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New Revolutions in Particle Physics: Standard Model  Lecture 4
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Title  New Revolutions in Particle Physics: Standard Model  Lecture 4 
Title of Series  Lecture Collection  Particle Physics: Standard Model 
Part Number  4 
Number of Parts  10 
Author 
Susskind, Leonard

License 
CC Attribution 3.0 Germany: You are free to use, adapt and copy, distribute and transmit the work or content in adapted or unchanged form for any legal purpose as long as the work is attributed to the author in the manner specified by the author or licensor. 
DOI  10.5446/15085 
Publisher  Stanford University 
Release Date  2010 
Language  English 
Content Metadata
Subject Area  Physics 
Abstract  (February 1, 2010) Professor Leonard Susskind continues his discussion of group theory. 
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00:05
Stecher University I'm gonna do
00:09
something I've wanted to do for years and I really wanted that OK but gentlemen people how the audience viewing this from afar we never stopped for a commercial break I want alter the life by my book The Black followed water my battle with Stephen Hawking make the world safe poor mechanics air lock you will learn about the cruelty of theoretical physicist your own only learned about showed interest cap but you will learn about source dunes fresh she and her flushed man the pilot for years use a how little bottle removed you may remember that when we last saw allies and mom Alice was crying barely because the world long was seen her at the low end world theory Alice dared to say is not finished Bob is not finished were delighted with the way we will see Alice suffer more smoke 880 questions about the group theory for the moment a the battled back story means storyline and any questions about group theory where were not finished with we really ripped from yacht used to parts of your expose the route it seems to be some that at that time the logic of Asian and there's also the board on group act of various projects that reduce the jewel flout they they're completely related to the abstract notion but so let's take the case of rotation of a spent there's a perfect Kaster for study busy abstract completely abstract notion of a rotation basis but don't tell you what I just tell you that is an operation that you canoe physics has spilled respect symmetry under the rotations of a space and sell we must then say how does the rotation of space act on the quantities which describe which are physically relevant if we were talking about classical physics we might just say that the rotation of space our rotates the coordinates of a particle in might rotate the direction of a magnet whatever it does we know how to describe that we describe that by we have described that in quantum mechanics states of the system always represented by a linear vector space busy abstract notion of the state of the system and the concrete representation of it in terms Our column vectors how big are those column vectors what see Steve number of entries in the combat what depends on system but basically it's the number of mutually orthogonal possibilities for about system if the system and questioned only consists of an electron spin and only 2 states told mutually orthogonal states ups and downs of its toll electrons and again were only interested the speed and they would be for states to up to Brown point bam bam up Our if we're talking about a more complex object for example the entire motion of an electron its position as well as its more them then there are an infinite number of states number of states that it takes to describe were bitter located a number of mutually orthogonal states of an electron speaking now about position is infinite it could be anywhere is in space or you could choose to describe it in terms of momentum so the column that there's describing electrons position are infinite dimensional an amplitude for every possible location offer every possible moment so they must be representations of the rotation group of this group of rotations which are infinite dimensional matrices they are not just how what are the matrices which act on column vectors which described the system if you're interested in the size of those matrices will depend on the number of mutual Allstate back so that's comeback of electron so everything else Avenue head spin of electron go by 2 matrices group elements are the abstract work stations they're represented by 2 by 2 matrices a spin 1 particle spin 1 particle has 3 mutually orthogonal states locations are represented by 3 by 3 meters use that those 3 by 3 matrices are familiar the Somerville they are the threedimensional matrices which makes up the components of free vectors comes after Springer at a threeday at that 1 half of Christians in 0 about 0 0 rotations don't act on at all along so it's completely trivial you could think of it in summer Mary useless sense that that rotations I'm act was 0 just leave it alone that corresponds to 1 by 1 major she's just the unit matrix of just didn't make expose nothing Otto by 2 that's perhaps 3 mightily that's been 1 for 4 that's been 3 halves and so forth so the sea abstract group elements the seem abstract symmetry operations may have many matrix representations on different dimensionality on that important fact delivery who has been more time but let's just a review for a moment what symmetries we talked about the symmetries of rotational space on a spin has been a half particle particular may have article are those those rotation operations become represented by major cities which I called you you'll because they unitary so these are 2 by Unitarian matrices I want father being more specific and just putting out their most don't like to unitary matrices act on stage vectors are spinner half again I won't be more specific and writing around and this is called a representation of the book but tomb onetoone correspondence with the group elements matrix multiplication group multiplication her onetoone correspondence but just Russia was it mean from matrix to unitary it means that its inverse is intermission conjugate but we know that number or degrees of freedom number of parameters in a unitary 2 by 2 matrix this holiday how did we surrender warm 8 To begin with 4 elements each complex as for elements of this His for equations that cut you down before but there are only 3 independent parameters the 3 or angles if if you like the 3 angles describing a wrote 3 things describing rotation mainly the rotation angle and the 2 parameters which determine the direction of rotation so there are only 3 independent parameters rotation these matrices have 1 too many parameters and get rid of that 1 extra parameter 1 more condition that's the determinant of you as represent that we ride is dead but I'll describe it as absolute value but it's not really absolute value the symbol for determinant but that is equal to 1 now a unitary simple fact that when you multiply matrices the determinants multiplied determinant of
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8 times turn the other product of any is the determinant of 8 times a determinant of beat a it is also true that the determinant of 80 her mission conjugated operate is complex conjugate of the determinant of the original operator solo is and say Let's take the determinant of both sides of the determinant of both sides of this His 1st of all the determinant of you dagger you but that's also equal so we determine interviewed dagger times a determinant of Yo was said that ended the war 1 is able toward I 1 here of course means the unit matrix old ones on the back of my now next thing is to determine interviewed dagger is just the determinant all of after well is determinant rum is a complex conduit to determine you and if the determined view was 1 we determine you daggers 1 and all things consistent don't point is that it is consistent that's a consistent conditions on you with that consistent I mean that's consistent with the law on matrix multiplication by if you take a bunch of Unitarian matrices every 1 of which has determined 1 any multiply the together you again get a thing with determine what so it sets a mathematical theorem which may not be too surprising from any of you that the special unit Terry matrices toe but 2 matrices which have 