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Supersymmetry & Grand Unification: Lecture 4

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on Steiger University 1st we
start with cemeteries Emil remind you of a few things about symmetries because supersymmetry if nothing else a asymmetry symmetries always relate things after what they do is they relate things which related by being transformed in some way a particular symmetry is left to right symmetry hand my right hand symmetric and her if the world an IED and everything about me has left-right symmetry than my right hand the ways exactly the same thing as my left hand would be exactly a strong my left-handed would be in every respect and the coca my left hand our parents in that and cemeteries relate to objects objects which may be different but symmetric in particular those objects related by a cemetery generally have the same maps so let's talk a little bit about the mathematics of cemeteries we weren't talked about some symmetries in particular things like rotations symmetry in space and now I'm thinking about continuous cemeteries where the supersymmetry is a continuous symmetry or not is a question which has no answer has no because it doesn't fall into the category really of things with the question actually makes sense but let's just review a little bit about symmetry symmetries Our operations on a system which it described quantum mechanically by 1st of all operations but operations described by operators operators on stage we stop state that their whatever happens to be I don't care what kind of system that is described by a state there and we symmetry operation the most well-known symmetry operation I think of is rotation actually I could think of an even simpler 1 just translation of space then the operation would simply more than objects from 1 place to another but let's take the case of rotation has a more representative example for the moment what what the mathematical symbol to represent their year transformation a unitary operator a unitary matrix or unitary operator this unitary operating here could represent rotations space you have to indicate what the axis was and you have to to indicate what angle wars but once you did so there would be a unitary operator that unitary operator acts on any state be given new ones which is a rotated version of the same thing did not states not state aid CASE other Asian of rotating an object it rotates at act object as I found out that some state but state mean collections of objects and this is the operation which rotates everything is rotates everything in space so take Toyota state is often do example the state might consist of the mathematical representation of a particle over here and a particle over here so is state which represents 2 particles 1 over here and 1 over here a rotation about an axis of rotation Bentley always take place about access but said about this axis it was simply give you a new state this particles been rotated years this want here and the new state with the particle your heart a series of both the Represa rotation Hilbert space that
represents the rotations based at got better both life but this is the rotation but we do want a Hilbert space now it's not strictly a rotation a unitary transformation which is similar to a rotation of both a transformation of the Hilbert base and represents the act of rotations base by now offer as I said you represents a rotation about an exodus about a certain angle there is particularly
transformations which are especially interesting to us the infinite has over before that let's talk about briefly about the properties that AT symmetry operation has dissatisfied most important is that has to be unitary military means that intermission conjugate own inverse this is a unique property which says that state vectors maintained that's actually just probability conservation a probability that this sector should have the same ruling in a Hilbert space sustained Norman the Hilbert space as the original for every sector that the conditions that there will you be unitary twice code unitary about score I never was a special family of fees which are very small rotations rotations by very small angles know every place I say rotation you could substitute other interesting group operations but let's step for the moment with just ordinary rotations be infinitely small rotations infinitely small means by infinitely small angle rotation by no angle at all words no rotation that's not represented by you equals 0 0 you equal zeros unitary its represented by you equals 1 so and infinitely small represent rotation you equals 1 well suppose you were rotate a little bit not quite 1 but you have the Eds something and if the angle very small would the angle issues epsilon epsilon is a small number then you have a epsilon trying something in order to make it unitary you have to and ii times a year permission opera what L L remembers the angle of rotation epsilon but failed it has to be indexed by direction of space so this could be rotation about the X axis for example thicker be rotation about a Y axis or it could be rotation about VX or it could be rotation about some of well if you want if you some them you want the getting a rotation about some axis which young intermediate exercise so this 3 basic ones out of which you could build all the small rotations an arbitrary axis is just a linear combination of rotation about ex-wife ex-wife CEO and it's easy to check that health is provisions the new his unit at least for the leading order in epsilon study things orders tourist orders Repsol now 1 of the most important things about a group of symmetries is that the can't said that algebra is a mathematical concept the commutator algebra on the generators physical the generators small they call the infinitesimal generators the elbows and they satisfy commutator algebra I'll tell you 1st were commutator algebra is and then I'll show you how it's connected to his transformation 1st of all mathematically workers a commutator of on it's basically you take the independent generators in this case rotation about the ex-wife c-axis Yukon Aqsa we will better yet just else body the commutator algebra is just a close relations that if you commute 81 of the elves with anyone on 1 of the aisles you get back something proportional Pullen elegant there some constants that appear here so for example let me write down exactly what you have Alex Hawaii you have Elzy here but that's not quite right were missing aren't you alright and this cycle through a wireless Eagles ILX and so forth but it is summarized the old by writing epsilon I THAT LOOK but this just as epsilon IJK is is just the symbol which is as you if any of the 2 of them same IGA and see plus 1 minus this 1 depending on whether this is even or odd permutation a 1 2 and 3 iron ore it says is L X with a Hawaii commutator LX Hawaii I L Z and so forth I now what's the meaning what speak what's the meaning