancel out only me you could check at a factor Michael you checked the sort of thing in the support case right exactly the same thing this object in that commutes With both Q and buyer this thing that happens to be true also of D offer which is debate minus ii lover data by Sigma derivative but only get be interested for today in 1 of 1 of them and I will say what kind of constraint what sort of simplification will be made on my super field if I insist that I have a very special property namely D on fire is equal to 0 you can't both do would
be bar and if you did B Bar and that would say the fire was equal to 0 but let's just take bond fight equals 0 I tell you what the answer is 1st of all this it would be a constraint on the components of the Super feel it might eliminate components of the Super field it might simplify the super field and I'm going to give you the general solution of this equation the general solution of this equation is if I try of course depends Rosa depend on it depends on X and feeder and the bar but the answer is that depends I would say armed Stewart to Boras for the still on the loose symbol 1st surpassing b bar was just by a bar without this over here well it's a peseta Debye dictator boron this gives 0 I would just say the fighters independently the baht judge Buddhist buys a function of X and data but not complicated this condition and D bar fight 0 means that fine depends on the thing about in a special way particularly simple special way how are you is it says the fine a function of X new plus body played a Sigma Nu beta end figure among words at only depends on the peculiar combination X full-sized faded boss signal Theta it doesn't depend independently does not independent some variation with respectively the body only way it depends on played is through this combination that's what this equations says Let's check it let's see if we can check at the doesn't quite say that I only depends on later it says it depends on bar in a particular way that comes in together with Maxwell assist jacket not here but Armed let's differentiate this with 1st with Hearst with respect to fade a bar what happens when you differentiate this with respect later a bar you get a derivative with respect to this argument here another was to remove the X kind earnings i.e. Sigma Theta is clear that the derivative with respect for played a bar of it is equal to the river notified with respect the X from simplifying it now kind the derivative of this variable with respect tool I'm sorry to the through strip a bar is the river will respect the acts of kindness on Sigma Nu that just understand their captor was operation of differentiating with respect to this variable here by differentiating with respect the whole thing and then finding derivative of this whole thing would respect the fate of that really and the other thing which comes in is my has I said no data the review respect the X forgot exactly cancels owes maybe function of the special form 5 of ex post I made a bastard with data and freighter is automatically annihilated by D. bar that's all always D. Barr says or be bar equals 0 says it says that this is a particularly simple form on another way to say it is to call this whole variable here White knew everything under here and then this the fire only depends upon Friday of white and played a bar is our that no independent dependents no independent dependence on the bar only through want of the meaning of this and the whole set of ideas year is it you could have a super field which has a simpler and it still is a representation of the supersymmetry algebra still transformed by a coherent way under the supersymmetry algebra I feel like this which has only this dependence it's called up rules super field it's called a Cairo's super field this is a cut incidentally Cairo super fields are always complex they're not real things they are generally complex and the way it could show that is you could see that a small variation of the Cairo super field by hitting it with a Q gives it an imaginary part of it it could have Meridian imaginary part of Cairo super field also or complex soup of complex in the center of the complex numbers it's a complex thing air but that so that makes a more complicated only depends on fate of foreign particularly simple way now I do not argue this is an serve on ending story are of formal manipulation but after like it's kind of fun to do a formal manipulations and to see how they work so let's do it the next formal manipulation which is to start calculating a grungy and 4 5 from super field capital
fired all before we do let's just see what this says this says a what's expand fight flight Mike these some scalar ordinary number object which depends on what comes next next comes plus sidebar times Slater sigh Barkin depend on white and if Ada From left anything out many components of fade to write member to scanner or component that means you can have made a square which really means they 1 thousand stated to solve you can have plus a thing which is always cold at 4 times why I'll call the status quo but it really means state one-time stated to Sunday things played minority but this is not will not be the bars by dependences all in the white North the bar dependent except what's in white so the super feel the Kyle super field has a limited the much smaller number of components than the generals of the field and makes it