3 parameters and special means that you value would cause warned that the group of those matrices identical to the group of Our rotations in threedimensional space now it's not quite true that all there is nothing that says that Germany has not been found that they just as he read lot cut right determined the unitary matrix must be a pure phase that's just from the fact that unitary reflected unitary but you could take any support you think in unitary matrix but suppose it's determinant is eagerly I you could then multiply that unitary matrix by armed William multiplied by Ito the minus side they go with 2 because when you calculate the determined to apply 2 elements together so he took you have multiplied by each of miners I failed to you would construct a special unit at the point is that the extra phase it's in is almost trivial you could always get rid of it by a redefinition of you and in fact the rotation group doesn't contain this PC it's just isomorphic matrices with our would determine 1 of our at tho Littlefair border said that as you too is exactly the same as the rotation group that's not exactly Tulu 1 correspondents instead of a onetoone correspondence unitary operators you and minus you if you was unitary operator you die you equals 1 soul minus you whether you it is in fact is it totaled 1 correspondents matrices you might you both correspond to the same rotations face but that's fine point that the barrier which is not the place any special role and then we say at least for the time being so acidic both statement which is almost rule is that there's a correspondence to live as a correspondent but a onetoone correspondence not quite true withstood correspondence between unitary matrices too late to make matrices special unitary matrices those would determine 1 and lead a group of rotations of without reason electron spin where electron wave function is called a doubler credits acted on by Timbuktu matrices let's talk a little bit about the generators 0 Group think we actually talk about but would be specific about again generators of the world when you have a symmetry and symmetry is represented by a set of makers she's like us then as a concept of the generators generators have to do with infinitesimal elements of the book and has element of the group means in the case of rotation a rotation about some taxes by by a very very small angle another words it's an element of the group which is very close to unity unity means no transformation elements which are very clause which is just very small shifts of angles what are called infinitesimal elements of the group and the generators of the group are very closely related to these infinite passed only small Our operation a number of reasons that infinitely small but elements are important and interesting is because you could build up any element of the sequence of small 1 another words if I want the rotate about some angle by a finite rotation I just think of it as a multiplication of a bunch of infinitesimal 1 so if you know the properties of the infinitesimal generators you couldn't put together the entire structure of the group are always going to do that American study that today but we are just going to define the generators of the group and the reason with undefined generators the group because because they really are conserved quantities they are the quantities our that are conserved and which played the role of conservation laws wants to know what the symmetry group OK so that's fine instantly and written here is nothing specific OSU till this could be SUO aid now where in this the number of elements special unitary and my in major cities there of course not all isomorphic rotations it's only the S U 2 group which is isomorphic rotations of space as curious to force you 5 muscle easy to visualize OK so so we have a definition analysis taken element which is very close to the identity it doesn't have to be issued to write a new is equal to 0 identity 1 class something small and let's put it right in forever simply notational simplicity will simplify patient a small parameter epsilon and a quantity which is often I'm not gonna call Gigi for generator because we've already use G 4 group element and so therefore I will use he White beats may I don't know so you'll is 1 plus I teach 1 let's find out Wickham learned about 280 we use a fact 1st of all you that you was war so you Daddy you that's 1 plus i Epsilon incidentally it's easy to prove that a few Daddy is 1 thing you you daggers 1 don't prove that for you that you will you that always commute with each other not operators but for unit carries so it doesn't matter which ordered multiplied fights which 1 plus I have sovereignty times oneliners I Epsilon dagger and if this is you and this is you dagger i.e. gets change the minus sign when you complex conjugate and this is posed a beagle don't want Let's retailer things are only for the lowest order in epsilon epsilon square is too small for our interest tonight so it's just keep things or Epsilon and that's says 1 times 1 cancels the 1 and it says that i Epsilon times T minus t dagger is equal to 0 the simplified it just as that team is equal to T that Watson
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operator sequel boots on country permission annihilation operators in quantum mechanics represent observables the Commission operators represent observable quantities the case rotations what are these sees what physical significance delay have there are 3 of them little 1st while the 3 of them are actually infinite number of them but there 3 linearly independent war explain why that is the reason is because the rotation group has only 3 parameters is only 3 independent ways that you can rotate Yukon rotate about the Xaxis so I could have made In infinitesimal a rotation about the XIX I could have made an infinitesimal rotation about the Y axis or the zaxis and in fact for an infinitesimal rotation if I wanted to rotate about some other accidents like a 1st bill was rotate a little about the axis and then a little about the axis in about Rosiak there's another way to say it Is the generators behave like vectors if I want of vectors and arbitrary direction her words of vector of rotation about an axis about fine arbitrary direction the way I could think of it is making it up out of the components of the component rotations about the x y taxis so there are free independent direction said you could rotate in 3 independently linearly independent he's bowled with just 1 point from back back up there is this condition here conditions the determinant is 1 of 5 southern Italian visual develop back the determined being 1 of the determinant any matrix which is of the form 1 class of scored epsilon times a small matrix determinant determinant of him it is a 1 plus epsilon times the trace of small determinant of a matrix which is close the identity is 1 of just identity and
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in a small deviation is epsilon times trace that something to probe gone that but we can see what that means immediately it means that trace R T 0 the trace of tea has to be 0 annoyer follow the determinant of you to be equal to more a 2nd conditions the trace of 0 and it should be her mission on Monday independent permission to by 2 matrices are there with trace equal to 0 the answers 3 of them and they correspond to the rotations about the but more abstractly in generating the generators of the Group of rotations are the angular momentum operator for a system that triangular momentum means in quantum mechanics is generated the generators of the rotation group enter your act with the angular momentum all 1 plus I epsilon Times e angular momentum to rotate system a small amount in the Tijuana rotator large America does do it over and over again so everything is contained all the information about the structure of the group was contained generators In fact it's contained in be commutation relations of the Jerry foes of important things I want going again we did commutation relations of angular momentum as want to connect things up a little before for you but the important thing is let me be conserved quantities associated with the symmetry on the generators the end of their time treated but reducing said there are 3 D a independent 2 to vie for the collect over mission yet that in a bowling alley did not say