geometric meaning of these commutator algebra what are they saying and why is it important that b generators close under commutation right close and the commutation which means the commutator ready to them is another 1 this is called a commutator out the wisest what Bush really has to do with as the following operation you take you take a rotation represented by a now you take another rotation about some other axis by some other angle start at you and then you multiply by me was that corresponds to a rotation about taxes followed by another rotation about another axis they knew what undo the first one so what so far what we've done is rotate rotate then rotate bakker Our view original access by minus the angle that we originally reported it and then do does that give back no rotation all does it give back no rotation not the answer is not to rotate about taxes and rotated run from other actors on undo the the first one in the end of the 2nd 1 is not the original configuration it is not equal to the identity is itself a small role another rotation just laughter you've seen a demonstration of this sort of thing what happens if you rotated thing about the X-axis the rotate about the why keep to you rotate about the
axis rotated otherwise taxes North forgot which we expect to do it again to come back Yukon back some of them could figure it said it will be erected since the complicated it back with a rush by let's check what you get 1 of those it again on the rise he inside is some other rotation that I will give it a that's another you it's of but let's see what happens if we do it for small rotations let's check in fact exactly what happened was more patients and let's take the case of a rotation about X a rotation about Hawaii reversed the retention about acts and reverse the 1 about why but each 1 of them being small rotations and let's see what we get nose well have mentioned that what we did yeah he walked lot us that it also gave them both those unless there is certain to be bad a theory right or you like we saw with you than we did meet and then we undated you I wrote backer you would like me to write you a nurse right now I know you do so white that's the same operation saying operation end incidentally of food is rotating about a given access but say 1 axes the X-axis a goal from X what what would have a taste of my acts of taste my mother XM under the first one in under the 2nd 1 model comeback no rotation of going so we do everything in 1 place we don't get a anything interesting it's if we do it in tow planes that we get something bad said the was b but see what happens we start with you that's 1 pass i Epsilon X that's rotation about about angle of epsilon then about the y axis and on the use a different angle from recall the angle don't our 1 prices I Delta L why they were there under low the first one that's 1 might ii epsilon LOX incidentally for small rotations infinitesimal be universe operator it is just 1 might party salons toward us and finally we under the last 1 1 hour minors by Delta L White Bear sweats see if we could figure out what we're going to get on this us terms when you multiply these all our end some of them get mixed up with some of them will be order 1 in fact only be 1 which is or 1 of its 1 plans 1 times 1 times won the next thing will be orders dealt them but see what's they ordered Delta just Delta by itself well there's gonna be I Delta of white times the rest of the 1 strike so Libya minus I don't white fans 1 depends 1 times 1 but then they'll be class are white the White Rose 1 turns 1 and 1 so they'll canceled the same fate of B I Epsilon XOB 1 from my side epsilon next 1 1 but then they'll be the opposite from this 1 over here so the thing the Delta and the things stored Epsilon will cancel think nothing left of them they were things are Of ordered Delta square and things of water epsilon square Iowa keep track of them we think you can keep track of them and of the day what I want to keep track of his things are of water epsilon kinds dealt the things which are sensitive to the fact that rotated about 2 different acts he's 1 of them the Epsilon kept track of the rotation about the X-axis the the kept try track thing about wire so let's see what they're worried means there's talk would the times epsilon you going that Dr. times epsilon always involved a white how many of them are there altogether built the kinds of salons are there is worn over here there's toward all here In earned wherever I have missed this times this right if this want times this
1 earlier the sources said a Thursday that for partisan forums before Jeremy yacht 3rd tier in terms of forced a he 1st together that her 1st term from 4 terms 1st times a 4th term it's getting too complicated maneuver cleanup orbiter to chosen to tow truck support yard right the good thank you he is dealt that friends and Epsilon with a summary which is why acts right a light times LAX fans minors I turned minors I think is my 1 right OK and then there is L Y times X Over here and that comes in with what size With the opposite side I like that parents to Athens that's fair cancels as does so in China opposites are probably me help me sailors are a good idea of how if fine right is that my plus linens where started my times what's over here and then we also have a white terms works over here the might of the might be the last time that I enough because of my life my history Mr. and now we have that's a lifetime's X now we have LX times a wife elects to left a White to the right now what we have that culprits twice she thought her get a two-term my assigned to terms of plus sorry the turned minus sign times LAX in the remote plus sign relics a wife so images jacket white kind is my ears transmontanus and and the like fudge acts Ms. plus terms plus those at the same time right then there's L Y L X Times l and that has the opposite side of the road to know why Alex's into LX wine if Bell X NL white commuted then this would cancel then this would cancel but the fact that these rotations don't cancel the a rotation another rotation Farber bad back again don't cancel are an indication that this should not be 0 another words that have between LX is the thing which keeps track of the fact that rotations in different orders don't cancel out as the the commutator really the commutator is none cancelation of operations which but which the Council that's what this is indicating he sewers is W. this tells you the commentator of this is a complicated twice the commutator Ave hello X With twice the commutator acts with a that's what's left it tough want is also the warm factor is a teacher I wonder her if so that you I think it's right that's what look yeah 1 with a 1 that's what I thought so was is it just 1 times yet but that seems right there are 3 with 1 signed not really 3 with 1 in 1 of the other order remind me just just like that if legal lies around the land of only Alexis civility and why you have to drive