simpler it's the simplest example of a superficial and only has 3 independent components flight a fermionic component of words a Fermi on field bows on field a Fermi field and there what looks like an Ovibos on-field refined it s is not really an independent bars on-field perplexed Turkey Garner yacht Walter and in 1 more time your wife is that like X new force I made a boss Sigma Nu Data Definition definition is not Exon why it's not just 1 just use the term white to denote this particular combination was awkward why the super feel just terminates after the 2nd that's a good thing it makes simpler however why is not acts and after all were really interested in things very with DEC's so at some point Rogan have translate back Texas OK let's talk about what grounds in this were broad use with some program views were set thinker last time recorder land ordinary Lagrangian I'll use the term and these will guardians depend on radar data bar exodus but ordinary Lagrangian would only depend on the exodus and where we do and we computed action ordinarily the action as an integral will grant density below ground density over the 4 directions a space we do have that D 4 x but also now in into both chief data dictator bars now comedy components of fatal are there took components of Haider was Camilla be read red lead Tuesday which means dictator 1 data and D to fit a bar a four-dimensional integral although the 4 components of the data that's the action if we don't need to go over 1st really get left with something which doesn't depend on the figures is anymore and just ordinary at the ordinary Lagrangian so that's our goal is to construct super Lagrange unions which have supersymmetry and then To do these goes and find out what kind of ordinary field theory with talking about friends out that the Lagrangian and is able grungy and construction of a grungy genes in this context is really quite simple simpler even then the things that we talked about last time even simpler when I give you some examples of summer Grosjean show you have a lead to find diagrams as I said the degree of abstraction there's really unusual you just up the folly in those follow the equations and in the end will pop out something barely recognizable and of all of this will be of barely recognizable structure a Lagrangian for firmly under the bows on some interactions some find diagrams but it will be guaranteed to do certain things that only supersymmetric Lagrand you don't talk about process OK so let's take who grant density bigger Lambda and explained that out of the power series of played a bar will have maybe terms the last of which will be proportional they say the squared their bask roughly the 1 thing to the 3 figure for what they 1 thing to a bar 1 thing to bar to all proper for independent status when I'd only integral all the components of later it simply pick out only that Western so the rule is very simple you construct of Grand Union's with all this crap all such a higher power was a failure but In you pick out only the last Turner's that's summer sounds stupid but it's very efficient and very clever the way works so that still a simple example undertake a very simple example it seems to it seems too simple to be interesting but lipstick anyway let's take Calamba God beta the Soper field Kyogle super feel 5 times its own
complex conjugate now
that really seems much through simple incidentally that's right down complex conjugate is Greece are OK when or erase this at all write down with the complex conjugate years the complex conjugate of the same thing it is a complex conjugate super field but it depends on the complex conjugate variables acts buying I made a boss Sigma Nu data and they bar just complex conjugate Everything everywhere complex conjugate flying complex conjugate the coordinate depends on the depends on peculiar complex coordinate here instead of depending on state it depends on the bar that's complex conjugate let's multiply those too together and Vinci of weaker what kind of Lagrangian it gives OK was fight depend on fight depends on X minus played a bar Sigma Theta I made a Boston Bader and fate bar that's started the war multiplying that by fiery which is far-right oral ex party leader bar Sigma Theta and just plain old later however argue that go with it we're going to graded on the fate is and also over the exodus D 4th Badeau more components of state Will before I do it any integration misses the action Rice's the action before I do any integrations I want to shift the x coordinate Idell 1 happens in an integral is interval of a function would you doing this calculating area of a function what happens if just shift the function if I shift this is a funk this could be a function of X by shift the function therefore all X plus anything plus a buzz of those that will change not so you could shift the argument function of X without changing be integral but must consistently shift the argument exercise shifted here I must shift at the same way he I can't possibly shift away both the RIA this thing and this thing because 1 of the fears minus sign women appears over past side but I can shift away 1 of them at the expense of the other ones so supposedly shift the x coordinate by shifting away the dependence over here what happens to it over here that it doubles it doubles Bergen Ellette X shifted by this amount here when you do so this will get doubled to 2 