how did you would do 3 by 3 major sees as a new variant is a traces world that there are directors River reverse normal that aid is a sane a blue ones indeed for try to a that that's a different admitted that secular different 8 not because it's and mathematically and associated with different text S U 3 symmetry a color but young mother Maddox OK so that was a little bit about road you'll if the sorrows of our the structure hours to more with pursuit 3 Jared as Italy the pattern is exactly the same except what we're talking about 3 by 3 matrices tribal militia of the river a much more to say about it other than doing some account can and finding out how many independent generators are so the the campaign is very easy 3 by 3 unitary matrix has harmony elements stops on 18 18 independent 9 independent complex numbers that's that's 18 but then we have a you dagger you equals 1 now this really is AT equation for each element This is a product of matrix product so you think of as high Janie and this is dealt by J. over here many equations is theirs is not quite so we have 18 lenders 9 but then 1 additional equation that the determinant you could want equations and so we can guess that 8 different if we start with EU response With the unit matrix and how many distinct directions in the group's space could before away from the identity and the answers any implication being there are really independent Kareem by 3 harm trace was mission matrices Fraser was mission matrices number parameters is 8 sell power there were going to come back trapped but before we go let's talk a little bit about how representations combining the form new representations now what we're talking about here it is combining systems together before systems amid cases I love figured we might be talking about combining more than 1 particle together the create some sort of composite for example and I can't pass it would also have a space that's not worry about promotion of the particles with just think of them as things that we staple Copper Beech others the nail bound to each other but each 1 has a spirit so it as they are supposed that we take to such particles we put them on top of each other to enhance the particles what do we get well we give for a possible stage but those for a possible states can be thought of as a column a basis in there for particle states can be thought of as 0 object and the 3 In
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1 objects that about half the 2 have spins can make it we take have been we can certainly make us in 1 why come lineup the spins in the same direction so busy component of spin Comey 1 minus 1 and a committee 0 abuse 0 and to wait right but still has been 1 particle only has 3 states so 1 of those 2 ways of making 0 0 must not be part of the spin 1 multiple what can it be it can only be spin 0 why spin 0 because it gives nothing for it to transform into a server 3 combinations which formed the spin 1 multiple it and 1 which forms the spin 0 multiple when you combine spins together that way there are 2 possible COBOL spins you could make some if you look at again Adam made out of a may have to have been particles and ignore the orbital motion altogether 1 possibility is that the spins are in the same direction that would you use been 1 of possibilities is that they are an opposite direction in which case it would be spin 0 or Thorn power hydrogen particularly for the hydrogen atoms with 2 spins would be the protons and electrons on them so you combine it's important to know how the combine spins and to know what he get what possibilities you get when you combine together what uppish youth rate will exempt as 2 3 end discussed some very specific representations of S U 3 the important ones which we will discuss 1st of all there is a triplet Triplette it is simply if you'll is a 3 by matrix then hit has backed on a thing with 3 components In the present context the 3 components represent the 3 colors what so what talking now phenomenon 1 quart of 3 quarks 1 quart and that work comedienne red yellow or red bull while grain Italy red blue or green misses the amplitude that red this is the amplitude of is the amplitude of green and what does the group stood it mixes them up that mixes up red blowing green takes any particular state and sends it to some quantum superposition of states are mixed colors shall we say so we took a red quarter that would be like a up Spain 0 0 1 only rotated by Mrs. 3 operations we would get something of a mixed character which would have amplitudes of being red blue or green some quantum superposition that's that's the meaning of these operations but it's interesting and important to know what happens if you combine representations together we go back a step in the back yard how we want talk about various representations of this year's rate representations of SU tool on the spin states spin 0 at half 1 3 have tool 5 heads and so for those other representations of rapacious want something about the representation of S U 3 and I could tell you very much just a very little bit their rather towing 3 important representation for our purposes 1 of them is just a threedimensional representation and they can be thought of as the 3 components that 3 things that go here could be love as the 3 field operators for a quarter red war move court agreed we can I don't think of these matrices as mixing up the state's or a single quark all we could think of it as mixing up the field operators they create single quarks red blue and green now next visit all this representation like 3 by 3 matrices that acts on the quarks that representation is called the 3 In the same language you would go if he used the same language would call the hats things particle OSU tool would call it to tool for Tuesday sir so what the 3 is a representation that's identified with the quirk fuel itself all with quarks now they're also impact works in bank works midfielder operators are the complex conjugate of the quark operators say that is south if there a matrix which acts on cue will give some new we could complex conjugate everything we could complex conjugate everything that we could say Let's such right if that's scholarly you you'll was a 3 by 3 matrix that acts on the column that you I'm now being a schematic view was a 3 but we make that skewers a column vector and acts to give us court you climb a rotated Q then what is it that acts on the complex conjugate emission conjugate field operator the object which represents quirks Willis quite obvious it's going to be the complex conjugate video not demolition Khan said olive elements if you would use
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a bunch of elements you start is nothing but the complex conjugate elements stock quotes started so there is a 2nd representation of S U 3 which is the son of matrices whichever complex conjugate represent which the complex conjugate registered but every matrix you there is a complex conjugate and that defines a 2nd representation called the complex conjugate representation it is the transformation properties of entire quarks and that representations called the 3 this is very abstract but what is still in the case of electrons was is correspond to encase of Elektra it is simply the fact that if the electron wave function gets multiplied by the hour later than the positron wave function is multiplied by each of the minus side they'd be transformed with complex conjugate group elements sir you would say the electron positron complex conjugate representations a B U 1 group that's the language the quark nearby Quark are complex conjugate conjugate representations of the group SU 3 With 5 other cases you too young Meraux ASU tool is a special degenerate case where we as you do with a complex conjugate is equivalent to the original matrices by another matrix a special case call a real group but this is this is not so important for us now be 30 and the free bars representation quite different and they look very similar to each other but they are distinct they should not be thought of as the same representation your 3 quite different than fact the generators the generators Of the 3 bot representations are the negatives of the generators Of these 3 representation that's not too proved that they that means they carry the opposite color if a quirk carries a red color the nearby caught carries a minus red color