at the Taliban fuel and form just ready for you I still want I arrived as this 1 and then there are 3 others who is a quarter the life that at 1 be yes are very good enough for the match I I'm blind by now I can't do it anyway this in order to get you get something proportional to commutator here so in other words the he answers epsilon times Delta epsilon times Delta not look at LAX times LYT or the commutator LX times a white as becomes 1 plus all right Epsilon Delta els e another words this combination rotations about X Y and back again along the X factor again the Gong why give you a small rotation by an Anglo Epsilon times Delta about the c axis this is what the commentators you so when have a group of transformation of particular continuous group of transformations the whole structure of the group is contained in these infinite test will traders and especially In their commutator algebra commutator algebra is so the group multiplication table but in the form of these infinite test will generators right that's a if you lose you better be far that new guy n was still work to do for the 3 water the disease there before does get more complicated but it does get more complicated but if you know the commutator algebra here you can In principle figure out how compound together many many little Our transformation is just 1 example you could obviously buildup any transformation of what a little ones so if you know how the little ones combine that very much knowing the properties of the group knowingly algebra or structure group right that's that's for commutator algebra is a convenient representing the whole structure of the group and how the operations intertwined among themselves now next next statement things that all remind you of Ahmad visited later in the case of rotations L is of course just the angular momentum I take it just there are other transformations of space beside rotation Bentley translations based on early fell out the rest of the year the rest of the group of transformations of US-based these other rotations and the rotation for example about the the origin of the all-or-nothing you you could do was spaces to translate translations are generated again by you their unitary transformations but where the infinitesimal generators translation a special translation the momentum as well momentum is a translation but say
about the axis will you push everything along the the x-axis by infinitesimal amount again the form 1 Plus I times a parameter the parameter now would correspond to the distance that you push things just as they you translate again with quite epsilon some angle now it's a distant times the x component of momentum this would be a translation along the x-axis translation along the y-axis 1 plus I Epsilon PYE himself so this would be the group of translations this will be the group of rotation about commutate is between rotations and translation armed would you expect them to commute or not Rick South about commutation postal among just translation you translate something this way you translate this way he translated back this way and new translator practice were translation along X Y invert the X in the back when you get look at when you trade slaves or what wrote source of wealth again when talking about not talking about literally moving things of talking about just imaginary transformation no no force while they do do it did happen and do what did today below at 4 after all the years for about a year there's under a deal actual motions of a system that just asking what With the state of the world be like if you displaced everything I'm not saying displaced at best what would the state believe you did this place go little bit of a difference now now mathematics just a mathematical description of the displays thing if you this place along an axis in the new displaced along another axis of new displays back along the perspectives of new display back along with 2nd you come back the same place that's a property of Euclidean space so what does that tell you about coming skaters moment 0 commutate a moment or 0 came now while let's let's try let's try to what happened Va therefore you data point 1st translated translate by 1 unit in this direction now will rotate by an angle so an angle theta I'm sorry rotate from he up to say all he to settle data the angle theta and they end we translate Back translate along the same access back and then on to do with the rotation where we come to agree combatants in point now after we come back to about Over he is someplace so much clearer the translations and rotation don't commute translations and rotations I knew that if you work it out it's not hard to work out to do little translation little infinitesimal translations in you could work out 1 the commutation relations are between the rotation generators and the translation generate I'll tell you what they are they just commutator Our Ehrlich eggs well let's just 2 in 2 dimensions are LIA with P. J ii epsilon JKE PKE but that is not what of this number for what's important is the idea that everything is kept track of buying these commutation relations I that's the 1st thing about groups groups are really commutate algebras in the skies or vise versa next thing about groups Enquirer can extra solar symmetries but they tell you something about the energies of different configurations naively it's clear mainly it's clear what they say they say that if you perform 1 of these symmetry operations and you really have a should not change the energy of the system if you have an object whatever that object is has a certain energy and you just rotated so should change its energy off just translated that should change its energy so they have a legitimate armed symmetry then the conclusions should be back when you perform a symmetry operations energy of a state shouldn't change but what that says to mathematically approves that says for the requirement or case so let's suppose we have some state any state ever happens that have energy each as energy summer eigenvectors of the energy has a definite energy works quality and let's applies a small transformation to it but college you where state EU fuel is it does that this more just the transformation of the appropriate kind translation rotation isospin transformation anything we corresponds to a symmetry of the system and now as with the energy of the resulting stated we find the energy of the resulting state posts 1st or say what this means what this means