and now we kinda read over here you don't have to do this but it did simple visit trick a check which is unnecessary but simplifies the calculation of good deals on OK now let's plug-in now we have to do some work although this up till now was binges just formal manipulation it's still formal manipulation but now we could do a little bit of work were going to write that our hard work with right authority is equal to a on-field plus sidebar data sidebar data a plus AFP square but that's is a function of wife another words is a function of X plus the thing he I want right as a function of X so let's see what I have to do I have Taylor series expand Inc Powell Of the please To a exhibit precisely how the thing depends on so let's do that let's start with this five-year which is a function of this variable we can write it as fight 8 5 x misses fire everything depends on white and wife Hawaii all not Roy doing wrong it's not white it is just whatever this thing is why was just X foresight paid about signify misses acts forced to wife a boss potatoes worldwide our legal Prime right so this is this begins with fire of X as just a bow Sonic piece of suffice next we have to differentiate with respect X P Friday by D X is just a mosaic PCO be fired by ID X times the amount that shifted X switches to 0 I made a Boston that there is another term which is the 2nd derivative with respect to x under a simplified notation and disk August the 2nd derivative with respect experts squared off 5 times this thing's Quayle I'm not gonna bother writing up that things square discover notice that it contains 2 powers of data and to pack the powers of
Dana and 2 powers of a bar is only 1 expression contains 2 powers of data and to powers of a bath it's a product of old for a components of than anything else will have redundancies so let's just call this that for
and not get a barber keeping track of the numerical coefficients the next tournament a power series expansions of little five-year little fires the bows on a piece of big will contain higher powers will contain more powers later and therefore will be 0 so that's the end of the expansion of the 1st turn super field next to the Super field involves side I X times but I have is that that the derivative of sidebar with respect to x times this shift here which is used to getting this tool made a Sigma Theta times data Riverside vexed times than we them we differentiated sidebar with respect X a multiplied by this levy simplifying and just right later bar and has 2 powers of data I want out of state a bar the sister of all the crap and his rise to powers of data and want power bar just b schematic and being schematic Mel what we write that down where we did some Burgos erased it wrote them a fine of why is equal little 5 wiII plus sigh data of wiry plus F favor data right now I'm still think the argument of from writing that wine that is why I'm writing the subject only here right X plus was amount so about means I have to to shift the fire if the law fire by doing a Taylor series expansion this 1 Haynes it played a faded faded bar if I try and another derivative I will have too many pages and will kill they have 3 powers of beta or something 3 powers of beta cells nothing after that and then there's f force F failure squared do I have to start Taylor expanding this but Taylor series expand this it would give me a derivative of F tie-ins status in the data as too many powers of fate and don't anything with Mrs. just cataracts salary thing depends on acts but we have 1 2 3 4 5 6 turns coming from expanding it out in the Taylor series expansion EX dependents but trunk aiding it when I get to higher powers of data which 1 All-Star yes they squared thank you for your shop Over here I don't have to do very much that this summer with in 1 bracket and the other bracket not so hard it's just fi star little five-star the are part of X plus God side later Boris the rate of shy of X plus but f star take a bath a bar I don't know any shifting in this 1 because it's only a function of X 5 undone the shift in here by and stuck it in that year now the trick is the it's very complicated looking for remember always want out of it is the last turn in the expansion the fate of the walk so let's see what their toll on the fate of let's start with five-star yet what Pfizer has no favors so I have to find something with at the former senator multiplied by using only thing with fatal for them it was gonna be terms which is 5 star of X times a 2nd derivative of some sort X squared 5 now we're on real have gained the greatest with respect X resist look familiar but still a bungee interest scalar field Lagrange infra scalar field would marry Lagrangian from massless scalar field we can't go on loan amount of effort to get a bronze in 1st gala field malady see what else is Our let's take this 1 here but that has the multiply Tuesday deserve a figure bar right over here salt that gives us something with 2 faded To date is bar that gives us plus something like bar With respect that next time society now this isn't quite sensible the index alpha and index Bader Annan index New so what really has to go here had I kept all indices simmer matrix signal new Alpha Beta guess what this is all around you this just the firmly on the ground gives the Dirac equation when you work out the aura of a grunge equations for it this 1 gives a Klein Gordon equation when you work out variation on that much keeping track of numerical constants because I've long ago learned not to do that on a blackboard 1 less thing is here it is F star at 4 of X Everex now but scalable was on