has the opposite value for the generators I is our work and the and work now what happens if you take look work and antiquark don't works all are Corrigan backward just like it could take 2 electrons and make spin 1 which was Triplette not capital of history but because the trip you to decide you comic Osborn wondrous been 0 you could do some similar things with me and Reebok you take took Watson put together Of course we kind of Reebok started 3 times 3 quarks off 3 this is just a notation 3 times arranged quarks sometimes is represented by a circle to indicate that unite multiplying numbers is simply building the space which is now a space of 2 quarts it has 9 independent states Red red red blue and green and so forth and if you work out what representations of S U 3 of here when you multiply 3 by 3 the answer it is 1st of all there isn't In that Quark representation of proved to be interesting that if you take 2 quarks you make something or whose color is the same as in and backcourt that's a little but I think but strong straw and 1 more representation called the 6 dimensional representation of S U 3 whenever they need 6 dimensional representation but there is a 6 dimensional representation surgery on I can't tell you what these things correspond to act if you think Of the quirks is just indices color indices then you could combine the to quota think about them combined the 2 what states symmetrically already record great you could combine them symmetrically R and by symmetrically and Dyson symmetrically gives you the 3 bars symmetrically gives you the 6 but this is not the point is if you make to Quark somehow laboratory of the transformation properties Of the 2 quirks system under this issue 3 1 turned in 1 way of combining them together 3 independent states of that system which will be like and the other 1 will be this mysterious 6 which we don't give a damn about force you are now what happens if we take a quirk and get we take any Ekwok dike work where we get is it different than what we get what we get work quart we get again 6 3 is 9 states was not the the same 9 possibilities 1st of all we make an object which is awarded was a 1 that's a thing which is completely invariant under the collar operations roughly speaking at a Red Plus that ready and I read Quantum superposition ready by red blue and red blue and green and agree rats 1 thing and that's a single representation and simply does not transformed under the astute 3 group it's called a single at an S U 3 singly and 1 way of saying it is edits the analog of combining together 2 electrons to make a spin 0 state is like a spin 0 state it does not transform under the rotation of the transformer offers is 1 state so how could transform was nothing for the transformed into so you don't Kwok an impact court their 9 states 1 of which 1 linear combination ready and I read plus blue whereby bloopers greedy and integrity that is forms a single and then there are 8 others and I was 8 others Our 3 8 dimensional representation averse to read that each incident we think about that aid for a moment we see in a another place it was a number of generators just like the number of angular momenta when you rotate space the angular momentum rotate into whichever when you rotate in color space me a generator's get mixed up with each other the way they 8 generators transformers called dimensional representation of 3 this is sometimes called the adjoint representation of the words don't matter it's just a
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representation that transforms the same way generators and that's what you get when you multiply 3 cross 3 finally won the other operation when he'll get if you take 3 quarks 3 cross 3 cross terrain take 3 quirks 27 states while you can do pieces ukase AFI combined together these 2 quarks what I will get is either a 1 Surrey is either a bar which is in court or 86 Ireland Charles these quirks together will necessarily he said interbank work or 86 the Sixers among them is a peculiar representation rotisserie won't care about her much now what happens if you take this father and now cross it again with rate and other words we see what we get when we combine took works is a we get and then we combine to begin with another court we will ease E get what a appearance if we take a quark Anthony antiquark only on that it is that's 1 plus the posse it just means it ordered you Eva get all won the singly or in a toe starts a new weaving it's been 1 or spin 0 as with the plus me or William multiply 6 times the I'd be you get a plus but this again is not very interesting for us we don't care very much about the interesting thing in the most interesting thing is what we multiply 3 quarks together we get the possibility again I'll be sitting right a state which is completely cholera arrests it has no transformation property the color neutral it's just like taking a plus charge in my discharge urged cancel each other we get a single which has known as you 3 transformation of all Kenekham before lover as a combination which simply is invariant under the astute 3 group oblivious to 3 of the U. mixing up the the quarks What is it can we guess what it is like up 21 it's just a red green and blue quarter but you have to be careful 2 red green and blue quark entice symmetrise import of red green and blue or red green and blue quark are at the center of the city and there combinations red red green and so forth they formed the remaining states here but the most interesting 1 up purposes will be same but he so I see no only take a quirk in the APEC or just under guard to the colors you come combine the colors in a particular way to get a single class stuff you take 3 quarks you can combine them together because work force of a job is probabilities of things alone has the question is what the energy of them in turn will tell you the energy of the incident but will come back wrecked the energy of them is in there and so they don't appear Anthony inspectors balcony energy of the the ins and outs that was a Greek puzzle which we now understand your to talk about that right but now there is 1 other kind of particle in nature arrears many other kinds of particles in nature but is 1 of the particle In quantum chromodynamics we're talking about quantum chromodynamics QCD theory 5 quirks and glue won but of course we left out is did I the blonde is an object which I think I've told you before behaves as if it were a work in an APEC work with respect to the symmetry that doesn't behave like a quirk in the entire corpora error with respect to what happens if you hated attorney just with respect for color symmetry of color properties on the same as at work and the entire court but only be ate it transforms as they eat dimensional representation transforms the same way as the generators of the group so again it is this Ark kept which is separate from a single year it's the remaining states here and that's what the blue waters of on a thing with 2 indices like a quirk in the APEC work except the quirk in backcourt arranged into this head arrest you 3 source not neutral with itself was not her quirk postulates feel like of quantum crumb of QCD S U 3 symmetry which I will write down quark transforms as a 3 sorry and die quarter transforms as 3 bar and the blue 1 for the gaunt CEO as an 8 Christie's equations don't really mean anything particles can equal numbers particles of particles numbers are numbers just means the quark field transforms and backward as the entire trip with complex conjugate 1 seat 8 which is a P E C O Corrigan back for this remaining possible 1 doesn't exist in nature doing a white completely but the important point about it is it doesn't mix up with the other components of throwing away is a consistent mathematically consistent thing to do because it doesn't get mixed up with the other components were new transformed into that are that group theory as applied the quantum climber dynamics of light quarks and gluons are another postulate of quantum chromodynamics and it's not really a postulated dynamical Of the theory but let's take it as a postulant and explore what it means in a moment of and that pass through it is all Of the particles of nature of real particles nature not what will ones which have never seen a single are always seen in composites that the particles which can be separated often examined individually can of particles of finite master real things of current laboratory are all particles we'll