stated given energy means that that that's eigenvectors the Hamiltonian the Eigen value that's the definition in quantum mechanics of saying thing as a definite energy but the Hamiltonian acts on just give me Eigen value back eigenvectors of energy ballots ask about this state if the original state was an eigenvectors energy knowing of symmetry and the resulting state must also be an eigenvectors the energy right so that I must say that Hamiltonian on this day must be equal to the energy firms Assange state you will eat but this is also a quote 0 age you what just I just said that energy firms you tires EU could we replaced by but they Hamiltonian over here why while because each on a as he I said I just subtract H U minus you what age on E equals 0 that's what the conclusion is if therefore a symmetry operation doesn't change the energy but the flow of the truth is that if they do that our parents go through it again bigger through again this is tricky we start by saying We haven't eigenvectors the energy H part he course E on this says that the eigenvectors he eigenvectors Hamiltonian with energy said No. 1 right now I assert that you would he has exactly the same energy white because you would is a symmetry ever rotates does something with just a really according to translator rotates about knowing important to the system this must have the same energy and so this must also be an eigenvectors of Amoco but right that H. you'll eat must equal the E. U the drug he is just a number of except the sits in the vector of that 8 times vector here must it will eat turned the same OK that's the that's the statement they Utah disease also eigenvectors of energy OK now eases the numbers Michael said so it can be brought inside he doesn't matter which side of you it's are just the number you'll find each there enough I didn't do anything I just to change the numerical number EU with operating but now he comes the victory is H firms the rectory so I can't say that this is you'll age eat it causes former here but chip and now come of the marvelous conclusion by Trans boson H times you minus you'll times H. acting on edict is equal to 0 for an eigenvectors of energy but this has to be true not fun 1 particular eigenvectors energy but full eigenvectors of the energy for any eigenvectors the energy this must be true if the energy Eigen states are complete basis of states that follows mathematically that H. times you might pursue times H must itself be 0 is something is true of something gives on every sector for every eigenvectors of a particular operator than that thing must be 0 so the conclusion is if you have a symmetry it means that the military operations commune with a Hamiltonian types of Aceh signal formats the conclude that conclusion that for weather a set of transformations like you of symmetries unmarked they must commute with a humble target I still hear every symmetry where'd is but commutes the Hamiltonian and that's equivalent the saying that the symmetry operations do not change the energy of the state but still has an example as an example we figures a figure state of spin pointing in a certain direction rotated what happens the energy of the answer is nothing why not because rotations a cemetery can't spin in the magnetic field what happens if you rotate spin and magnetic field it will change its energy but the reason of course is because you're not really doing the symmetry operations you're rotating 1 thing relative to a thing which is not rotating will happen if you did a truce symmetry operation that would mean rotate the spin and also rotate the magnetic field that would not change the energy after same thing for example you have particles on the same electron beams of electrons in the field of a proton what happens if you translate electron as it does the energy of the electrons say the disdain of cars not like to do work to move it against a proton so translation of the
electrons not a symmetry not translation electron by itself was not a symmetry but translation of the whole thing is a cemetery act warm translation bigger symmetry rotation beings symmetry says that those operations commuter the Hamiltonian now if every you'll commutes of the Hamiltonian that's come back for the generators that says that the generators commute from here yes every operator of the former 1 plus i Epsilon Eyal commute the Hamiltonian 1 commutes with everything commutator of
1 with anything is 0 IN epsilon matter here it says that all the components Of the generators of a cemetery commute with the Hamiltonian but what else could use a The commuting with the
Hamiltonian means if you go back quantum mechanics young the time derivative are thing which commute the Hamiltonian 0 sell 1 says cemeteries comply conservation laws or conservation laws implies symmetries it's not important work but this is this is this guy have simply repeated everything go we talked about before about symmetries but most were to remind you of that momentum conservation angular momentum conservation also trees and of course there are plenty of other cemeteries around now the picture of deaths in the fire feel has the symmetry or you have to rotate the external field right right OK but important thing is that the rotation does not change the energy or not rotation but betray operations don't change the energy that's the reason that the different components say we have electrons in orbit around the idea in a particular orbit around the island nucleus OK so the orbiter characterized by a total angular momentum and busy component of angular momentum to different compose components of angular momentum could be rotated into each other they get mixed up with each other on the rotation total angular momentum Eyal doesn't change under a rotation of total magnitude of the angular momentum but the various see components of the angular momentum dual rotated to whichever other this symmetry under rotation tells you that all the different states of an Adam at a given L but a different in every window in this for anatomy and Mr. magnetic or number really corresponds busy component of angular momentum the different components components have the same energy as a consequence of rotational symmetry bite or the symmetries that we discussed up till now have 1 characteristic in common when they act on HBO's islands they give back bows on when they act upon me and they give back for me what happens to LBOs on what happens to an electron if you rotate it stays electron doesn't turn into photo Our quirks have symmetries the color transformations remember the color transformations mixes up the 3 colors of court they took wanted to acquire it doesn't take a quirk in the low blow on her blue ones about was on a court for me In fact although the symmetries we talked up to to talk about that now don't change the spin of a particle it would take a scalar particles scalar particles uh half spin particles perhaps particles generally speaking the symmetries we've talked about don't do anything spins don't do anything the type of particle don't do anything to the charges article keep the Charter of particle unchanged and especially don't do anything to a massive particles in some way the symmetries tell you that there are it qualities between masses of different states for example they do tell you that the various Zeke components of angular momentum all have the same energy because you can rotate to the other Ichabod of angular momentum and Adam as well as added was not an external magnetic field for electric supersymmetry is no kind of symmetry it's a really crazy kind of symmetry which when it acts