direct like a formula and something trivial this thing is extremely trivial it has no derivatives and in fact what what its equation emotion be there would be a partial derivative of the this is all about this whole thing here is new ordinary Lagrangian integral of the ordinary Lagrangian ahead if I have nothing with no derivatives decide what the he equation of motion for it just do reveal grungy with respect ethical 0 there is no the removal of grungy in with respect to the derivatives of F this jet deal by IDF equal 0 or F star is equal to 0 and start is equal to 0 start never complex conjugate act is trivial it doesn't do anything here has no interest in the equation emotion is another way of thinking about it on begin think about it in terms of proper Gators and in terms of down with grounds of course lead to find diagrams be connected and were ground use like just lead the proper this guy over here is going to give us up perfectly conventional propaganda OK with for fighting nothing would be unusual fires going cover propagator which propagates from want whatever the ordinary scalar field propagator is missile give this is going to be a perfectly conventional firmly they also have a conventional propagator With but what about this summer he at this is a little bit trivial more than a little bit trivia was not derivatives and it is not derivatives and that means that if it were a particle associated with it so that particle would not move from 1 point to another is not terminal which couples neighboring point of space it would give propagator that propagate it would be totally trivial under drop picture Florida tell you what it deprived the property it would not be 0 quite would be something a little bit less West trivial 0 let's draw a like that squiggly lines from 1 point or another but does it really propagate from 1 point to another not so you care guess what might be it's a delta function the prodigal come with the particle which is not really a particle comes in and goes out exactly the same point doesn't get anywhere so the propagator of Florida's very very simple will grant you would know derivatives and is just Delta fortune be propagated begins and In at the same point and the particle King anyway that's the structure of the you find propagator that come out of this terminal around here now this in itself is not an interesting Lagrangian mildly interesting bows on the Fermi on exactly as we expect for a supersymmetric theory the both massless particles have man's and they have no interactions but no interaction self let's go on the interactions but dog interactions and see if we can find some interesting find and diagrams ever yes at we took 5 times the North interesting this Abdullah Burundi and they even have any derivatives and the soup burns in itself was just by star fight it the workers extremely simple and didn't have any derivatives of the Super field where did get its derivatives from it came From the fact that we had to shift the argument of the Super field ever was shift we never would have generated any interest in motion of the particle from 1 point to another so supersymmetry does something interesting it takes an extremely simple Lagrangian our or super Lagrangian which seems to have no structure to it 3rd propagation from 1 point to another but because of the shift of the argument this shift of the argument over X there they really does shift particles from 1 . 1 spreads OK let's at something else simple and certify star 5 let's end in just 5 squared and while I'm doing that our away Tri-Star square but that's just the kind the complex kind square of a figure out what 5 square that I will also have figured out what 5 stars are just complex conjugate will be as when he saw but concentrate on that's just played by squared what does it do not only need to do is a square fight with square fight but before we do little delivered integrate D 4 acts can also would only be integrated with respect but something interesting simplified happens here simplified compared to here this is missus capital 5 Chipperfield capital stance was so perfect let's remember what we have array of capital flight Of that explosive I stayed about cigna data and data plans another fight of exactly the same kind the show Renoir here I started with 5 start the firm's 580 I know those integrated with respect to x source shifted the X integration a shift that X I shifted X to get rid of me out of the state independence here but what do here it doubled over here I'm