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particles Barrows nothing not real about work so I hate to when the real particles media unconfined particles of free particles the liberated particles the free particles are always transforming there no 1 another words they piercing with all the real particles and interest it was only 2 independent ways play with the ways of creating Singh what sort of mathematical fact there are only 2 fundamental ways of making single its Out of quarks 1 of them is to take what impact court that creators and the other is to take 3 quarks now you go no other things you could take a quarter and that work and juxtapose it would freak works by the elements of basic or you could take 3 quarks and another 3 quarks 3 quarks in a single and 3 other quirks in the same way that will also be in the same way but all of the objects that you could make up half the quantum numbers of combinations of things made up of 3 quirks or a quarter by quarter was want the little ones themselves although the 1st will be forwarded a skinny sings names was an object made out of 3 quarks for Barry Bonds From our errors right Faria oasis have half spin and have been when I say have been there can be worn have careers have white figures made a free hand been particles what about these quirky and corpses this is feasible ones of course in the 3 3 crossed the across 3 that Triplette sorry singlet call Barry on the singlet and 3 cross 3 bond with where they called quirk in backward as arms now we left out 1 thing can we make things out of the blue wall of course make things I warm yacht there's a thing that you can make just out school warned that if you take each Crowson 8 echoed you roughly what happens if you combine quirks together you could ask what happens if you combine little ones together Wurgler ones with quarks and that go go through the whole list of multiplication tables it you get when you combine all sorts of things but there's certainly at least 1 more interesting thing and that's a cross in a this is simply to walk Kabul on stars not out 63 states altogether 8 across 8 1 of them is a single 63 64 states 63 there are 60 4 states and a crusty and there are some forms 63 things which are a combination of a lover 8 other things aid appears a number of times I forget which I don't remember which representations of 63 of the state's none of them are saying just 1 way of combining together blue ones performed so you might ask his airway particle which is composed of 2 1 was just to blue ones bound together into an astute 3 the answer is yes they called walls are not made of courts don't have any court current Internet are there are fundamentally just a pair of blue ones you could also make 3 glue warmed to eventually possible also possible to get the ball complicated things but there are no walls so the spectrum of Haiti drowned the spectrum of strongly interacting particles which comment from quirks and warns consist of barrier islands as offerings include balls all of them are colored single collar watchword no another fact another fact given check this yourself easily I think we've actually checked it before a mail or we can make it is quirky quark quark what work who 1 1 end combinations of those what kind of electric charges do we get if we combine 3 quarks but I'd be going it's Sarah spend arguing it charged too but ceiling nearterm 0 1 until we need to 0 by taking 203 up quarks 3 down quarks Iowa with 3 bank or are 3 down corpses raise 1 3 down quirks America's warned your minus 1 0 1 into the important point is you cannot make a fractional charge with Rick works no way the take 3 quarks and make a fractional charge what about a quirk in the attack work will again take on top I think I opt for twothirds of minus a 3rd not twothirds onethird in or Europe story and you could again make integer charges but this is a user charges that were agreed that it is would group them yeah it's a same as a down although the quartering up like a damn right and they come in 3 families be up to them the trauma strange the top and bottom Burkett 3 replicas of the same thing so or you could make it integer charged particles and so from the point of view of color the fact that you only have integer charged particles nature is the same status as all the particles combined always into S U 3 single cover Singh it's like like hangover lecturer and instead east here got all of the way you dress well that is the way you make such Bates as to entangle them very definitely are certainly entangled doesn't have any air pressure get separated I don't like that well if you can't separate them because they get stuck together but the separator as far as you like and is cluster a lot of energy sewing principle yes there and Pangle learned entangled after it's not easy to manipulate too would want to try to experiment that type all of us said it was felt that way
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off but their heads 3 feet with his groove rate of representation theory OK so let's let me let me tell you more mathematically where is going on here but now let's take you to 1st right and take 2 and a half so we can think of the toast perhaps in terms of the field operators which create their particles that's the easiest way to think about the size of just call the field operator signed and they ought to components to it the side which creates 8 up spin and the sigh which creates bounced then let's call them society sign on a well or I take on 2 values up or down now supposing we want to create too particles signed creates a particle or sigh dagger does not cite creates a particle utterly create to pop likely it's a particle we can think of Of the 2 components we could think of sigh 1 Saito was tied up inside and the S 2 operations typewriters now that's 1 particle had only make
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2 particles well we're twice with sorry what we connect with sigh about twice that creates tulip particles to connect the side down twice that creates too damn particles would do it 1 of each stalwarts right sigh sign J in fact these 2 operators don't really have to correspond to the same particle it's even know we could even think of 1 of them was being electron beams of protons and hydrogen atoms for example but see that's even give them different names are sigh and fire they might be the same they might not be the votes so allow them to be different from what they saw this now is an object with To win some object was to indices it's kind of like that cancer if we were thinking about vectors threedimensional space cancers Our objects with 2 sector indices Asante this like he can serve the amended Harrah's repeated indices now ASU token operate on its big assisted to operate some site and Facts on File This is a collection of 4 objects pop popped up on downed bound up and down there so we could lay out those 4 sigh upside up Friday out side flight bound so for someone with lay them out literary and then ask what happens when you would act with S U 2 on this composite object here will only get mixed up with each other again mixed up with each other for the simple reason that the sun is in mixed up with each other and the fines get mixed up with each other as certainly beside times fines if we get mixed up with each other who's going to be a form by 4 matrix which represents the same protection as just going through all signs and 5 separately and rotating them by the group action OK now you look at these high at the way these things mixup up with each other you look at the way they up with each other and you can't go discovery you discover that a combination of 2 orders sigh up farright down my ears society that only the farright Bob doesn't mix with anything else on this operation it just goes into itself it just goes into itself this is an easy thing to check you just see how S U 2 and acts on site up that mixes it was side down to see how acts on down a new check what happens to this particular combination in theaters nothing so this is a single it 1 of the other 3 possibilities for Ave they are sigh of farright Darren plus side Darren this is 0 5 I'd have not put in squares of 2 with you could put a squirt of 2 you wanted this is orthogonal this and then this sigh of fire end side down Friday what happens when you do the S U 3 rotations on these states they mix into each other In particular ways survey will mix and rich over this city it does not get mixed with these 3 states at all and these 3 states mixup among themselves as if they were spin 1 object they get mixed among each other and this is called the 3 in 1 has 3 states these 3 make some among themselves this dozen mix up with anything so when you combine to have been states that aura combined to have been states you make a 1 or you make a afraid depending on which combination take Nickel 1 or you make the 3 links 1 sorry mixed in 0 been 1 that's a sense in which these equations of being used here are when you combine together quark and antiquark 9 different ways to do it 1 of the linear combinations behaves In this way he it's a single and the others transform into each other 8 objects 8 additional object transform into which other threats meaning of these equations here on many of each kind of combination to get them out of there is a lot of cases will be buzz was I got