it takes a firming in tow bows on our bows on affirmed just like the rotations tell you that there have to be these multiplex with different components of angular momentum component of angular momentum which is plus a half the rotations symmetry tells you that has to be 1 of my staff of the same energy if is a spin off with 1 unit then that's enough to tell you the biggest and bound with 1 unit but if you look carefully at the mathematics you'll also find out best to be 1 of 0 in between all of exactly the same manner as supersymmetry does for your off as I say supersymmetry is a very strange no construction which when it acts fixed firmly on bows on the mathematics of it is very weird very strange very nonclassical extremely nonclassical and absolutely impossible to visualize and value a little bit about the mathematics but what is important incidentally it's important to be cost if have a theory that enjoys the symmetry has this supersymmetry then it will always be for every both on with exactly the same of every bows on a formula exactly that's exactly the situation that we're looking for where both sons of Fermi might be able to act exactly cancel out there were Infiniti use or a large large renormalization effects that we discussed a couple times a ready this very very hard to to motivate his hard to motivate because it is so completely on to it very intuitive symmetry that would pick from me the bows on the generators of them cold Cuba but Is it killed 2 of several generators called Culotta score queue for the moment and they take a bows on but doubling of these Cuomo's works figure at how Sabres right at OK alkalic make an operator which takes a formula until bows iron bows are they were firmly let's think about that for a moment kill is an operator which could take a firm until bows on the bows from what kind of operator could do that well very seriously easy actually has suffered a series of church well it has was enough to do if you have a bows on it has removed the blows on and put back a for me the this is a fairly and that it has removed from the Bekaa bows out construct Europe you does exactly that we of creation and annihilation operators for me under both arms right that what we want Q which takes a out to a and it has to annihilate elbows on so let's put in annihilation operator but see I was forget a creation a a daggers creation right so it is a violation of this annihilate a bows on so did just that this supposed this annihilator bows on any annihilation operated and they move followed by the creation operated for affirming what we call creation operas from music see see that right see see dagger but it is an apparatus which never acts on a both sides will always give a friendly right when Adams of lights from on Bridges nothing right my from aren't so that's a had something to it let's get to it 80 dagger a kind see this nite violates a fermionic and create a bows out right so where's the subject of Phobos well this each the bows on and spits out a friendly on sewer creates a firming up what about this piece of this case annihilate firmly on but there is no firm on so the Teresa is dead at this relationship is correct what happens if the same operator acts on a boat on a dagger this is a from Iran plus see better 8 what does that do on a firming well this work and eyelids that there is no bars
nothing this 1 and violates the fermionic put back above the trumpeted bar so yes we can make operators which Dole have the property that they intertwine between firm eons of HBO's the revisions Flanders Physicians Park well that acts on superposition will produce a superposition of acts on bows on the formula for back from Umberto bowls and it won't make a superposition or itself is not that he will it is not there is operated Our excellent general state will make a superposition of attacks both admits a formula for of a firmer makes bars what happens if it acts on the vacuum it was a tour on record for a luscious if war if creditors right here does make it is 0 1 leg 0 let's try on a miraculous a photo of a NATO violates a bows out but there is no blogs are see annihilate firmly but there is no formula adjusted 0 not the vacuum which is 0 0 so the 1 exception to the rule what the rule being that way acts on a fermionic creates a bows on the backside of both creative firmly on past the 1 exception to this rule out of exception will roll justify the exception is it acts on the vacuum it just gives 0 so these it has a class of operators which has the properties of supersymmetry generator supersymmetry generators are of this former may involve for meals and HBO's arms are in a way that is mixed together for years now the question is it we can we can have a low manager said at a place there if I was a little of both classes this is a message that is a very safe at bay what do a casual composed took a bronze yes yes so for example yes so often example got good good good they're here you you know telling me a little more about the supper ridicule sorrow let's suppose it acts on In the bows on the spins euro be creates a firmly on has been a half have particle has 2 components so does a create please speed up or does it create spend down will instead must be be if we don't make any sense out there so that they can be just a single queue that has to be toward them 1 which takes the bows on into the opt Leland and 1 which takes the bows on bound for there has to be another index Q was always have indices and the index is always there have been index itself was a ahead index because it has to be able to change the speed by half a unit that what happens if 2 0 with a spin off acts of a fermionic after a meal also has however the index the upward down so we ought to put their we are put indices on a formula that will see what happens Force it acts foreign aid F J. well basically the is all I use the same J. makes the bars on fire is not the same as as Jay makes so whenever these operators are he'll symmetry operate generators these other generators of supersymmetry Our generators of supersymmetry not only the they fermionic their fermionic In the sense that they have an odd number of firmly operators and Beirut on a number of per every interesting quantity that we are experiencing up till now things it we measure and so forth always have even number of Premier operators and silver generators have odd number firmly on operators because got to change the formula number and they also have indices indices are spin indices are really understanding supersymmetry and from on and fermionic fermionic symmetry fermionic symmetries because the generators themselves have odd numbers from the arms In the end they changed for me Uncle Bob's on really understanding that the the mathematics of it really does entail a generalization of arithmetic of the notion