not in this case what happens if I shift the argument x shift away made a blossom data over here if it does it both places in fact it this way ATP how of fiery not without any fly stars for a Our firewood no-fly stars you adjust shifted away every where's one's pretty simple but drive out old old old but now we have a problem where a problem with the data integration Daedalus data indeed to fade a bar is nothing about dependency laughter a fine integrate B. theta Beauregard something with no bar dependents is 0 he is 0 so on the basis that this looks like 0 but the food is very simple it turns out to be a completely supersymmetric thing to do when there's nothing the pot dependents at all is to just integrate over data that needs to be justified that needs to be justified that when you have a Lagrangian a function which doesn't depend on the bar or just integrate over later and you still maintained the supersymmetry say that needs to be justified but let's assume a justified this is the thing that you do when there's no bar dependent at all but so what what that means is that we should expand things out of power squared and they squared terms which will give us the answer around your right put he what's current him over here In the latest squared it they will be Forex but still we have to fight and fight square excuse me if I of X a small fire wrecks plus sidebars data plus after they've squared square that with just writer side by side formulaic square things is to write side by side by side greater cost at favor squared end war on a computer is determined this which is proportional to square what we get well 1st of all we get 5 times f twice flight times ever for the year and a half times fired over here so visitors without keeping track of the constants which will look like eft times 5 but also be the complex conjugate because was supposed to have complex conjugate but just 4 B F times 5 is 1 thing that only 1 other terms for use by sigh squared but before sigh bar squared Schreiber brought but sigh bar which are brought but both Surrey Novick said I think yup that's it but that's all there is From this terminal around f times 5 plus sidebar sidebar OK 1st of all let's look at the Dirac part of it it's In the sidebar Siad Barre that's turned from Dirac particle romance times a product of 2 sides with no derivatives it is a master Calder my run a mess terms it's a rather different slightly different match the kind of about passed but it is it's a strike comes in the Messiah goes out was a coefficient and it is when combined with Riyadh with a conventional piece of the Dirac Lagrangian here it gives rise to wave equation which has a so this is a message the Dirac particle firmly armed but this M F 5 looks a little peculiar it's not like anything we've seen before but if we think about it for a minute it will make a fine diagram in which Our flight particle the line becomes an F for now an F Webster's draw that therefore that we will have property area Watts of fiber comes in at 4 of 5 becomes an effort that nothing it doesn't do anything it just sits there but you can have a fine and by ran to 2nd order with the F becomes a fight again that will involve square this general a grand unified gives you a transition from a flight 2 and a half but that is not a real particle doesn't go anywhere is a better get back a fight that will involve 2 powers of this interaction and in squared and give us off easy enough I out another words it's completely indistinguishable From having and squared I square milligram and squared fly squared is just on has turned for the bows on we do this another way we can do it in a more the more sophisticated technical way without diagrams let's look at everything in this Lagrangian which depends on the 8 deaths 1st of all we have ever star and then we have a plus In F 5 also the complex conjugate and their star five-star well as I said and occurs in this awfully trivial way in the foreground and its equation of motion is just gotten by differentiating this with respect to F and setting equal to 0 no derivatives so they're quite an
emotion that half is just as star plus and 580 equals 0 if that's the equation of motion for it OK let's solve this equation of motion for F F he because mine this M a fiery and then while get
back in the Act Anthony run and see what we get for what's at star ever going to do have started off well just hours after death is equal to minors and star as is gonna give and squared fi stuff I and Square Firestone What about in flight f at Taylor give my M squared Friday they canceled but as 1 more leftover here and that once also M squared Tri-Star Friday so that's another way you could be see the thing diagrammatically that 1 this f thing here does finding diagram is give something which is indistinguishable from and Square if I squared off over you can solve for F plug it back in and you see the effect of the Lagrangian just and five-star 5 is also the bows on and in fact the mass of the bows on is exactly the same as the rest of the rumored bows on equations right and squared firmly on equations you right an hour so server firmly on and the bows on both get In the past and in both cases that mask is just em again we see the power of this symmetry taking place making sure that the bows on the front and have exactly the same maps that cholera of this really becomes significant we could just aren't respect taking a bath on-field from young man field giving him the same maps and free field theories no interactions their masses would never change there would be nothing very interesting it would be supersymmetric power of the particles interact so let's go 1 step further and gave a 3rd another turn Lagrangian times flight Q. fight FIQ is more interesting to Vader saying game but we have to put 1 more sidebar later plus F. Davis Square let's see what kind of things we get from here remember we're looking for the coefficient of data squares but all accounts In the Lagrangian and that is the status quo so let's see what we combine we can find out finally laughter quotes from foreign flight times