1:04:56
a free because if you have to to know what the representations of us 3 are we did a bit of work to find out what the representations of astute to war and they will almost in half and spend integer spin articles we could spend From weeks studying to 3 and find out what b representations of this 2 3 are what are what matrix representations are then not they are not matrix representations of every dimensionality the matrix representations for S U 2 0 happened to be the threedimensionality matrix representation restrict press you 3 not so there a saying that there's a bears the trip but and I trip there that BAT 10 15 20 the whole bunch them but not every number appears in the possible matrices which have the same multiplication table as you so that's a that's a better work to prolong within there I just went there by saying used to get end but but the real interesting point is a single it's that appears the ways it Neukom combining quirks together to get combinations which don't rotate or which only analog electrically neutral while electrically neutral because electrically neutral are exactly the objects which don't transform under that you 1 of the things which the phase cancels out of Member back again to electrodynamics things which are electrically neutral on combinations were all the phases cancel out an electron hasn't eaten by beta a positron many of the minus sign made an electron and positron combination of them are neutral don't transform Kohl electrons at its in each of the 2 white later so it's not called neutral or staff called for a transformed from the group so Stiglitz unimportant but you should just think of them as neutral objects With respect the leader of a transformation properties iron there is buzzes figures show that single that trades revision just what example of forward hero said at bar it is a little too much but stunned yeah I 1 simple way to see if the puerile our free right just right the subject here for Europe's Nigerians Timbuktu matrix which is a mentor symmetric matrix Young on contracting the sees this way but that equal to this that's book and it's easy to prove that this is a whole look at this the spend the moment that in every dimension now matron matrix dimensionality there always in epsilon symbol which has a certain number of indices Dubai 2 matrices the epsilon symbol and Epsilon symbol is enticed a fully and isometric matrix composed zeros and ones Epsilon is so by 2 matrix what about what about Ferreira 3 by 3 matrices With its epsilon educate requires 3 indices suffered a right now if take issue 3 epsilon IJK and let's call it what we call a quirk fields Q Q J Q then you make the 3 threequarter singly which has no Caller ID Katie half but will be different from each other otherwise this 0 so that's why said a red green the blue make single OK we wanna our suffered too much around for what cap look bad that Peter crossing sequels 1 of them had 84 1st lady means when you when you take weights and put together were still warns infrasonic a single rest and whereas again a prostate prostate that would as means an end for stand 10 day I saw is I write a tennis 29 made a decision maker while on a transit 64 its members plus 9 8 last month said there a plus 27 was anorexic Citroen
1:10:18
I can't tell 64 is a 64 I'm so it be it clear information this is a fact I know I know the representations of this you free up to 27 if there is a 35 member harbor end we don't do enough to succeed not so what this means is 1st of all there is 1 in only 1 combination quantum Quantum superposition states of 2 blue ones which makes a similar 1 only 1 amounts of global bid turns out there are 2 orthogonal ways of making architects tool that's 16 states altogether but they formed 2 groups and the 1st group mixes into themselves mixing into a 2nd the 2nd group mixes and itself without mixing 1st and both of them formed architects different combinations different patterns of symmetry of the wave function of the of the state of the 2 core of the 2 warned of alarm system there something called it can begin Varrato distinct ways of combining it will once together to make ends but OK has about things because they don't appear in nature anyway and then as a 27 dimensional representation among threecourse for across all of those same let them know we have a little Bamberger remember if there is a 4 what happened we change the basis do they we of activist stated it is the same as the triplets after earlier Colorado College Yale and are that news for things is bases that major described operation that users will be pitch him as 1 0 years and then they see as a lawyer for the move loses the will be so that they viewed it is that there's major if you just work in terms of the bases vectors are up up back up all them and bam bam there is a certain set of the 4 by 4 matrices which represent the but but if you work in terms Of these linear combination this 1 this 1 this 1 this 1 vendor matrices exactly the property they there's a 1 over here and and that the sale of Jewish was not as major the Big Eight the big 64 by 64 the matrices break up and a bunch of box block diagonal in a particular bases rights in the basic steps but I would like to say that if it exists savage that which this tough bus by line that's right so there 64 estates so ESU 3 operation has to be represented on the 2 under 2 1 system a 64 by 64 dimensional matrices but in some bases so 64 than 60 fourdimensional matrices are won an 8 by 8 matrix over he and another 8 by a matrix over hiya at 10 by can never have enough room another 10 by 10 matrix anemic 27 matrix barrier in a particular place that's exactly like each other as they feel that's
1:14:42
right for he's each a flight track between the right places they don't get mixed this 1 doesn't get mixed with this 1 doesn't get mix with this 1 they only get mixed together as image it is a little a name that Sosa stops that caused after a piece of it like this armed Oh when you motor nothin Sherwood toward her latenight buzzword but I think you're looking for a particular which escaping me right now now aren't I don't such turn exist but I know to on on cited they are for Peters says you could try here they care about them but they want appears Real particles in a laboratory where talk about what your throat creates this peculiar situation but you can't have a freak work by itself but we can't so let's talk about the incident would quantum chromodynamics idea goes back a long ways goes back a song I know was it noted Campbell Geronimo end mammal had the idea choir of 1962 quite a bit good 10 years before require chromodynamics became the standard theory of of quirks in blue ones factor that the whole idea of colored goes back him Megan Court Colorado goalie called it and the idea of ones also owns and his ideal woes fairly simple he understood that if you made color single you would not get fractionally charged particles that was as motivation What do you have a hand in the quantum Carmelite and Teller the theory of quarks in order to have some dynamics which will forbid particles fracture charge so he realized bad if the works carried this kind of color quantum number end if the color always had to be a single at the end you would never get you will never get fractionally charged particles Our had an