of numbers but just remind ourselves 1 thing about fermionic fields and bows on fields or just Beyond the operators creation and annihilation operators good enough fields made up creation of violation apparatus but does remind us of what we know about the creation and annihilation operators blows on creation annihilation operators for blows up bows arms have commutation relations the commutation relations are kind of made up of 80 with any and this could be any creation operated any both on creation up the annihilation operative what's about what's being commutation relations with an 0 the effect it could be different that it could be for different ball on even itself the commute with each other about a dagger with a dagger where about 80 with a dagger or is it a dagger with a big upset worry was negative 1 but that's the algebraic properties Our bows out like in particular this could cut to this particular relation over here of from over here this could actually Stanford to a different creation operators 1 let's say for a particle of 1 momentum 1 particle another momentum off particle 1 place the article waste let's take this relationship yes end create a particle state Tobosa on 1 we label I in 1 label J said Labor line label Jacob represent momentum they could represent position they could represent anything you like Pollyanna creator with them were going to create by just applying the creation operators this is the state of 2 Bozo Wandsworth type II in lot of package houses is differs From a state where the same exact particles but on which they been introduced in the opposite order a dozen different exactly the same thing it's exactly the same thing and mathematically the commutation relations he'll tell you that a better guy a dagger J is the same as a baggage AIT their equal to each other I'm busy it the symmetry of wave functions under interchange Of those are when you interchange the arguments the arguments of the wave function of our labels labeling the
state's is related to each other despite the changing I it does not multiplies the state the No. 1 that's the character of both sons and the character bows as indicated by commutation relations between a between the between fields between the things that make up the fields where about our seas OK but other commutator 2 seats but but Toyota when he 1st Ciera c bad is that were not the subway we do things with firm Iran's with freely on circuits and bicarbonate data was and I commutatum mean means dagger plus see Dennis C. C. dagger plus seed Agassi not miners let's see what that means is that every place where you saw commutator can replace a buyer and Viacom was commutatum means and die commutator of 2 things a would be means a B plus VAT approved a B plus BAT OK 1st let's see what this says what say about particle state 1st suppose is only 1 kind of firmly on just a CD with no label it hero of sea dagger was seen that it just means seed Agassi dagger policy dynasty that Disney's twice C. Davis a dagger that's equal to 0 sell it just says when you multiply see Decker by itself you get 0 but doesn't mean the sea daggers itself 0 C dagger creates a firmly on what does it mean to say that when you've square it period 0 it simply means you can't put to affirming islands into the same state is simply can't but the from and the same state 5 as probably exclusion principle so the Polly exclusive principle is simply obtained from a country that from is that commutation relations now supposing you happened that have 2 different kinds of from and they might be the same kind of firmly on electron but they might be electron 1 place electron another place of electron 1 moment the drama of the moment the or they might just be different the state of electron but suppose again to labels would make them firmly on now see Degas seem jail How does that relate the state dagger JT c dagger 5 where interchange the Iron Jack well we can't die commutation relations state it seemed that I see dagger Placido Vijay Singh I equal 0 another way of saying it is that see baggage is my next seabag adjacent that this is the enticed symmetry of the wave function of what they hit by symmetry of firmly on states with respect the interchange of the labels of the Fermi so sold the character of formulas in the character bows arms is encoded in these commutation Orient commutation relation on whether family dealers or whatever the mathematics of affirming is related to the mathematics of HBO's owned by replacing commutatum abide impact but particularly the property all of the school where I'm from you operated being 0 is very very different than the square was on offer it at the buzzer fields both field when you build up a large number of HBO's arms behaved very classically with what we have now the large number of 5 times we think of these fields electric and magnetic field we think of them as ordinary numbers reading of ordinary numbers 5 an orderly numbers Camille what really numbers commute what about firmly on whether frontiers you never build up large numbers state so field operators of fermions never get big never get big because this square a zero-sum alchemy get big armed as a hold generalization of arithmetic which up Khalil a little bit about just a little bit in which the numbers arithmetic are replaced by numbers which in commute instead of commute and these numbers are extremely useful they really just bookkeeping devices the bookkeeping devices the same way that me and fields a bookkeeping devices they keep track of various relationships is not real numbers is not numbers that you can measure numbers that day but the experiment can be entered the 3rd victim be inches to experiments the bookkeeping devices because grass numbers of theory of no kind of numbers are as I said it's a generalization of arithmetic and tell you a little bit about it because firmly on fields all really grasp the numbers really grasp the numbers the generators Of these fermionic symmetries these things which take reveals the bowlers islands also grasp graph the numbers have the property that they can't just a little bit about that said drop practices now they really just a curiosity but they really are at the heart of row of what supersymmetry is so we have a collection of ordinary numbers begin let's call them what numbers article the alphasub I I just labels which number with talking about just labeling the officer body is something which is true of a pair of ordinary numbers officer by office Jay's equal subjecting officer 3 times 5 is 5 times now that these new numbers breast call say that instead of Alpha Beta I don't know how many there are only 1 of them might be 2 of within the 3 of them how many there are they have the properties that they are in Dyker commutes With David J. another they I made a jetty at quarter minus David J. later when you interchange when you interchange then they change sigh that's a carrot that's 1 of the characters of grass numbers and honestly what the value of the ideas that 7 not 6 not doesn't 3rd doesn't have any ordinary numerical value another thing you could say about them is that squares as Iraq faces which is a good said but they 1 squared 1 squared is just 1 favor 1 but they once 1 is just 1 half of 81 favor 1 peseta 1 thing 1 and better once they want us they don't want us 1 just be an icon of fade away paperwork so they squared are equal to 0 out now let's suppose the simplicity now let's suppose is only 1 grasped the number was just caught data 1 grass number as like just having 1 number an arithmetic the No. 7 will also end an ordinary but we