of flight times an but was flight times fight terms as comedy terms like that 3 terms like that right and visitors an evolving we can ever find time aims times sidebar flight times times sidebar 5 times sidebar tires sidebar terms like that is also also 3 North 1st of all your terms GET all times the coupling constant g g is a cup when entrusted 1 of this German fly sidebar sidebar it's interaction in which To comedians and 1 bows on goes off me and comes in a bows on goes off in other words it corresponds to a diagram with 2 firmly on both sides it has a coefficient G it's a new direction for trimmer Korea love complicated find them diagrams is fine and that has become endlessly complicated What about this 1 over here 3 fly square gaffe OK 3 5 squared have let let's forget the 3 the 3 is not be interesting thing here it gives us a G with 2 5 is now 1 of peace and F propagate euro rather have left America a bigger nothing is just not particle associated with it together can make that up to something else but you don't know the thing to connect up with is a mother a version of the same thing it was a vertex with to find become an there's a vertex were have becomes to fight what is this what kind of thing is it's remember
S doesn't go anywhere it just really just corresponds to a local vertex where 2 flies coming to flies go out from the same place another words in looks like fight before really it's Fi squared times five-star squared it's a fine and diagrams with 4 Bos arms coming in the vertex and it has a G squared with the Jews could come from because they want to vertices 1 where the 2 flies became an ethanol with became to fight rights of this thing that repeated twice and scour coefficient G square notice the really the His dusting thing here it is we could have written down exactly these kind of things what has the supersymmetry done it is made the particles have exactly the same and this is related numerically the value of AT 1 kind of coupling constant to firmly on the bows on it with another coupling constant which is for HBO's arms and fact told us that Texas and his text of his verdict was the square of the coupling constant as the square of this 1 that a tight constraints if you could do do experiments with these particles you could check by scattering by detainees and so forth whether we are finding that a tow bows arms coming in scattering is the same as defined in diagram for To beyond come in and produce a bows on that's the kind of thing the supersymmetry always does gives you equality of messes and relationships between different coupling constants that just before I did finnish just show you simply without actually doing any calculation but just what the implications our relationship between coupling constants like this are we talked about it before but with got to it carefully but said about the self energy diagram of the bows on he is oneself energy diagram so the lowest order and another 1 is bowlers on becomes a pair of firm eons which become bows on again resume but you could make of these vertices here a 5 4th vertex over here for Pfizer over Texas and his fireside time what's the numerical coefficient because of this 1 G squared Watson America look office in the Gaza this 1 also G squared G from here energy from here that the rest of the calculational course involves calculation of the integrals find diagrams of so for our but you can imagine now that we calculated carried them out a showing you that the coefficients however saying the integrals are also in the intervals are also the same and this Mrs. stopped all owners every diagram the matter how how can complicated which may also have been answered here it is all ways a collection of other findings diagrams which have exactly the same value except for the opposite sign which cancel out the opposite sign it would remember that firmly on loops involves on groups have the opposite sign so this kind of construction while rather tedious long-winded then uh you're just fix forever to explain has a tremendous amount of power if your goal is to write down theories which have exact supersymmetry supersymmetry meaning a bows I'm from on exact now I think we're finished with mathematical formalism of supersymmetry I him personally Cypriot or receive again but we still have some things the finish up some strings or speak to pile up 1 of them as the breaking of supersymmetry supersymmetry is not a real symmetry of nature and how that may happen what it does overtures particles around the other thing is the idea of unification unification of the forces and how it influenced by supersymmetry I think glee have 1 more classes Oh my God from 1 or less so we will talk a little bit about the spontaneous symmetry breaking of supersymmetry for its implications are what it says about the particle spectrum and then we'll talk a little bit about reunification of forces as you to reprocess Udo chorus you won the Standard Model becomes as you find and how it is influenced by the surgery
for Miller please visit abstained like down 80
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Titel Supersymmetry & Grand Unification: Lecture 8
Serientitel Particle Physics 3: Supersymmetry & Grand Unification
Teil 8
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 10.5446/14973
Herausgeber Stanford University
Erscheinungsjahr 2012
Sprache Englisch

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