answer to the question of why particles show only and Watson when something like this he said but right now we only need postulated and the other postulate is a connection or the interaction between quarks and lowered the postulate is red glow lawns play the same role in quantum chromodynamics that photons electrodynamics but photons couple the electric charge source Of the Photon Field is electric charge gone there 8 of them what other sources Of the ones what is the source of war must there must be 8 quantities 8 conserved quantities which you could think of as being similar could charge in each 1 of them is a source Of that particular glue 1 of that particular species of glue on the ATP it will ones correspond the fields which are similar in some sense to the photons of the electromagnetic field electromagnetic field is sought by electric charge what is the source of the cornfield while 1 8 quantities that we have available we 8 generators of S U 3 which are analogous to angular momentum Harris you too angular momentum was a conserved quantity it's a bit dedicated if you have to object to air their angular momentum and collar really means when you speak of the cholera than object speaking about how it transforms under group but the actual which radiates thing which radiates the blue 1 Merrifield is the color itself or the color itself or the generators of Ocala 8 of them and each 1 can make the appropriate kind of Blue 1 the 1st meeting of this is that if you take a colored objects such as Quark is going to be surrounded by glow on field don't much like collector barracks we quantum chromodynamics replaces electromagnetic field with a quantum crumbled with Cromwell dynamic field but the only thing about it is that they're 8 such fields and they 8 such fields transform the same way that's associated with the aid glue to have a 1st firstrate now the 2nd thing is the most something about there about the dynamics of lecture From magnetism but let's begin with a charge and charge the charges made charge attract each other the meaning of that is if you bring them closer together the energy goes down so charge an Olympic Charter of less energy if the close together the and that far apart as Freud might as well be barring because he energy of the neutral system is less than the energy of the tool charged the energy of a neutral system is lower than the energy of a charge system as a rule for example the energy of go class charges is much more than the energy of a plus charge my stride how I know that because the blast charges repel each other if you have to do work to push the boss charges together you get worked out of the running of plus charge full Towards a minus charge so he is an example where a system with net charge has more energy than a system in which the charges cancel out a lake to think about it is if you have to pass charges be lines of forests have to go somewhere else and net field out beyond the object that in in this field has feel the energy field energy is
1:22:06
always positive is proportional to the square of electric field Faso's sealed energy storage In the fact that they to pass charges here if it was a plus charge discharge they might be fielding in between them and the end of supposed you brought them very close together an ideal dipole field in there but there would be no field and outside it would be no large field energy on outside and so the field and agility field energy stored in a neutral system is less than the field energy stored NATO charge system tho the launching of the charge Of the electron and be more profitable it is like I have a neutral system energetically for the less profitable more expensive it is in energy there have a net charge system does big any electric charge if we had the electric charge of the electron was a thousand times bigger than it is the self energy of the electrons due to electromagnetic field would be much bigger than it is in fact the repulsive force between 2 plus charges would be much much stronger areas in the amount of energy would take to assemble to pass charges would be very much bigger the and the energy needed for plus charge minus charge their huge charge of electron would make it very very proliferative have net charge and very very efficient low energy however it will charges cancel out so it if that coupling constant from Electra antics were large very large will be very used to the idea that the low energy particles of nasa's energy of course will messed particles in nature the lowmass objects in nature would be electrically neutral if the charge was big enough we might never have discovered individual electrically charged particles they would be confined who just be too prohibitive and energy to pull them out of an Adam for example so now said it is look if I have this quantum chromodynamics and if cholera the colored generator of colored degree of freedom is the source of the glue field then if we make the chart numerical coupling constant Riyadh analog of electric charge if we make it large enough then it will be simply impossible the polar pipe these quarks entombed nuns singlet which states a singlet stayed busy analog of neutral polling them apart would be leftover would net amounts of color which would cost you a lot so we understood that 1 systems were in the color singlet state they were fully attractive pull themselves together when they warrant it was always an element of repulsion that a man could cost the enormous amount of energy pull together so and then those explanations quantum chromodynamics has a natural way of making sure that the real particles in nature are digit charged and the quirks always calm you knew the quirky and back or combination the 3 core colonization power mental essentially correct it turned out to be correct but the dynamics is a little more interesting so let's talk about the dynamics and why the dynamics as long as a questions of at my user dynamics more readers that the reason that then more interesting and that is because the analog of the photons the glue 1 is charged carries color at it glued on itself it is not neutral with respect to the year a neutral piece was his 9th member that we threw away we ain't glue themselves are not neutral they transformed under the eat it's as if they had a color charge that is why Luwanda's don't appear in nature of those objects color they they themselves are color because the colored they have a big field around the end their prohibitive in energy but this is interesting what it says is that the blue 1 itself is charged in a way that the full time was not the fact that photons and not charged it don't interact with each other and not sources of photons not source of another 4 electrodynamics of classical electrodynamics is a completely new theory meaning to say that the electromagnetic waves just passed through each other as because electromagnetic wave itself was not charged and in dozens of doesn't influence other electromagnetic waves solved electrodynamics the photon and electromagnetic wave are very simple and linear putting quantum chromodynamics it's much more complicated analog of the full time itself is charged that means that the analog electromagnetic waves intersecting another electromagnetic waves will be a serious interact from In fact even just a single wave of the blue 1 1 part of it will interact with the other 1 part of it may have some color and the other part of it may have some color those colors will interact with each other and the dynamics of global on his farm and they need not just cool ones but we field analogous to be electromagnetic from actual field boss for more complicated and dynamically interesting but also much more difficult swarms dinner that train much the important things that happens which is new to me summarize fairly easily last summarize it for you on the blackboard in terms of pictures the equations that go with it still are equations but but the pictures I think equation it is a quarter and and there it's quantum chromodynamics called glue 1 field around now in electrodynamics with different pieces Of the Photon Field of electromagnetic field do not interact with each other and so does not send it which 1 field line repel tracts of the few lines they just just completely independent they don't interact with each other and quantum chromodynamics the 1 field itself is not neutral and therefore the