can have both would never be numbers and grasp the numbers in the same arithmetic I should tell you about the OK so we should we take an ordinary commuting number the court Alpha Ruelas ordinary numbers commute with Is that I had that Camille with favors command I Mueller themselves as but with 4 4 yelling thing that intake commutes is aggressive numberless soft grass man no grass mob last year that no respite kind smoke the vast graphs graphs graphs graphs but yet again
quest grass mud Crestmont stability that the German mathematician 19th century that was clearly
smoking something worthy of their ahead if you're a half what that would aggressors number times an ordinary numbers a grasp the number right this is that's right we could easily prove that the grass the numbers times ordinary numbers are ordinary numbers what about Western number time grasp and those of grass on the grass numbers ordinary what about ordinary Pendergrast To grass right now so ordinary times Nari's ordinary would nearly transgress news grass and rest times ordinary appears I can't even numbers it all are times odds is even even even even the odd times Our art that that however but all OK so removed back face that's a property of grass numbers of what's OK so it's our Rio Our get a real really is a real real times a areas imaginary imaginary have demanded Israel the torso true a vegetarian real numbers but it's particularly true now what is what is unusual is that the square of any grass the numbers 0 because they have that commutes with itself 1 look set it I don't know Europe OK yacht across product of 8 times B's minus the cross product because they struck for Barbara overdue Mounger goes too far but yes that's right OK if there is a generalization taste nation on numbers the a generalization numbers we replace commuting these postulates of arithmetic is that multiplication commute he Pierre the postulate of arithmetic is is multiplication and by commutes are let's take the case of only 1 grasped the number the number and think about functions let's think about functions now of grass numbers will have grasped the numbers an ordinary members ordinary members of ordinary but there's only 1 grass number OK only 1 rest these functions of grass the numbers are very limited but what kind of functions we can build a grasp numbers biologist that to build polynomials of polynomial might be an ordinary number any ordinary number but caught Alpha plus another point you could multiply grasp the numbers by year by ordinary members so the next EU commandos is state at times a grass and number field is only 1 breast and number now now that straddle the quadratic polynomial called after beta gamma status square but say this good as so we run out there linear polynomials and back it was the most general functions of a single grasp the number please basic functions is not very rich consists of basically still functions constant and a linear function Of the grass and number no more than normal functions and that so it's a pretty damn simple arithmetic are pretty you they accuse stated time status squared the data squared 0 you 0 so everything beyond data Our on data 0 that's it is no there's not much it was not much of a space of functions OK let's say suppose that there are 2 0 grass variable straightaway and fated to them what kind of functions are very well again as Alpha Beta Theta 1 beta 151 quality we can then it was too to wanted embedded to adjust our ordinary numbers annually numbers could be complex numbers so important at the moment with the complex numbers of real numbers whatever they happen be and then we can have plus again they 1 favored to pay a one-time Sato is not 0 they won squared is 0 because it that commutes with self faded there are 5 it's a commuting number but it's surely not ordinary why is it ordinary number because it is not but it's square 0 there while they 1 the faded to 0 so it surely not ordinary number Hey it's not 0 but square both does have the property commute commutes with Russell that's it that's the last that's the biggest function the most complicated function you could make of incidentally this it is not equal to our Our Spadea 1 favor 1 1st base to failure to pass gamma slated to be the 1 point because they and die commute that means To make them equal you would have to put a minus sign you would never talk data a 3rd they what they did not say what she seems like a would review different all there's no you could multiply later bye have cup but real but young before multiplier for everyone so what we say we should say that favors arms or we told them opt to they really former two-dimensional vector space are apart from apart from multiplication by ordinary number really we toured an appraisal of the space of functions is very limited there's not much structure to it if there are 3 thinkers well but it's a little bit longer you go up to beta 3 favor 3 you could have 1 thing to within 1 3 2 3 and finally made a 1 2 3 and that's a sorrowful our polynomials in this a fury on all the functions of a polynomial functions of data are very limited and they are
nowhere Bob poor mental function picked whereby exponential data of interest to question but 2 exponential who let's see if he could maintain that say there's only 1 I think we can do this with just 1 favor eaten Lee a favor Ito be over OK let's that 1st eatonii made Huntington Beach is that equal to each of the 8 plus BDAT cheated on her with the things in exponent Arts Arts a 5 eatonii anything that is 1 plus anything that past ADD square meters square with 2 factorial plus a cube ADQ over 3 factorial but with dead after I left the data so he could be a favorite is just 1 later Ito the beach data is 1 plus beef data the product of these is 1 plus a fate and Crosby data plus a B square but Lafayette Square 2 0 0 so this is 1 plus 8 plus be later and that he is just he need plus they so exponential 's exist the algebra of exponential is exactly the same as the algebra of exponential ordinary things but exponential no more complicated than that curious what works with 2 now they 2 different flavors which try Our case where eta eta favor 1 each of the each of faded to send burqa 1st we get 1 closest 801 times 1 plus they to which is equal to 1 they'll favor 1 to pass they won favor to I think that it is and that's it that's as far as we got OK let's see if that's equal EDI halted his Peter paid a warm plus this should be equal to 1 that no 1 had frustrated tool class Beta 1 plus faded school aired over 2 factorial right over factorial plus higher but the problem of hires elected again anything Teresa we get something from here it is because we have paid a one-time stated to they want square 0 faded to square to 0 but we deal have twice they fated to over 2 factorial which is exactly what appears here but now what happens when you go to favor 1 slated to cure that have things like they do want you 0 it's gonna have things like they do cube that 0 let's have paid a 1 square times they did to bad 0 because they don't want square 0 below is always given me some repetition of 1 of the ladies and a repetition means it's equal to 0 service things not there and yes it does repose reproduce that's interesting just an interesting fact armed exponential functions exist but they were extremely simple a class know I know it doesn't it doesn't doesn't I let's see why that's a good point with a white doesn't hung in favor 1 was slated to warn you know you may be right that Cup looks right 1 room they once square was slated to square up 31 faded to slated to favor 1 yacht know that their what's were calm a player date he said just a rest of requested young male armed will exponential spoke but always makes sense and are looks like it is a failure of multiplication of about my missing something that looks like the work fresh really know that traverse a 1st time River tried here was sent much of it the chip friend yeah tapetzella 0 but no but this is not 0 they 1 faded to is not 0 after that because I'm not he goes might stupid all I don't think so they were not don't think so I don't think so now I think if fails think really does failed culture kept our is union no wonder if they want was not an ordinary yeah lacked makes them realize that throughout throughout their looks like you can't exponentially and we get mixed I get jumped group from refuse by if Arafat Georgia could but Her visions they are they are the era I don't think they like anything else I don't think like they're like anything else their bookkeeping device which is useful but not like anything else regressed remembers grass the boxes of grass numbers can be differentiated and integrated riches a useful bookkeeping device that what I say they can be integrated I mean you could define the intervals a rest number and the derivatives that satisfy certain properties are if you have the derivation
derivative I'm a function progressed number is straightforward derivative with respect the data of 1 or a leader in each 1 or 80 it is equal to 0 derivative of data with respect the Bader it is 1 so for example if we have just 1 favorite a problem we have functioned as a Class B fate at the derivative of that with respect the data is very straightforward and it's just data at supposing we have more than 1 this is easy and this is just ordinary derivative and this is definition is a definition we derivative with respect they don't want Our function Alpha but there are 2 status just example to favors Alpha post Beta Theta 1 plus they don't faded to pass finally gamma 1 faded to try to go through it just as if the was an ordinary functions derivative with respect they want you find something from he and later 1 traffic from media nothing from media and from here you have gamma faded is exactly what you would expect the only surprise in the definition is that to differentiate with respect the faded to nothing from the Alpha nothing from beta to more about here you might expect paid 1 right a case in the rule a derivative operation is also like grasp the number end when your pettitte through another pressed the number changes site the rule for the vast numbers as get past them through each other they changed the derivative progressed the number is counted an edge a grass with respect to sign so that means when you pass through fatal want you change signed and this will give you might Schember so you treat derivatives the same way you treat the grass members themselves a derivative with respect for data is also grass and kind of variable I wanted passes to India other grass variable changes they'll only peculiar things about differentiation with respect to were the grasp remembers incidentally these rules sure of some of the rules of ordinary differentiation for example the issue of the rule that a derivative of a product is the 1st derivative the 2nd 1 plus the 2nd 1 times derivative 1st for the usual rule our our calculus for product if in chains assigned to get the trouble so the calculus of grass variables His again simple you don't do anything unusual access remember that whenever a derivative passes through a grasp and variable the thing changes side we have to do a integration of grass the variables which too late for now are we will be able to write a cable systems so few Glassman function the table of grass integrals will be about that long so will also find that intervals why am I doing this it is just to show you that there does exists a of mathematical algebraic framework in which Our which supersymmetry and which fermium operators fit even tho a new kind of arithmetic it is essential promised Stambaugh Mr. understands supersymmetry like a dress you're not they want too was 0 head a letter dated to where death on a a a null and it's normal food that the square of where it would be best asked him where it is there the productivity never know this now you see this is equal to favor 1 squared plus slated to square plus they know 1 thing plus they 231 right but quite true right distrust To act so this month that it was go 0 just as if they wanted to force they had to be the 1 equals 0 yes row of their one-time Slater was you might too they warm but they want to his medical but still puzzled about this exponential I had always thought that the I never used it fame and I don't know what's good for but I was hoping that exponential exist that they have son however it Jack multiplied the fadeaway they through it does look like you did left the like you do ordered or this he wears tire wear the that means this is at work right but that's not the same as it was simply not the case this is equal to that To bring a case but ah Gypsy mean that it threatened about 90 per cent of of all got me out definition I think that there may be some other there maybe some other differed after think about the bibulous definition art for more please visit
us at Stanford Dr. DU
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Titel Supersymmetry & Grand Unification: Lecture 4
Serientitel Particle Physics 3: Supersymmetry & Grand Unification
Teil 4
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 10.5446/14977
Herausgeber Stanford University
Erscheinungsjahr 2012
Sprache Englisch

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Fachgebiet Physik

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