onfield interacts with other pieces of blue 1 field and the result is very simple turns be fairly simple the lines of flocks basically attract each other they attract each other and they attract each other in a way which makes the rule about lines of Fox's they don't Indyk September the charge me lines of books coming out of corporate coming out of court charges attract each other and in the process of attracting each other energetically they want perform tool they want performed to observe flocks with flocks running right down to it is more or less to go I imagined that the wines of books are attracting each other all that empty space is repelling the lines of flocks of pushing them and squeezing them into it that's what that's pretty much effect of these nonlinear interactions between the 1 that it makes it energetically favorable take those lines of flocks and bungalow M and bundle into tool but suppose it's true now not supposing we have a quirk in nearby quark where McCorkle Andrea backcourt lines of books come out of a quirk in go into and if this were electrodynamics those lines of flocks which separated formed dipole pattern that were all familiar with anybody was overlooked magnet and right I feel the energy associated with this big dipole configuration is relatively small because the field its weak far away In this configuration there's a uniformed feels no matter how far you take these particles from each other there's a uniform field in between just exactly as if you took electro of electrical and magnetic field and pushed it to a tool it would be a uniform fueled the lines of flocks don't end and the cost is a uniform field busy uniform energy per unit length what's the result of that the result of that His if
1:31:48
you try to separator quarter from quark the energy just grows linearly with distance between them as you pull it out so everyone feels was really sticky nearly a Turkish Taffy except Steffi we get thinner and thinner as you would stretch that I was not quite like chewing gum of Turkish Taffy inside a kind of chewing gum that as you separated it always created more chewing gum in between so that never got thinner and thinner is always a fixed number of lines of flocks coming out of every recharge that jet and don't disappear so toward the goal of Fox Our folk stories and b what as a uniformed charge per unit on this when he charged object which means any object with nonzero Carla any none seemed any object which is not only a single areas it puts outlines of flocks of of floods bungle dollars and unless they are unless they and on object of opposite color that object is just gonna have an infinite charge because those lines of folks have go someplace that ended Infiniti the cost and energy will be infinite that's the basic dynamics of apart from dynamics linear flocks to like which hardly ingredients which keep quirks from flying around freely if you hit a quart real hard inside Amazon this is a picture of a year so extreme picture of Amazon quark and work you hit that core card was flying but simply can't get away we feel the energy just increases in increases increases linearly with distance until you've run out of kinetic energy in the because back or something else happens was the other thing that can happen right yacht right the other thing that can happen is that the strains can break if this is a quark and this is man that work they there spontaneously our during particle pair production created dike work over here and work over here but still have created a what you've created to Manzano song to give 1 quirky Bingham as Armageddon good shot you hit it really hard and tries to to escape and I either can't escape because it runs out of energy get pulled back but actually more likely will happen as a quirk accord agreed between discrete Tomas arms of this quirk over here is still going too fast it may still try to escape the may try to escape this 1 over here what will happen so what happens when you hit at work really hard news that insider as on news that a whole bunch of mayors are arms flying off forming where's called jet going up or are structural collegiate could get there as long as neutral this is a picture of a on the other ingredients for those in quantum chromodynamics said 3 quarks can exist as explained freak works the threequarter explained by mathematical fact that the flux lines can come together In 3 use quirk quirk port and go but that's not so surprising because 3 quarks and again make a neutral object 3 quirks again make a neutral object and that means that you have 3 quarks that being no net lines coming out of it so somehow lines of force command each 1 of them after be able to cancel each other that's quantum chromodynamics based on the S U 3 group call the beach theory indeed is simply any theory with the conserved quantities are couples Fulham Maxwell like field coupled to a field with the focus on it has a deeper meaning than that but for our purposes and Inc time there is a conserved quantity analogous to electric charge which functions as the source of a time like feel photon might feel the blue 1 like field that call the gage theory a gage theories are always based on symmetries based on symmetries biggest it based on the idea of conserved charge and conserve charge means a symmetry of some incidentally what's the connection between conservation and lines of flocks consolation lines of flocks are intimately connected if you have a rule for example electrodynamics every charged particle has to be the source of of lines of flocks and lines of Fox can only end on charged particles then charged has to be conserved is no way this lines of force go off to infinity well you could say I could make charge suddenly disappear if I allow the lines of flocks to simultaneously will disappear but that would violate the speed of light constrained you would be able to send a message arbitrarily far distance if by simply removing that charge you suddenly changed the field at infinity the fact that you cannot send the signal fester the speed of light does you that you can't remove this charge white because anchored by flux lines of infinity and if you could remove it either 1 of 2 things either you would send this signal far away that the charge was banned by removing the field or are you would be leftover with these lines of flocks that don't bend on charges of lines of flocks have been on charges the tantamount to saying that the sources of fuel conserved which In another language says there must be be a symmetry so some cemeteries are connected with fall carnallite field which give rise forms of flocks those theories of called gage theories and you can't even imagine without breaking the rules of the the theory what it would mean for the charge not be conserved because the rules of Harry patch charged particles To be wonderful for that's that's kind promote barracks overnight killing money depending on how you count you that twothirds or threequarters of the way through the Standard Model we've gone through Aug 19 generators if you're like a generators of S U 3 in what's the other 1 elected China and which is not generated a plane they elect him as a conserved quantities they coupled with 19 gage fields be blue ones and be broke current the Standard Model Harris 3 more generators to talk about that's we got 9 out of 12 3 quarters Our the other way of camp things to say we studied as 3 or 4 more dynamic we study you 1 that's electrodynamics standard models S U 3 process you to cross you 1 with missing yes to yes you to not not angular momentum is another ingredient in the Standard Model caucus you would pick up there will start to pick up pursued to next time we got a lot of ground turned their own thinkin about 2 nor lectures we could probably finish the basic structure Of the Standard Model as your 3 courses you to cross you want me 1 put it together into a package and rarely it could element that we need of course is the Higgs field Higgs field plays an important role where half but don't say within about 3 lectures we will have the working parts of the standard model we thought to explore what puzzles RYE white people think we have to go beyond the standard model and why people think they're going to be interesting new things discovered at having or get the bird flu guy for more please
1:41:28
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